text
stringlengths 16
172k
| source
stringlengths 32
122
|
|---|---|
In telecommunication, alongitudinal redundancy check(LRC), orhorizontal redundancy check, is a form ofredundancy checkthat is applied independently to each of a parallel group of bit streams. The data must be divided intotransmission blocks, to which the additional check data is added.
The term usually applies to a singleparity bitper bit stream, calculated independently of all the other bit streams (BIP-8).[1][2]
This "extra" LRC word at the end of a block of data is very similar tochecksumandcyclic redundancy check(CRC).
While simple longitudinalparitycan onlydetecterrors, it can be combined with additional error-control coding, such as atransverse redundancy check(TRC), tocorrecterrors. The transverse redundancy check is stored on a dedicated "parity track".
Whenever any single-bit error occurs in a transmission block of data, such two-dimensional parity checking, or "two-coordinate parity checking",[3]enables the receiver to use the TRC to detect which byte the error occurred in, and the LRC to detect exactly which track the error occurred in, to discover exactly which bit is in error, and then correct that bit by flipping it.[4][5][6]
International standardISO 1155[7]states that a longitudinal redundancy check for a sequence of bytes may be computed insoftwareby the following algorithm:
which can be expressed as "the 8-bit two's-complement value of the sum of all bytes modulo 28" (x AND 0xFFis equivalent tox MOD 28).
Many protocols use an XOR-based longitudinal redundancy check byte (often calledblock check characteror BCC), including the serial line interface protocol (SLIP, not to be confused with the later and well-knownSerial LineInternetProtocol),[8]theIEC 62056-21 standard for electrical-meter reading, smart cards as defined inISO/IEC 7816, and theACCESS.busprotocol.
An 8-bit LRC such as this is equivalent to acyclic redundancy checkusing the polynomialx8+ 1, but the independence of the bit streams is less clear when looked at in that way.
|
https://en.wikipedia.org/wiki/Longitudinal_redundancy_check
|
Thecyclic redundancy check(CRC) is a check of theremainderafterdivisionin thering of polynomialsoverGF(2)(thefinite fieldofintegers modulo2). That is, the set ofpolynomialswhere eachcoefficientis either zero or one, andarithmetic operationswrap around.
Anystring of bitscan be interpreted as the coefficients of a polynomial of this sort, and a message has a valid CRC if it divisible by (i.e. is a multiple of) an agreed-ongenerator polynomial. CRCs are convenient and popular because they have good error-detection properties and such a multiple may be easily constructed from anymessage polynomialM(x){\displaystyle M(x)}by appending ann{\displaystyle n}-bitremainder polynomialR(x){\displaystyle R(x)}to produceW(x)=M(x)⋅xn+R(x){\displaystyle W(x)=M(x)\cdot x^{n}+R(x)}, wheren{\displaystyle n}is the degree of the generator polynomial.
Although the separation ofW(x){\displaystyle W(x)}into the message partM(x){\displaystyle M(x)}and the checksum partR(x){\displaystyle R(x)}is convenient for use of CRCs, the error-detection properties do not make a distinction; errors are detected equally anywhere withinW(x){\displaystyle W(x)}.
In general,computation of CRCcorresponds toEuclidean divisionof polynomials over GF(2):
HereM(x){\displaystyle M(x)}is the original message polynomial andG(x){\displaystyle G(x)}is the degree-n{\displaystyle n}generator polynomial. The bits ofM(x)⋅xn{\displaystyle M(x)\cdot x^{n}}are the original message withn{\displaystyle n}zeroes added at the end. The CRC 'checksum' is formed by the coefficients of the remainder polynomialR(x){\displaystyle R(x)}whose degree is strictly less thann{\displaystyle n}. The quotient polynomialQ(x){\displaystyle Q(x)}is of no interest. Usingmodulo operation, it can be stated that
In communication, the sender attaches then{\displaystyle n}bits of R after the original message bits of M, which could be shown to be equivalent to sending outW(x)=M(x)⋅xn−R(x){\displaystyle W(x)=M(x)\cdot x^{n}-R(x)}(thecodeword). The receiver, knowingG(x){\displaystyle G(x)}, dividesW(x){\displaystyle W(x)}byG(x){\displaystyle G(x)}and checks that the remainder is zero. If it is, the receiver discardsR(x){\displaystyle R(x)}(the lastn{\displaystyle n}bits) and assumes the received message bitsM(x){\displaystyle M(x)}are correct.
Software implementations sometimes separate the message into its parts and compare the receivedR(x){\displaystyle R(x)}to a value reconstructed from the received message, but hardware implementations invariably find the full-length division described above to be simpler.
In practice CRC calculations most closely resemblelong divisionin binary, except that the subtractions involved do not borrow from more significant digits, and thus becomeexclusive oroperations.
A CRC is achecksumin a strict mathematical sense, as it can be expressed as the weighted modulo-2 sum of per-bitsyndromes, but that word is generally reserved more specifically for sums computed using larger moduli, such as 10, 256, or 65535.
CRCs can also be used as part oferror-correcting codes, which allow not only the detection of transmission errors, but the reconstruction of the correct message. These codes are based on closely related mathematical principles.
Since the coefficients are constrained to a single bit, any math operation on CRC polynomials must map the coefficients of the result to either zero or one. For example, in addition:
Note that2x{\displaystyle 2x}is equivalent to zero in the above equation because addition of coefficients is performed modulo 2:
Polynomial addition modulo 2 is the same asbitwise XOR. Since XOR is the inverse of itself, polynominal subtraction modulo 2 is the same as bitwise XOR too.
Multiplication is similar (acarry-less product):
We can also divide polynomials mod 2 and find the quotient and remainder. For example, suppose we're dividingx3+x2+x{\displaystyle x^{3}+x^{2}+x}byx+1{\displaystyle x+1}. We would find that
In other words,
The division yields a quotient ofx2+1{\displaystyle x^{2}+1}with a remainder of −1, which, since it is odd, has a last bit of 1.
In the above equations,x3+x2+x{\displaystyle x^{3}+x^{2}+x}represents the original message bits111,x+1{\displaystyle x+1}is the generator polynomial, and the remainder1{\displaystyle 1}(equivalently,x0{\displaystyle x^{0}}) is the CRC. The degree of the generator polynomial is 1, so we first multiplied the message byx1{\displaystyle x^{1}}to getx3+x2+x{\displaystyle x^{3}+x^{2}+x}.
There are several standard variations on CRCs, any or all of which may be used with any CRC polynomial.Implementation variationssuch asendiannessand CRC presentation only affect the mapping of bit strings to the coefficients ofM(x){\displaystyle M(x)}andR(x){\displaystyle R(x)}, and do not impact the properties of the algorithm.
These two variations serve the purpose of detecting zero bits added to the message. A preceding zero bit adds a leading zero coefficient toW(x),{\displaystyle W(x),}which does not change its value, and thus does not change its divisibility by the generator polynomial. By adding a fixed pattern to the first bits of a message, such extra zero bits can be detected.
Likewise, using a non-zero remainder detects trailing zero bits added to a message. If a CRC-protected messageW(x){\displaystyle W(x)}has a zero bit appended, the received polynomial isW(x)⋅x.{\displaystyle W(x)\cdot x.}If the former is divisible by the generator polynomial, so is the latter. Using a non-zero remainderS(x){\displaystyle S(x)}, appending a zero bit will result in the different remainderS(x)⋅xmodG(x){\displaystyle S(x)\cdot x{\bmod {G}}(x)}, and therefore the extra bit will be detected.
In practice, these two variations are invariably used together. They change the transmitted CRC, so must be implemented at both the transmitter and the receiver. Both ends must preset their division circuitry to all-ones, the transmitter must add the trailing inversion pattern to the result, and the receiver must expect this pattern when checking the CRC. If the receiver checks the CRC by full-length division, the remainder because the CRC of a full codeword that already includes a CRC is no longer zero. Instead, it is a fixed non-zero pattern, the CRC of the inversion pattern ofn{\displaystyle n}ones.
These inversions are extremely common but not universally performed, even in the case of the CRC-32 or CRC-16-CCITT polynomials. They are almost always included when sending variable-length messages, but often omitted when communicating fixed-length messages, as the problem of added zero bits is less likely to arise.
All practical CRC generator polynomials have non-zeroxn{\displaystyle x^{n}}andx0{\displaystyle x^{0}}coefficients. It is very common to convert this to a string ofn{\displaystyle n}binary bits by omitting thexn{\displaystyle x^{n}}coefficient.
This bit string may then be converted to a binary number using one of two conventions:
The msbit-first form is often referred to in the literature as thenormalrepresentation, while the lsbit-first is called thereversedrepresentation. It is essential to use the correct form when implementing a CRC. If the coefficient ofxn−1{\displaystyle x^{n-1}}happens to be zero, the forms can be distinguished at a glance by seeing which end has the bit set.
For example, the degree-16 CCITT polynomial in the forms described (bits inside square brackets are included in the word representation; bits outside are implied 1 bits; vertical bars designatenibbleboundaries):
All the well-known CRC generator polynomials of degreen{\displaystyle n}have two common hexadecimal representations. In both cases, the coefficient ofxn{\displaystyle x^{n}}is omitted and understood to be 1.
The msbit-first form is often referred to in the literature as thenormalrepresentation, while the lsbit-first is called thereversedrepresentation. It is essential to use the correct form when implementing a CRC. If the coefficient ofxn−1{\displaystyle x^{n-1}}happens to be zero, the forms can be distinguished at a glance by seeing which end has the bit set.
To further confuse the matter, the paper by P. Koopman and T. Chakravarty[1][2]converts CRC generator polynomials to hexadecimal numbers in yet another way: msbit-first, but including thexn{\displaystyle x^{n}}coefficient and omitting thex0{\displaystyle x^{0}}coefficient. This "Koopman" representation has the advantage that the degree can be determined from the hexadecimal form and the coefficients are easy to read off in left-to-right order. However, it is not used anywhere else and is not recommended due to the risk of confusion.
Areciprocal polynomialis created by assigning thexn{\displaystyle x^{n}}throughx0{\displaystyle x^{0}}coefficients of one polynomial to thex0{\displaystyle x^{0}}throughxn{\displaystyle x^{n}}coefficients of a new polynomial. That is, the reciprocal of the degreen{\displaystyle n}polynomialG(x){\displaystyle G(x)}isxnG(x−1){\displaystyle x^{n}G(x^{-1})}.
The most interesting property of reciprocal polynomials, when used in CRCs, is that they have exactly the same error-detecting strength as the polynomials they are reciprocals of. The reciprocal of a polynomial generates the samecodewords, only bit reversed — that is, if all but the firstn{\displaystyle n}bits of a codeword under the original polynomial are taken, reversed and used as a new message, the CRC of that message under the reciprocal polynomial equals the reverse of the firstn{\displaystyle n}bits of the original codeword. But the reciprocal polynomial is not the same as the original polynomial, and the CRCs generated using it are not the same (even modulo bit reversal) as those generated by the original polynomial.
The error-detection ability of a CRC depends on the degree of its generator polynomial and on the specific generator polynomial used. The "error polynomial"E(x){\displaystyle E(x)}is the symmetric difference of the received message codeword and the correct message codeword. An error will go undetected by a CRC algorithm if and only if the error polynomial is divisible by the CRC polynomial.
(As an aside, there is never reason to use a polynomial with a zerox0{\displaystyle x^{0}}term. Recall that a CRC is the remainder of the message polynomial timesxn{\displaystyle x^{n}}divided by the CRC polynomial. A polynomial with a zerox0{\displaystyle x^{0}}term always hasx{\displaystyle x}as a factor. So ifK(x){\displaystyle K(x)}is the original CRC polynomial andK(x)=x⋅K′(x){\displaystyle K(x)=x\cdot K'(x)}, then
That is, the CRC of any message with theK(x){\displaystyle K(x)}polynomial is the same as that of the same message with theK′(x){\displaystyle K'(x)}polynomial with a zero appended. It is just a waste of a bit.)
The combination of these factors means that good CRC polynomials are often primitive polynomials (which have the best 2-bit error detection) or primitive polynomials of degreen−1{\displaystyle n-1}, multiplied byx+1{\displaystyle x+1}(which detects all odd numbers of bit errors, and has half the two-bit error detection ability of a primitive polynomial of degreen{\displaystyle n}).[1]
Analysis using bitfilters[1]allows one to very efficiently determine the properties of a given generator polynomial. The results are the following:
|
https://en.wikipedia.org/wiki/Mathematics_of_cyclic_redundancy_checks
|
Simple file verification(SFV) is a file format for storingCRC32checksumsof files to verify the integrity of files. SFV is used to verify that a file has not beencorrupted, but it does not otherwise verify the file'sauthenticity. The.sfvfile extensionis usually used for SFV files.[1]
Files can become corrupted for a variety of reasons, including faultystorage media, errors intransmission, write errors duringcopyingor moving, andsoftware bugs. SFV verification ensures that a file has not been corrupted by comparing the file'sCRChashvalue to a previously calculated value.[1]Due to the nature of hash functions,hash collisionsmay result infalse positives, but the likelihood of collisions is usually negligible with random corruption. (The number of possible checksums is limited though large, so that with any checksum scheme many files will have the same checksum. However, the probability of a corrupted file having the same checksum as its original is exceedingly small, unless deliberately constructed to maintain the checksum.)
SFV cannot be used to verify the authenticity of files, as CRC32 is not acollision resistanthash function; even if the hash sum file is not tampered with, it is computationally trivial for an attacker to cause deliberate hash collisions, meaning that a malicious change in the file is not detected by a hash comparison. In cryptography, this attack is called acollision attack. For this reason, themd5sumandsha1sumutilities are often preferred inUnixoperating systems, which use theMD5andSHA-1cryptographic hash functionsrespectively.
Even a single-bit error causes both SFV's CRC and md5sum's cryptographic hash to fail, requiring the entire file to be re-fetched.
TheParchiveandrsyncutilities are often preferred for verifying that a file has not been accidentally corrupted in transmission, since they can correct common small errors with a much shorter download.
Despite the weaknesses of the SFV format, it is popular due to the relatively small amount of time taken by SFV utilities to calculate the CRC32 checksums when compared to the time taken to calculate cryptographic hashes such as MD5 or SHA-1.
SFV uses aplain textfile containing one line for each file and its checksum[1]in the formatFILENAME<whitespaces>CHECKSUM. Any line starting with a semicolon ';' is considered to be a comment and is ignored for the purposes of file verification. The delimiter between the filename and checksum is always one or several spaces; tabs are never used. A sample SFV file is:
An example of anopen-sourcecross-platformcommand-line utilitythat outputs crc32 checksums is7-Zip.[2]
Many Linux distributions include a simple command-line toolcksfvto verify the checksums.
|
https://en.wikipedia.org/wiki/Simple_file_verification
|
Innumber theory, theKronecker symbol, written as(an){\displaystyle \left({\frac {a}{n}}\right)}or(a|n){\displaystyle (a|n)}, is a generalization of theJacobi symbolto allintegersn{\displaystyle n}. It was introduced byLeopold Kronecker(1885, page 770).
Letn{\displaystyle n}be a non-zero integer, withprime factorization
whereu{\displaystyle u}is aunit(i.e.,u=±1{\displaystyle u=\pm 1}), and thepi{\displaystyle p_{i}}areprimes. Leta{\displaystyle a}be an integer. The Kronecker symbol(an){\displaystyle \left({\frac {a}{n}}\right)}is defined by
Foroddpi{\displaystyle p_{i}}, the number(api){\displaystyle \left({\frac {a}{p_{i}}}\right)}is simply the usualLegendre symbol. This leaves the case whenpi=2{\displaystyle p_{i}=2}. We define(a2){\displaystyle \left({\frac {a}{2}}\right)}by
Since it extends the Jacobi symbol, the quantity(au){\displaystyle \left({\frac {a}{u}}\right)}is simply1{\displaystyle 1}whenu=1{\displaystyle u=1}. Whenu=−1{\displaystyle u=-1}, we define it by
Finally, we put
These extensions suffice to define the Kronecker symbol for all integer valuesa,n{\displaystyle a,n}.
Some authors only define the Kronecker symbol for more restricted values; for example,a{\displaystyle a}congruent to0,1mod4{\displaystyle 0,1{\bmod {4}}}andn>0{\displaystyle n>0}.
The following is a table of values of Kronecker symbol(kn){\displaystyle \left({\frac {k}{n}}\right)}with 1 ≤n,k≤ 30.
The Kronecker symbol shares many basic properties of the Jacobi symbol, under certain restrictions:
On the other hand, the Kronecker symbol does not have the same connection toquadratic residuesas the Jacobi symbol. In particular, the Kronecker symbol(an){\displaystyle \left({\tfrac {a}{n}}\right)}forn≡2(mod4){\displaystyle n\equiv 2{\pmod {4}}}can take values independently on whethera{\displaystyle a}is a quadratic residue or nonresidue modulon{\displaystyle n}.
The Kronecker symbol also satisfies the following versions ofquadratic reciprocitylaw.
For any nonzero integern{\displaystyle n}, letn′{\displaystyle n'}denote itsodd part:n=2en′{\displaystyle n=2^{e}n'}wheren′{\displaystyle n'}is odd (forn=0{\displaystyle n=0}, we put0′=1{\displaystyle 0'=1}). Then the followingsymmetric versionof quadratic reciprocity holds for every pair of integersm,n{\displaystyle m,n}such thatgcd(m,n)=1{\displaystyle \gcd(m,n)=1}:
where the±{\displaystyle \pm }sign is equal to+{\displaystyle +}ifm≥0{\displaystyle m\geq 0}orn≥0{\displaystyle n\geq 0}and is equal to−{\displaystyle -}ifm<0{\displaystyle m<0}andn<0{\displaystyle n<0}.
There is also equivalentnon-symmetric versionof quadratic reciprocity that holds for every pair of relatively prime integersm,n{\displaystyle m,n}:
For any integern{\displaystyle n}letn∗=(−1)(n′−1)/2n{\displaystyle n^{*}=(-1)^{(n'-1)/2}n}. Then we have another equivalent non-symmetric version that states
for every pair of integersm,n{\displaystyle m,n}(not necessarily relatively prime).
Thesupplementary lawsgeneralize to the Kronecker symbol as well. These laws follow easily from each version of quadratic reciprocity law stated above (unlike with Legendre and Jacobi symbol where both the main law and the supplementary laws are needed to fully describe the quadratic reciprocity).
For any integern{\displaystyle n}we have
and for any odd integern{\displaystyle n}it's
Ifa≢3(mod4){\displaystyle a\not \equiv 3{\pmod {4}}}anda≠0{\displaystyle a\neq 0}, the mapχ(n)=(an){\displaystyle \chi (n)=\left({\tfrac {a}{n}}\right)}is a realDirichlet characterof modulus{4|a|,a≡2(mod4),|a|,otherwise.{\displaystyle {\begin{cases}4|a|,&a\equiv 2{\pmod {4}},\\|a|,&{\text{otherwise.}}\end{cases}}}Conversely, every real Dirichlet character can be written in this form witha≡0,1(mod4){\displaystyle a\equiv 0,1{\pmod {4}}}(fora≡2(mod4){\displaystyle a\equiv 2{\pmod {4}}}it's(an)=(4an){\displaystyle \left({\tfrac {a}{n}}\right)=\left({\tfrac {4a}{n}}\right)}).
In particular,primitivereal Dirichlet charactersχ{\displaystyle \chi }are in a 1–1 correspondence withquadratic fieldsF=Q(m){\displaystyle F=\mathbb {Q} ({\sqrt {m}})}, wherem{\displaystyle m}is a nonzerosquare-free integer(we can include the caseQ(1)=Q{\displaystyle \mathbb {Q} ({\sqrt {1}})=\mathbb {Q} }to represent the principal character, even though it is not a quadratic field). The characterχ{\displaystyle \chi }can be recovered from the field as theArtin symbol(F/Q⋅){\displaystyle \left({\tfrac {F/\mathbb {Q} }{\cdot }}\right)}: that is, for a positive primep{\displaystyle p}, the value ofχ(p){\displaystyle \chi (p)}depends on the behaviour of the ideal(p){\displaystyle (p)}in thering of integersOF{\displaystyle O_{F}}:
Thenχ(n){\displaystyle \chi (n)}equals the Kronecker symbol(Dn){\displaystyle \left({\tfrac {D}{n}}\right)}, where
is thediscriminantofF{\displaystyle F}. The conductor ofχ{\displaystyle \chi }is|D|{\displaystyle |D|}.
Similarly, ifn>0{\displaystyle n>0}, the mapχ(a)=(an){\displaystyle \chi (a)=\left({\tfrac {a}{n}}\right)}is a real Dirichlet character of modulus{4n,n≡2(mod4),n,otherwise.{\displaystyle {\begin{cases}4n,&n\equiv 2{\pmod {4}},\\n,&{\text{otherwise.}}\end{cases}}}However, not all real characters can be represented in this way, for example the character(−4⋅){\displaystyle \left({\tfrac {-4}{\cdot }}\right)}cannot be written as(⋅n){\displaystyle \left({\tfrac {\cdot }{n}}\right)}for anyn{\displaystyle n}. By the law of quadratic reciprocity, we have(⋅n)=(n∗⋅){\displaystyle \left({\tfrac {\cdot }{n}}\right)=\left({\tfrac {n^{*}}{\cdot }}\right)}. A character(a⋅){\displaystyle \left({\tfrac {a}{\cdot }}\right)}can be represented as(⋅n){\displaystyle \left({\tfrac {\cdot }{n}}\right)}if and only if its odd parta′≡1(mod4){\displaystyle a'\equiv 1{\pmod {4}}}, in which case we can taken=|a|{\displaystyle n=|a|}.
This article incorporates material from Kronecker symbol onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.
|
https://en.wikipedia.org/wiki/Kronecker_symbol
|
Inalgebraic number theorythen-th power residue symbol(for an integern> 2) is a generalization of the (quadratic)Legendre symbolton-th powers. These symbols are used in the statement and proof ofcubic,quartic,Eisenstein, and related higher[1]reciprocity laws.[2]
Letkbe analgebraic number fieldwithring of integersOk{\displaystyle {\mathcal {O}}_{k}}that contains aprimitiven-th root of unityζn.{\displaystyle \zeta _{n}.}
Letp⊂Ok{\displaystyle {\mathfrak {p}}\subset {\mathcal {O}}_{k}}be aprime idealand assume thatnandp{\displaystyle {\mathfrak {p}}}arecoprime(i.e.n∉p{\displaystyle n\not \in {\mathfrak {p}}}.)
Thenormofp{\displaystyle {\mathfrak {p}}}is defined as the cardinality of the residue class ring (note that sincep{\displaystyle {\mathfrak {p}}}is prime the residue class ring is afinite field):
An analogue of Fermat's theorem holds inOk.{\displaystyle {\mathcal {O}}_{k}.}Ifα∈Ok−p,{\displaystyle \alpha \in {\mathcal {O}}_{k}-{\mathfrak {p}},}then
And finally, supposeNp≡1modn.{\displaystyle \mathrm {N} {\mathfrak {p}}\equiv 1{\bmod {n}}.}These facts imply that
is well-defined and congruent to a uniquen{\displaystyle n}-th root of unityζns.{\displaystyle \zeta _{n}^{s}.}
This root of unity is called then-th power residue symbol forOk,{\displaystyle {\mathcal {O}}_{k},}and is denoted by
Then-th power symbol has properties completely analogous to those of the classical (quadratic)Jacobi symbol(ζ{\displaystyle \zeta }is a fixed primitiven{\displaystyle n}-th root of unity):
In all cases (zero and nonzero)
All power residue symbols modnareDirichlet charactersmodn, and them-th power residue symbol only contains them-throots of unity, them-th power residue symbol modnexists if and only ifmdividesλ(n){\displaystyle \lambda (n)}(theCarmichael lambda functionofn).
Then-th power residue symbol is related to theHilbert symbol(⋅,⋅)p{\displaystyle (\cdot ,\cdot )_{\mathfrak {p}}}for the primep{\displaystyle {\mathfrak {p}}}by
in the casep{\displaystyle {\mathfrak {p}}}coprime ton, whereπ{\displaystyle \pi }is anyuniformising elementfor thelocal fieldKp{\displaystyle K_{\mathfrak {p}}}.[3]
Then{\displaystyle n}-th power symbol may be extended to take non-prime ideals or non-zero elements as its "denominator", in the same way that theJacobi symbolextends the Legendre symbol.
Any ideala⊂Ok{\displaystyle {\mathfrak {a}}\subset {\mathcal {O}}_{k}}is the product of prime ideals, and in one way only:
Then{\displaystyle n}-th power symbol is extended multiplicatively:
For0≠β∈Ok{\displaystyle 0\neq \beta \in {\mathcal {O}}_{k}}then we define
where(β){\displaystyle (\beta )}is the principal ideal generated byβ.{\displaystyle \beta .}
Analogous to the quadratic Jacobi symbol, this symbol is multiplicative in the top and bottom parameters.
Since the symbol is always ann{\displaystyle n}-th root of unity, because of its multiplicativity it is equal to 1 whenever one parameter is ann{\displaystyle n}-th power; the converse is not true.
Thepower reciprocity law, the analogue of thelaw of quadratic reciprocity, may be formulated in terms of theHilbert symbolsas[4]
wheneverα{\displaystyle \alpha }andβ{\displaystyle \beta }are coprime.
|
https://en.wikipedia.org/wiki/Power_residue_symbol
|
Inmathematics, adiffeomorphismis anisomorphismofdifferentiable manifolds. It is aninvertible functionthat maps one differentiable manifold to another such that both the function and its inverse arecontinuously differentiable.
Given two differentiable manifoldsM{\displaystyle M}andN{\displaystyle N}, acontinuously differentiable mapf:M→N{\displaystyle f\colon M\rightarrow N}is adiffeomorphismif it is abijectionand its inversef−1:N→M{\displaystyle f^{-1}\colon N\rightarrow M}is differentiable as well. If these functions arer{\displaystyle r}times continuously differentiable,f{\displaystyle f}is called aCr{\displaystyle C^{r}}-diffeomorphism.
Two manifoldsM{\displaystyle M}andN{\displaystyle N}arediffeomorphic(usually denotedM≃N{\displaystyle M\simeq N}) if there is a diffeomorphismf{\displaystyle f}fromM{\displaystyle M}toN{\displaystyle N}. TwoCr{\displaystyle C^{r}}-differentiable manifolds areCr{\displaystyle C^{r}}-diffeomorphic if there is anr{\displaystyle r}times continuously differentiable bijective map between them whose inverse is alsor{\displaystyle r}times continuously differentiable.
Given asubsetX{\displaystyle X}of a manifoldM{\displaystyle M}and a subsetY{\displaystyle Y}of a manifoldN{\displaystyle N}, a functionf:X→Y{\displaystyle f:X\to Y}is said to be smooth if for allp{\displaystyle p}inX{\displaystyle X}there is aneighborhoodU⊂M{\displaystyle U\subset M}ofp{\displaystyle p}and a smooth functiong:U→N{\displaystyle g:U\to N}such that therestrictionsagree:g|U∩X=f|U∩X{\displaystyle g_{|U\cap X}=f_{|U\cap X}}(note thatg{\displaystyle g}is an extension off{\displaystyle f}). The functionf{\displaystyle f}is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth.
Testing whether a differentiable map is a diffeomorphism can be made locally under some mild restrictions. This is the Hadamard-Caccioppoli theorem:[1]
IfU{\displaystyle U},V{\displaystyle V}areconnectedopen subsetsofRn{\displaystyle \mathbb {R} ^{n}}such thatV{\displaystyle V}issimply connected, a differentiable mapf:U→V{\displaystyle f:U\to V}is a diffeomorphism if it isproperand if thedifferentialDfx:Rn→Rn{\displaystyle Df_{x}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}is bijective (and hence alinear isomorphism) at each pointx{\displaystyle x}inU{\displaystyle U}.
Some remarks:
It is essential forV{\displaystyle V}to besimply connectedfor the functionf{\displaystyle f}to be globally invertible (under the sole condition that its derivative be a bijective map at each point). For example, consider the "realification" of thecomplexsquare function
Thenf{\displaystyle f}issurjectiveand it satisfies
Thus, thoughDfx{\displaystyle Df_{x}}is bijective at each point,f{\displaystyle f}is not invertible because it fails to beinjective(e.g.f(1,0)=(1,0)=f(−1,0){\displaystyle f(1,0)=(1,0)=f(-1,0)}).
Since the differential at a point (for a differentiable function)
is alinear map, it has a well-defined inverse if and only ifDfx{\displaystyle Df_{x}}is a bijection. Thematrixrepresentation ofDfx{\displaystyle Df_{x}}is then×n{\displaystyle n\times n}matrix of first-orderpartial derivativeswhose entry in thei{\displaystyle i}-th row andj{\displaystyle j}-th column is∂fi/∂xj{\displaystyle \partial f_{i}/\partial x_{j}}. This so-calledJacobian matrixis often used for explicit computations.
Diffeomorphisms are necessarily between manifolds of the samedimension. Imaginef{\displaystyle f}going from dimensionn{\displaystyle n}to dimensionk{\displaystyle k}. Ifn<k{\displaystyle n<k}thenDfx{\displaystyle Df_{x}}could never be surjective, and ifn>k{\displaystyle n>k}thenDfx{\displaystyle Df_{x}}could never be injective. In both cases, therefore,Dfx{\displaystyle Df_{x}}fails to be a bijection.
IfDfx{\displaystyle Df_{x}}is a bijection atx{\displaystyle x}thenf{\displaystyle f}is said to be alocal diffeomorphism(since, by continuity,Dfy{\displaystyle Df_{y}}will also be bijective for ally{\displaystyle y}sufficiently close tox{\displaystyle x}).
Given a smooth map from dimensionn{\displaystyle n}to dimensionk{\displaystyle k}, ifDf{\displaystyle Df}(or, locally,Dfx{\displaystyle Df_{x}}) is surjective,f{\displaystyle f}is said to be asubmersion(or, locally, a "local submersion"); and ifDf{\displaystyle Df}(or, locally,Dfx{\displaystyle Df_{x}}) is injective,f{\displaystyle f}is said to be animmersion(or, locally, a "local immersion").
A differentiable bijection isnotnecessarily a diffeomorphism.f(x)=x3{\displaystyle f(x)=x^{3}}, for example, is not a diffeomorphism fromR{\displaystyle \mathbb {R} }to itself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0). This is an example of ahomeomorphismthat is not a diffeomorphism.
Whenf{\displaystyle f}is a map between differentiable manifolds, a diffeomorphicf{\displaystyle f}is a stronger condition than a homeomorphicf{\displaystyle f}. For a diffeomorphism,f{\displaystyle f}and its inverse need to bedifferentiable; for a homeomorphism,f{\displaystyle f}and its inverse need only becontinuous. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.
f:M→N{\displaystyle f:M\to N}is a diffeomorphism if, incoordinate charts, it satisfies the definition above. More precisely: Pick any cover ofM{\displaystyle M}by compatiblecoordinate chartsand do the same forN{\displaystyle N}. Letϕ{\displaystyle \phi }andψ{\displaystyle \psi }be charts on, respectively,M{\displaystyle M}andN{\displaystyle N}, withU{\displaystyle U}andV{\displaystyle V}as, respectively, the images ofϕ{\displaystyle \phi }andψ{\displaystyle \psi }. The mapψfϕ−1:U→V{\displaystyle \psi f\phi ^{-1}:U\to V}is then a diffeomorphism as in the definition above, wheneverf(ϕ−1(U))⊆ψ−1(V){\displaystyle f(\phi ^{-1}(U))\subseteq \psi ^{-1}(V)}.
Since any manifold can be locally parametrised, we can consider some explicit maps fromR2{\displaystyle \mathbb {R} ^{2}}intoR2{\displaystyle \mathbb {R} ^{2}}.
Inmechanics, a stress-induced transformation is called adeformationand may be described by a diffeomorphism.
A diffeomorphismf:U→V{\displaystyle f:U\to V}between twosurfacesU{\displaystyle U}andV{\displaystyle V}has a Jacobian matrixDf{\displaystyle Df}that is aninvertible matrix. In fact, it is required that forp{\displaystyle p}inU{\displaystyle U}, there is aneighborhoodofp{\displaystyle p}in which the JacobianDf{\displaystyle Df}staysnon-singular. Suppose that in a chart of the surface,f(x,y)=(u,v).{\displaystyle f(x,y)=(u,v).}
Thetotal differentialofuis
Then the image(du,dv)=(dx,dy)Df{\displaystyle (du,dv)=(dx,dy)Df}is alinear transformation, fixing the origin, and expressible as the action of a complex number of a particular type. When (dx,dy) is also interpreted as that type of complex number, the action is of complex multiplication in the appropriate complex number plane. As such, there is a type of angle (Euclidean,hyperbolic, orslope) that is preserved in such a multiplication. Due toDfbeing invertible, the type of complex number is uniform over the surface. Consequently, a surface deformation or diffeomorphism of surfaces has theconformal propertyof preserving (the appropriate type of) angles.
LetM{\displaystyle M}be a differentiable manifold that issecond-countableandHausdorff. Thediffeomorphism groupofM{\displaystyle M}is thegroupof allCr{\displaystyle C^{r}}diffeomorphisms ofM{\displaystyle M}to itself, denoted byDiffr(M){\displaystyle {\text{Diff}}^{r}(M)}or, whenr{\displaystyle r}is understood,Diff(M){\displaystyle {\text{Diff}}(M)}. This is a "large" group, in the sense that—providedM{\displaystyle M}is not zero-dimensional—it is notlocally compact.
The diffeomorphism group has two naturaltopologies:weakandstrong(Hirsch 1997). When the manifold iscompact, these two topologies agree. The weak topology is alwaysmetrizable. When the manifold is not compact, the strong topology captures the behavior of functions "at infinity" and is not metrizable. It is, however, stillBaire.
Fixing aRiemannian metriconM{\displaystyle M}, the weak topology is the topology induced by the family of metrics
asK{\displaystyle K}varies over compact subsets ofM{\displaystyle M}. Indeed, sinceM{\displaystyle M}isσ{\displaystyle \sigma }-compact, there is a sequence of compact subsetsKn{\displaystyle K_{n}}whoseunionisM{\displaystyle M}. Then:
The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space ofCr{\displaystyle C^{r}}vector fields (Leslie 1967). Over a compact subset ofM{\displaystyle M}, this follows by fixing a Riemannian metric onM{\displaystyle M}and using theexponential mapfor that metric. Ifr{\displaystyle r}is finite and the manifold is compact, the space of vector fields is aBanach space. Moreover, the transition maps from one chart of this atlas to another are smooth, making the diffeomorphism group into aBanach manifoldwith smooth right translations; left translations and inversion are only continuous. Ifr=∞{\displaystyle r=\infty }, the space of vector fields is aFréchet space. Moreover, the transition maps are smooth, making the diffeomorphism group into aFréchet manifoldand even into aregular Fréchet Lie group. If the manifold isσ{\displaystyle \sigma }-compact and not compact the full diffeomorphism group is not locally contractible for any of the two topologies. One has to restrict the group by controlling the deviation from the identity near infinity to obtain a diffeomorphism group which is a manifold; see (Michor & Mumford 2013).
TheLie algebraof the diffeomorphism group ofM{\displaystyle M}consists of allvector fieldsonM{\displaystyle M}equipped with theLie bracket of vector fields. Somewhat formally, this is seen by making a small change to the coordinatex{\displaystyle x}at each point in space:
so the infinitesimal generators are the vector fields
For a connected manifoldM{\displaystyle M}, the diffeomorphism groupactstransitivelyonM{\displaystyle M}. More generally, the diffeomorphism group acts transitively on theconfiguration spaceCkM{\displaystyle C_{k}M}. IfM{\displaystyle M}is at least two-dimensional, the diffeomorphism group acts transitively on the configuration spaceFkM{\displaystyle F_{k}M}and the action onM{\displaystyle M}ismultiply transitive(Banyaga 1997, p. 29).
In 1926,Tibor Radóasked whether theharmonic extensionof any homeomorphism or diffeomorphism of the unit circle to theunit discyields a diffeomorphism on the open disc. An elegant proof was provided shortly afterwards byHellmuth Kneser. In 1945,Gustave Choquet, apparently unaware of this result, produced a completely different proof.
The (orientation-preserving) diffeomorphism group of the circle is pathwise connected. This can be seen by noting that any such diffeomorphism can be lifted to a diffeomorphismf{\displaystyle f}of the reals satisfying[f(x+1)=f(x)+1]{\displaystyle [f(x+1)=f(x)+1]}; this space is convex and hence path-connected. A smooth, eventually constant path to the identity gives a second more elementary way of extending a diffeomorphism from the circle to the open unit disc (a special case of theAlexander trick). Moreover, the diffeomorphism group of the circle has the homotopy-type of theorthogonal groupO(2){\displaystyle O(2)}.
The corresponding extension problem for diffeomorphisms of higher-dimensional spheresSn−1{\displaystyle S^{n-1}}was much studied in the 1950s and 1960s, with notable contributions fromRené Thom,John MilnorandStephen Smale. An obstruction to such extensions is given by the finiteabelian groupΓn{\displaystyle \Gamma _{n}}, the "group of twisted spheres", defined as thequotientof the abeliancomponent groupof the diffeomorphism group by the subgroup of classes extending to diffeomorphisms of the ballBn{\displaystyle B^{n}}.
For manifolds, the diffeomorphism group is usually not connected. Its component group is called themapping class group. In dimension 2 (i.e.surfaces), the mapping class group is afinitely presented groupgenerated byDehn twists; this has been proved byMax Dehn,W. B. R. Lickorish, andAllen Hatcher).[citation needed]Max Dehn andJakob Nielsenshowed that it can be identified with theouter automorphism groupof thefundamental groupof the surface.
William Thurstonrefined this analysis byclassifying elements of the mapping class groupinto three types: those equivalent to aperiodicdiffeomorphism; those equivalent to a diffeomorphism leaving a simple closed curve invariant; and those equivalent topseudo-Anosov diffeomorphisms. In the case of thetorusS1×S1=R2/Z2{\displaystyle S^{1}\times S^{1}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}}, the mapping class group is simply themodular groupSL(2,Z){\displaystyle {\text{SL}}(2,\mathbb {Z} )}and the classification becomes classical in terms ofelliptic,parabolicandhyperbolicmatrices. Thurston accomplished his classification by observing that the mapping class group acted naturally on acompactificationofTeichmüller space; as this enlarged space was homeomorphic to a closed ball, theBrouwer fixed-point theorembecame applicable. Smaleconjecturedthat ifM{\displaystyle M}is anorientedsmooth closed manifold, theidentity componentof the group of orientation-preserving diffeomorphisms issimple. This had first been proved for a product of circles byMichel Herman; it was proved in full generality by Thurston.
Since every diffeomorphism is a homeomorphism, given a pair of manifolds which are diffeomorphic to each other they are in particularhomeomorphicto each other. The converse is not true in general.
While it is easy to find homeomorphisms that are not diffeomorphisms, it is more difficult to find a pair of homeomorphic manifolds that are not diffeomorphic. In dimensions 1, 2 and 3, any pair of homeomorphic smooth manifolds are diffeomorphic. In dimension 4 or greater, examples of homeomorphic but not diffeomorphic pairs exist. The first such example was constructed byJohn Milnorin dimension 7. He constructed a smooth 7-dimensional manifold (called nowMilnor's sphere) that is homeomorphic to the standard 7-sphere but not diffeomorphic to it. There are, in fact, 28 oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere (each of them is the total space of afiber bundleover the 4-sphere with the3-sphereas the fiber).
More unusual phenomena occur for4-manifolds. In the early 1980s, a combination of results due toSimon DonaldsonandMichael Freedmanled to the discovery ofexoticR4{\displaystyle \mathbb {R} ^{4}}: there areuncountably manypairwise non-diffeomorphic open subsets ofR4{\displaystyle \mathbb {R} ^{4}}each of which is homeomorphic toR4{\displaystyle \mathbb {R} ^{4}}, and also there are uncountably many pairwise non-diffeomorphic differentiable manifolds homeomorphic toR4{\displaystyle \mathbb {R} ^{4}}that do notembed smoothlyinR4{\displaystyle \mathbb {R} ^{4}}.
|
https://en.wikipedia.org/wiki/Diffeomorphism
|
Incryptography,homomorphic secret sharingis a type ofsecret sharingalgorithmin which the secret is encrypted viahomomorphic encryption. Ahomomorphismis a transformation from onealgebraic structureinto another of the same type so that the structure is preserved. Importantly, this means that for every kind of manipulation of the original data, there is a corresponding manipulation of the transformed data.[1]
Homomorphic secret sharing is used to transmit a secret to several recipients as follows:
Suppose a community wants to perform an election, using a decentralized voting protocol, but they want to ensure that the vote-counters won't lie about the results. Using a type of homomorphic secret sharing known asShamir's secret sharing, each member of the community can add their vote to a form that is split into pieces, each piece is then submitted to a different vote-counter. The pieces are designed so that the vote-counters can't predict how any alterations to each piece will affect the whole, thus, discouraging vote-counters from tampering with their pieces. When all votes have been received, the vote-counters combine them, allowing them to recover the aggregate election results.
In detail, suppose we have an election with:
This protocol works as long as not all of thekauthorities are corrupt — if they were, then they could collaborate to reconstructP(x) for each voter and also subsequently alter the votes.
Theprotocolrequirest+ 1authorities to be completed, therefore in case there areN>t+ 1authorities,N−t− 1authorities can be corrupted, which gives the protocol a certain degree of robustness.
The protocol manages the IDs of the voters (the IDs were submitted with the ballots) and therefore can verify that only legitimate voters have voted.
Under the assumptions ont:
Theprotocolimplicitly prevents corruption of ballots.
This is because the authorities have no incentive to change the ballot since each authority has only a share of the ballot and has no knowledge how changing this share will affect the outcome.
|
https://en.wikipedia.org/wiki/Homomorphic_secret_sharing
|
Inmathematics, amorphismis a concept ofcategory theorythat generalizes structure-preservingmapssuch ashomomorphismbetweenalgebraic structures,functionsfrom a set to another set, andcontinuous functionsbetweentopological spaces. Although many examples of morphisms are structure-preserving maps, morphisms need not to be maps, but they can be composed in a way that is similar tofunction composition.
Morphisms andobjectsare constituents of acategory. Morphisms, also calledmapsorarrows, relate two objects called thesourceand thetargetof the morphism. There is apartial operation, calledcomposition, on the morphisms of a category that is defined if the target of the first object equals the source of the second object. The composition of morphisms behave like function composition (associativityof composition when it is defined, and existence of anidentity morphismfor every object).
Morphisms and categories recur in much of contemporary mathematics. Originally, they were introduced forhomological algebraandalgebraic topology. They belong to the foundational tools ofGrothendieck'sscheme theory, a generalization ofalgebraic geometrythat applies also toalgebraic number theory.
AcategoryCconsists of twoclasses, one ofobjectsand the other ofmorphisms. There are two objects that are associated to every morphism, thesourceand thetarget. AmorphismffromXtoYis a morphism with sourceXand targetY; it is commonly written asf:X→YorXf→Ythe latter form being better suited forcommutative diagrams.
For many common categories, an object is aset(often with some additional structure) and a morphism is afunctionfrom an object to another object. Therefore, the source and the target of a morphism are often calleddomainandcodomainrespectively.
Morphisms are equipped with apartial binary operation, calledcomposition. The composition of two morphismsfandgis defined precisely when the target offis the source ofg, and is denotedg∘f(or sometimes simplygf). The source ofg∘fis the source off, and the target ofg∘fis the target ofg. The composition satisfies twoaxioms:
For a concrete category (a category in which the objects are sets, possibly with additional structure, and the morphisms are structure-preserving functions), the identity morphism is just theidentity function, and composition is just ordinarycomposition of functions.
The composition of morphisms is often represented by acommutative diagram. For example,
The collection of all morphisms fromXtoYis denotedHomC(X,Y)or simplyHom(X,Y)and called thehom-setbetweenXandY. Some authors writeMorC(X,Y),Mor(X,Y)orC(X,Y). The term hom-set is something of a misnomer, as the collection of morphisms is not required to be a set; a category whereHom(X,Y)is a set for all objectsXandYis calledlocally small. Because hom-sets may not be sets, some people prefer to use the term "hom-class".
The domain and codomain are in fact part of the information determining a morphism. For example, in thecategory of sets, where morphisms are functions, two functions may be identical as sets of ordered pairs (may have the samerange), while having different codomains. The two functions are distinct from the viewpoint of category theory. Thus many authors require that the hom-classesHom(X,Y)bedisjoint. In practice, this is not a problem because if this disjointness does not hold, it can be assured by appending the domain and codomain to the morphisms (say, as the second and third components of an ordered triple).
A morphismf:X→Yis called amonomorphismiff∘g1=f∘g2impliesg1=g2for all morphismsg1,g2:Z→X. A monomorphism can be called amonofor short, and we can usemonicas an adjective.[1]A morphismfhas aleft inverseor is asplit monomorphismif there is a morphismg:Y→Xsuch thatg∘f= idX. Thusf∘g:Y→Yisidempotent; that is,(f∘g)2=f∘ (g∘f) ∘g=f∘g. The left inversegis also called aretractionoff.[1]
Morphisms with left inverses are always monomorphisms, but theconverseis not true in general; a monomorphism may fail to have a left inverse. Inconcrete categories, a function that has a left inverse isinjective. Thus in concrete categories, monomorphisms are often, but not always, injective. The condition of being an injection is stronger than that of being a monomorphism, but weaker than that of being a split monomorphism.
Dually to monomorphisms, a morphismf:X→Yis called anepimorphismifg1∘f=g2∘fimpliesg1=g2for all morphismsg1,g2:Y→Z. An epimorphism can be called anepifor short, and we can useepicas an adjective.[1]A morphismfhas aright inverseor is asplit epimorphismif there is a morphismg:Y→Xsuch thatf∘g= idY. The right inversegis also called asectionoff.[1]Morphisms having a right inverse are always epimorphisms, but the converse is not true in general, as an epimorphism may fail to have a right inverse.
If a monomorphismfsplits with left inverseg, thengis a split epimorphism with right inversef. Inconcrete categories, a function that has a right inverse issurjective. Thus in concrete categories, epimorphisms are often, but not always, surjective. The condition of being a surjection is stronger than that of being an epimorphism, but weaker than that of being a split epimorphism. In thecategory of sets, the statement that every surjection has a section is equivalent to theaxiom of choice.
A morphism that is both an epimorphism and a monomorphism is called abimorphism.
A morphismf:X→Yis called anisomorphismif there exists a morphismg:Y→Xsuch thatf∘g= idYandg∘f= idX. If a morphism has both left-inverse and right-inverse, then the two inverses are equal, sofis an isomorphism, andgis called simply theinverseoff. Inverse morphisms, if they exist, are unique. The inversegis also an isomorphism, with inversef. Two objects with an isomorphism between them are said to beisomorphicor equivalent.
While every isomorphism is a bimorphism, a bimorphism is not necessarily an isomorphism. For example, in the category ofcommutative ringsthe inclusionZ→Qis a bimorphism that is not an isomorphism. However, any morphism that is both an epimorphism and asplitmonomorphism, or both a monomorphism and asplitepimorphism, must be an isomorphism. A category, such as aSet, in which every bimorphism is an isomorphism is known as abalanced category.
A morphismf:X→X(that is, a morphism with identical source and target) is anendomorphismofX. Asplit endomorphismis an idempotent endomorphismfiffadmits a decompositionf=h∘gwithg∘h= id. In particular, theKaroubi envelopeof a category splits every idempotent morphism.
Anautomorphismis a morphism that is both an endomorphism and an isomorphism. In every category, the automorphisms of an object always form agroup, called theautomorphism groupof the object.
For more examples, seeCategory theory.
|
https://en.wikipedia.org/wiki/Morphism
|
Ingroup theory, given agroupG{\displaystyle G}, aquasimorphism(orquasi-morphism) is afunctionf:G→R{\displaystyle f:G\to \mathbb {R} }which isadditiveup to bounded error, i.e. there exists aconstantD≥0{\displaystyle D\geq 0}such that|f(gh)−f(g)−f(h)|≤D{\displaystyle |f(gh)-f(g)-f(h)|\leq D}for allg,h∈G{\displaystyle g,h\in G}. The least positive value ofD{\displaystyle D}for which this inequality is satisfied is called thedefectoff{\displaystyle f}, written asD(f){\displaystyle D(f)}. For a groupG{\displaystyle G}, quasimorphisms form asubspaceof thefunction spaceRG{\displaystyle \mathbb {R} ^{G}}.
A quasimorphism ishomogeneousiff(gn)=nf(g){\displaystyle f(g^{n})=nf(g)}for allg∈G,n∈Z{\displaystyle g\in G,n\in \mathbb {Z} }. It turns out the study of quasimorphisms can be reduced to the study of homogeneous quasimorphisms, as every quasimorphismf:G→R{\displaystyle f:G\to \mathbb {R} }is a bounded distance away from a unique homogeneous quasimorphismf¯:G→R{\displaystyle {\overline {f}}:G\to \mathbb {R} }, given by :
A homogeneous quasimorphismf:G→R{\displaystyle f:G\to \mathbb {R} }has the following properties:
One can also define quasimorphisms similarly in the case of a functionf:G→Z{\displaystyle f:G\to \mathbb {Z} }. In this case, the above discussion about homogeneous quasimorphisms does not hold anymore, as the limitlimn→∞f(gn)/n{\displaystyle \lim _{n\to \infty }f(g^{n})/n}does not exist inZ{\displaystyle \mathbb {Z} }in general.
For example, forα∈R{\displaystyle \alpha \in \mathbb {R} }, the mapZ→Z:n↦⌊αn⌋{\displaystyle \mathbb {Z} \to \mathbb {Z} :n\mapsto \lfloor \alpha n\rfloor }is a quasimorphism. There is a construction of the real numbers as a quotient of quasimorphismsZ→Z{\displaystyle \mathbb {Z} \to \mathbb {Z} }by an appropriate equivalence relation, seeConstruction of the reals numbers from integers (Eudoxus reals).
|
https://en.wikipedia.org/wiki/Quasimorphism
|
Inabstract algebra, thefundamental theoremon homomorphisms, also known as thefundamental homomorphism theorem, or thefirst isomorphism theorem, relates the structure of two objects between which ahomomorphismis given, and of thekernelandimageof the homomorphism.
The homomorphism theorem is used toprovetheisomorphism theorems. Similar theorems are valid forvector spaces,modules, andrings.
Given twogroupsG{\displaystyle G}andH{\displaystyle H}and agroup homomorphismf:G→H{\displaystyle f:G\rightarrow H}, letN{\displaystyle N}be anormal subgroupinG{\displaystyle G}andϕ{\displaystyle \phi }the naturalsurjectivehomomorphismG→G/N{\displaystyle G\rightarrow G/N}(whereG/N{\displaystyle G/N}is thequotient groupofG{\displaystyle G}byN{\displaystyle N}). IfN{\displaystyle N}is asubsetofker(f){\displaystyle \ker(f)}(whereker{\displaystyle \ker }represents akernel) then there exists a unique homomorphismh:G/N→H{\displaystyle h:G/N\rightarrow H}such thatf=h∘ϕ{\displaystyle f=h\circ \phi }.
In other words, the natural projectionϕ{\displaystyle \phi }isuniversalamong homomorphisms onG{\displaystyle G}that mapN{\displaystyle N}to theidentity element.
The situation is described by the followingcommutative diagram:
h{\displaystyle h}is injective if and only ifN=ker(f){\displaystyle N=\ker(f)}. Therefore, by settingN=ker(f){\displaystyle N=\ker(f)}, we immediately get thefirst isomorphism theorem.
We can write the statement of the fundamental theorem on homomorphisms of groups as "every homomorphic image of a group is isomorphic to a quotient group".
The proof follows from two basic facts about homomorphisms, namely their preservation of the group operation, and their mapping of the identity element to the identity element. We need to show that ifϕ:G→H{\displaystyle \phi :G\to H}is a homomorphism of groups, then:
The operation that is preserved byϕ{\displaystyle \phi }is the group operation. Ifa,b∈im(ϕ){\displaystyle a,b\in {\text{im}}(\phi )}, then there exist elementsa′,b′∈G{\displaystyle a',b'\in G}such thatϕ(a′)=a{\displaystyle \phi (a')=a}andϕ(b′)=b{\displaystyle \phi (b')=b}. For thesea{\displaystyle a}andb{\displaystyle b}, we haveab=ϕ(a′)ϕ(b′)=ϕ(a′b′)∈im(ϕ){\displaystyle ab=\phi (a')\phi (b')=\phi (a'b')\in {\text{im}}(\phi )}(sinceϕ{\displaystyle \phi }preserves the group operation), and thus, the closure property is satisfied inim(ϕ){\displaystyle {\text{im}}(\phi )}. The identity elemente∈H{\displaystyle e\in H}is also inim(ϕ){\displaystyle {\text{im}}(\phi )}becauseϕ{\displaystyle \phi }maps the identity element ofG{\displaystyle G}to it. Since every elementa′{\displaystyle a'}inG{\displaystyle G}has an inverse(a′)−1{\displaystyle (a')^{-1}}such thatϕ((a′)−1)=(ϕ(a′))−1{\displaystyle \phi ((a')^{-1})=(\phi (a'))^{-1}}(becauseϕ{\displaystyle \phi }preserves the inverse property as well), we have an inverse for each elementϕ(a′)=a{\displaystyle \phi (a')=a}inim(ϕ){\displaystyle {\text{im}}(\phi )}, therefore,im(ϕ){\displaystyle {\text{im}}(\phi )}is a subgroup ofH{\displaystyle H}.
Construct a mapψ:G/ker(ϕ)→im(ϕ){\displaystyle \psi :G/\ker(\phi )\to {\text{im}}(\phi )}byψ(aker(ϕ))=ϕ(a){\displaystyle \psi (a\ker(\phi ))=\phi (a)}. This map is well-defined, as ifaker(ϕ)=bker(ϕ){\displaystyle a\ker(\phi )=b\ker(\phi )}, thenb−1a∈ker(ϕ){\displaystyle b^{-1}a\in \ker(\phi )}and soϕ(b−1a)=e⇒ϕ(b−1)ϕ(a)=e{\displaystyle \phi (b^{-1}a)=e\Rightarrow \phi (b^{-1})\phi (a)=e}which givesϕ(a)=ϕ(b){\displaystyle \phi (a)=\phi (b)}. This map is an isomorphism.ψ{\displaystyle \psi }is surjective ontoim(ϕ){\displaystyle {\text{im}}(\phi )}by definition. To show injectivity, ifψ(aker(ϕ))=ψ(bker(ϕ)){\displaystyle \psi (a\ker(\phi ))=\psi (b\ker(\phi ))}, thenϕ(a)=ϕ(b){\displaystyle \phi (a)=\phi (b)}, which impliesb−1a∈ker(ϕ){\displaystyle b^{-1}a\in \ker(\phi )}soaker(ϕ)=bker(ϕ){\displaystyle a\ker(\phi )=b\ker(\phi )}.
Finally,
henceψ{\displaystyle \psi }preserves the group operation. Henceψ{\displaystyle \psi }is an isomorphism betweenG/ker(ϕ){\displaystyle G/\ker(\phi )}andim(ϕ){\displaystyle {\text{im}}(\phi )}, which completes the proof.
The group theoretic version of the fundamental homomorphism theorem can be used to show that two selected groups are isomorphic. Two examples are shown below.
For eachn∈N{\displaystyle n\in \mathbb {N} }, consider the groupsZ{\displaystyle \mathbb {Z} }andZn{\displaystyle \mathbb {Z} _{n}}and a group homomorphismf:Z→Zn{\displaystyle f:\mathbb {Z} \rightarrow \mathbb {Z} _{n}}defined bym↦mmodn{\displaystyle m\mapsto m{\text{ mod }}n}(seemodular arithmetic). Next, consider the kernel off{\displaystyle f},ker(f)=nZ{\displaystyle {\text{ker}}(f)=n\mathbb {Z} }, which is a normal subgroup inZ{\displaystyle \mathbb {Z} }. There exists a natural surjective homomorphismφ:Z→Z/nZ{\displaystyle \varphi :\mathbb {Z} \rightarrow \mathbb {Z} /n\mathbb {Z} }defined bym↦m+nZ{\displaystyle m\mapsto m+n\mathbb {Z} }. The theorem asserts that there exists an isomorphismh{\displaystyle h}betweenZn{\displaystyle \mathbb {Z} _{n}}andZ/nZ{\displaystyle \mathbb {Z} /n\mathbb {Z} }, or in other wordsZn≅Z/nZ{\displaystyle \mathbb {Z} _{n}\cong \mathbb {Z} /n\mathbb {Z} }. The commutative diagram is illustrated below.
LetG{\displaystyle G}be a group withsubgroupH{\displaystyle H}. LetCG(H){\displaystyle C_{G}(H)},NG(H){\displaystyle N_{G}(H)}andAut(H){\displaystyle \operatorname {Aut} (H)}be thecentralizer, thenormalizerand theautomorphism groupofH{\displaystyle H}inG{\displaystyle G}, respectively. Then, theN/C{\displaystyle N/C}theorem states thatNG(H)/CG(H){\displaystyle N_{G}(H)/C_{G}(H)}is isomorphic to a subgroup ofAut(H){\displaystyle \operatorname {Aut} (H)}.
We are able to find a group homomorphismf:NG(H)→Aut(H){\displaystyle f:N_{G}(H)\rightarrow \operatorname {Aut} (H)}defined byg↦ghg−1{\displaystyle g\mapsto ghg^{-1}}, for allh∈H{\displaystyle h\in H}. Clearly, the kernel off{\displaystyle f}isCG(H){\displaystyle C_{G}(H)}. Hence, we have a natural surjective homomorphismφ:NG(H)→NG(H)/CG(H){\displaystyle \varphi :N_{G}(H)\rightarrow N_{G}(H)/C_{G}(H)}defined byg↦gC(H){\displaystyle g\mapsto gC(H)}. The fundamental homomorphism theorem then asserts that there exists an isomorphism betweenNG(H)/CG(H){\displaystyle N_{G}(H)/C_{G}(H)}andφ(NG(H)){\displaystyle \varphi (N_{G}(H))}, which is a subgroup ofAut(H){\displaystyle \operatorname {Aut} (H)}.
|
https://en.wikipedia.org/wiki/Fundamental_theorem_on_homomorphisms
|
Ring homomorphisms
Algebraic structures
Related structures
Algebraic number theory
Noncommutative algebraic geometry
Free algebra
Clifford algebra
Inmathematics, aring homomorphismis a structure-preservingfunctionbetween tworings. More explicitly, ifRandSare rings, then a ring homomorphism is a functionf:R→Sthat preserves addition, multiplication andmultiplicative identity; that is,[1][2][3][4][5]
for alla,binR.
These conditions imply that additive inverses and the additive identity are also preserved.
If, in addition,fis abijection, then itsinversef−1is also a ring homomorphism. In this case,fis called aring isomorphism, and the ringsRandSare calledisomorphic. From the standpoint of ring theory, isomorphic rings have exactly the same properties.
IfRandSarerngs, then the corresponding notion is that of arng homomorphism,[a]defined as above except without the third conditionf(1R) = 1S. A rng homomorphism between (unital) rings need not be a ring homomorphism.
Thecompositionof two ring homomorphisms is a ring homomorphism. It follows that the rings forms acategorywith ring homomorphisms asmorphisms(seeCategory of rings).
In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.
Letf:R→Sbe a ring homomorphism. Then, directly from these definitions, one can deduce:
Moreover,
Injective ring homomorphisms are identical tomonomorphismsin the category of rings: Iff:R→Sis a monomorphism that is not injective, then it sends somer1andr2to the same element ofS. Consider the two mapsg1andg2fromZ[x] toRthat mapxtor1andr2, respectively;f∘g1andf∘g2are identical, but sincefis a monomorphism this is impossible.
However, surjective ring homomorphisms are vastly different fromepimorphismsin the category of rings. For example, the inclusionZ⊆Qwith the identity mapping is a ring epimorphism, but not a surjection. However, every ring epimorphism is also astrong epimorphism, the converse being true in every category.[citation needed]
|
https://en.wikipedia.org/wiki/Ring_homomorphism
|
Ininformation theory, theasymptotic equipartition property(AEP) is a general property of the output samples of astochastic source. It is fundamental to the concept oftypical setused in theories ofdata compression.
Roughly speaking, the theorem states that although there are many series of results that may be produced by a random process, the one actually produced is most probably from a loosely defined set of outcomes that all have approximately the same chance of being the one actually realized. (This is a consequence of thelaw of large numbersandergodic theory.) Although there are individual outcomes which have a higher probability than any outcome in this set, the vast number of outcomes in the set almost guarantees that the outcome will come from the set. One way of intuitively understanding the property is throughCramér's large deviation theorem, which states that the probability of a large deviation from mean decays exponentially with the number of samples. Such results are studied inlarge deviations theory; intuitively, it is the large deviations that would violate equipartition, but these are unlikely.
In the field ofpseudorandom number generation, a candidate generator of undetermined quality whose output sequence lies too far outside the typical set by some statistical criteria is rejected as insufficiently random. Thus, although the typical set is loosely defined, practical notions arise concerningsufficienttypicality.
Given a discrete-time stationary ergodic stochastic processX{\displaystyle X}on theprobability space(Ω,B,p){\displaystyle (\Omega ,B,p)}, the asymptotic equipartition property is an assertion that,almost surely,−1nlogp(X1,X2,…,Xn)→H(X)asn→∞{\displaystyle -{\frac {1}{n}}\log p(X_{1},X_{2},\dots ,X_{n})\to H(X)\quad {\text{ as }}\quad n\to \infty }whereH(X){\displaystyle H(X)}or simplyH{\displaystyle H}denotes theentropy rateofX{\displaystyle X}, which must exist for all discrete-timestationary processesincluding the ergodic ones. The asymptotic equipartition property is proved for finite-valued (i.e.|Ω|<∞{\displaystyle |\Omega |<\infty }) stationary ergodic stochastic processes in theShannon–McMillan–Breiman theoremusing the ergodic theory and for anyi.i.d.sources directly using the law of large numbers in both the discrete-valued case (whereH{\displaystyle H}is simply theentropyof a symbol) and the continuous-valued case (whereH{\displaystyle H}is the differential entropy instead). The definition of the asymptotic equipartition property can also be extended for certain classes of continuous-time stochastic processes for which a typical set exists for long enough observation time. The convergence is provenalmost surein all cases.
GivenX{\displaystyle X}is ani.i.d.source which may take values in the alphabetX{\displaystyle {\mathcal {X}}}, itstime seriesX1,…,Xn{\displaystyle X_{1},\ldots ,X_{n}}is i.i.d. withentropyH(X){\displaystyle H(X)}. The weaklaw of large numbersgives the asymptotic equipartition property withconvergence in probability,limn→∞Pr[|−1nlogp(X1,X2,…,Xn)−H(X)|>ε]=0∀ε>0.{\displaystyle \lim _{n\to \infty }\Pr \left[\left|-{\frac {1}{n}}\log p(X_{1},X_{2},\ldots ,X_{n})-H(X)\right|>\varepsilon \right]=0\qquad \forall \varepsilon >0.}since the entropy is equal to the expectation of[1]−1nlogp(X1,X2,…,Xn).{\displaystyle -{\frac {1}{n}}\log p(X_{1},X_{2},\ldots ,X_{n}).}
The strong law of large numbers asserts the stronger almost sure convergence,Pr[limn→∞−1nlogp(X1,X2,…,Xn)=H(X)]=1.{\displaystyle \Pr \left[\lim _{n\to \infty }-{\frac {1}{n}}\log p(X_{1},X_{2},\ldots ,X_{n})=H(X)\right]=1.}Convergence in the sense of L1 asserts an even strongerE[|limn→∞−1nlogp(X1,X2,…,Xn)−H(X)|]=0{\displaystyle \mathbb {E} \left[\left|\lim _{n\to \infty }-{\frac {1}{n}}\log p(X_{1},X_{2},\ldots ,X_{n})-H(X)\right|\right]=0}
Consider a finite-valued sample spaceΩ{\displaystyle \Omega }, i.e.|Ω|<∞{\displaystyle |\Omega |<\infty }, for the discrete-timestationary ergodic processX:={Xn}{\displaystyle X:=\{X_{n}\}}defined on theprobability space(Ω,B,p){\displaystyle (\Omega ,B,p)}. TheShannon–McMillan–Breiman theorem, due toClaude Shannon,Brockway McMillan, andLeo Breiman, states that we have convergence in the sense of L1.[2]Chung Kai-laigeneralized this to the case whereX{\displaystyle X}may take value in a set of countable infinity, provided that the entropy rate is still finite.[3]
The assumptions of stationarity/ergodicity/identical distribution of random variables is not essential for the asymptotic equipartition property to hold. Indeed, as is quite clear intuitively, the asymptotic equipartition property requires only some form of the law of large numbers to hold, which is fairly general. However, the expression needs to be suitably generalized, and the conditions need to be formulated precisely.
Consider a source that produces independent symbols, possibly with different output statistics at each instant, for which the statistics of the process are known completely, that is, the marginal distribution of the process seen at each time instant is known. The joint distribution is just the product of marginals. Then, under the condition (which can be relaxed) thatVar[logp(Xi)]<M{\displaystyle \mathrm {Var} [\log p(X_{i})]<M}for alli, for someM> 0, the following holds (AEP):limn→∞Pr[|−1nlogp(X1,X2,…,Xn)−H¯n(X)|<ε]=1∀ε>0{\displaystyle \lim _{n\to \infty }\Pr \left[\,\left|-{\frac {1}{n}}\log p(X_{1},X_{2},\ldots ,X_{n})-{\overline {H}}_{n}(X)\right|<\varepsilon \right]=1\qquad \forall \varepsilon >0}whereH¯n(X)=1nH(X1,X2,…,Xn){\displaystyle {\overline {H}}_{n}(X)={\frac {1}{n}}H(X_{1},X_{2},\ldots ,X_{n})}
The proof follows from a simple application ofMarkov's inequality(applied to second moment oflog(p(Xi)){\displaystyle \log(p(X_{i}))}.Pr[|−1nlogp(X1,X2,…,Xn)−H¯(X)|>ε]≤1n2ε2Var[∑i=1n(log(p(Xi))2]≤Mnε2→0asn→∞{\displaystyle {\begin{aligned}\Pr \left[\left|-{\frac {1}{n}}\log p(X_{1},X_{2},\ldots ,X_{n})-{\overline {H}}(X)\right|>\varepsilon \right]&\leq {\frac {1}{n^{2}\varepsilon ^{2}}}\mathrm {Var} \left[\sum _{i=1}^{n}\left(\log(p(X_{i})\right)^{2}\right]\\&\leq {\frac {M}{n\varepsilon ^{2}}}\to 0{\text{ as }}n\to \infty \end{aligned}}}
It is obvious that the proof holds if any momentE[|logp(Xi)|r]{\displaystyle \mathrm {E} \left[|\log p(X_{i})|^{r}\right]}is uniformly bounded forr> 1 (again byMarkov's inequalityapplied tor-th moment).Q.E.D.
Even this condition is not necessary, but given a non-stationary random process, it should not be difficult to test whether the asymptotic equipartition property holds using the above method.
The asymptotic equipartition property for non-stationary discrete-time independent process leads us to (among other results) thesource coding theoremfor non-stationary source (with independent output symbols) andnoisy-channel coding theoremfor non-stationary memoryless channels.
T{\textstyle T}is a measure-preserving map on the probability spaceΩ{\textstyle \Omega }.
IfP{\textstyle P}is a finite or countable partition ofΩ{\textstyle \Omega }, then its entropy isH(P):=−∑p∈Pμ(p)lnμ(p){\displaystyle H(P):=-\sum _{p\in P}\mu (p)\ln \mu (p)}with the convention that0ln0=0{\displaystyle 0\ln 0=0}.
We only consider partitions with finite entropy:H(P)<∞{\textstyle H(P)<\infty }.
IfP{\textstyle P}is a finite or countable partition ofΩ{\textstyle \Omega }, then we construct a sequence of partitions by iterating the map:P(n):=P∨T−1P∨⋯∨T−(n−1)P{\displaystyle P^{(n)}:=P\vee T^{-1}P\vee \dots \vee T^{-(n-1)}P}whereP∨Q{\textstyle P\vee Q}is the least upper bound partition, that is, the least refined partition that refines bothP{\textstyle P}andQ{\textstyle Q}:P∨Q:={p∩q:p∈P,q∈Q}{\displaystyle P\vee Q:=\{p\cap q:p\in P,q\in Q\}}WriteP(x){\textstyle P(x)}to be the set inP{\textstyle P}wherex{\textstyle x}falls in. So, for example,P(n)(x){\textstyle P^{(n)}(x)}is then{\textstyle n}-letter initial segment of the(P,T){\textstyle (P,T)}name ofx{\textstyle x}.
WriteIP(x){\textstyle I_{P}(x)}to be the information (in units ofnats) aboutx{\textstyle x}we can recover, if we know which element in the partitionP{\textstyle P}thatx{\textstyle x}falls in:IP:=−lnμ(P(x)){\displaystyle I_{P}:=-\ln \mu (P(x))}Similarly, the conditional information of partitionP{\textstyle P}, conditional on partitionQ{\textstyle Q}, aboutx{\textstyle x}, isIP|Q(x):=−lnP∨Q(x)Q(x){\displaystyle I_{P|Q}(x):=-\ln {\frac {P\vee Q(x)}{Q(x)}}}hT(P){\textstyle h_{T}(P)}is theKolmogorov-Sinai entropyhT(P):=limn1nH(P(n))=limnEx∼μ[1nIP(n)(x)]{\displaystyle h_{T}(P):=\lim _{n}{\frac {1}{n}}H(P^{(n)})=\lim _{n}E_{x\sim \mu }\left[{\frac {1}{n}}I_{P^{(n)}}(x)\right]}In other words, by definition, there is a convergence in expectation. The SMB theorem states that whenT{\textstyle T}is ergodic, there is convergence in L1.[4]
Theorem(ergodic case)—IfT{\textstyle T}is ergodic, thenx↦1nIP(n)(x){\displaystyle x\mapsto {\frac {1}{n}}I_{P^{(n)}}(x)}converges in L1 to the constant functionx↦hT(P){\textstyle x\mapsto h_{T}(P)}.
In other words,Ex∼μ[|limn1nIP(n)(x)−hT(P)|]=0{\displaystyle E_{x\sim \mu }\left[\left|\lim _{n}{\frac {1}{n}}I_{P^{(n)}}(x)-h_{T}(P)\right|\right]=0}
In particular, since L1 convergence implies almost sure convergence,hT(P)=limn1nIP(n)(x){\displaystyle h_{T}(P)=\lim _{n}{\frac {1}{n}}I_{P^{(n)}}(x)}with probability 1.
Corollary(entropy equipartition property)—∀ϵ>0,∃N,∀n≥N{\textstyle \forall \epsilon >0,\exists N,\forall n\geq N}, we can partition the partition∨k=0n−1T−kP{\textstyle \vee _{k=0}^{n-1}T^{-k}P}into two parts, the “good” partG{\textstyle G}and the “bad” partB{\textstyle B}.
The bad part is small:∑b∈Bμ(b)<ϵ{\displaystyle \sum _{b\in B}\mu (b)<\epsilon }
The good part is almost equipartitioned according to entropy:∀g∈G,−1nlnμ(g)∈hT(P)±ϵ{\displaystyle \forall g\in G,\quad -{\frac {1}{n}}\ln \mu (g)\in h_{T}(P)\pm \epsilon }
IfT{\textstyle T}is not necessarily ergodic, then the underlying probability space would be split up into multiple subsets, each invariant underT{\textstyle T}. In this case, we still have L1 convergence to some function, but that function is no longer a constant function.[5]
Theorem(general case)—LetI{\textstyle {\mathcal {I}}}be the sigma-algebra generated by allT{\textstyle T}-invariant measurable subsets ofΩ{\textstyle \Omega }, -x↦1nIP(n)(x){\displaystyle x\mapsto {\frac {1}{n}}I_{P^{(n)}}(x)}converges in L1 tox↦E[limnIP|∨k=1nT−kP|I]{\displaystyle x\mapsto E\left[\lim _{n}I_{P|\vee _{k=1}^{n}T^{-k}P}{\big |}\;{\mathcal {I}}\right]}
WhenT{\textstyle T}is ergodic,I{\textstyle {\mathcal {I}}}is trivial, and so the functionx↦E[limnIP|∨k=1nT−kP|I]{\displaystyle x\mapsto E\left[\lim _{n}I_{P|\vee _{k=1}^{n}T^{-k}P}{\big |}\;{\mathcal {I}}\right]}simplifies into the constant functionx↦E[limnIP|∨k=1nT−kP]{\textstyle x\mapsto E\left[\lim _{n}I_{P|\vee _{k=1}^{n}T^{-k}P}\right]}, which by definition, equalslimnH(P|∨k=1nT−kP){\textstyle \lim _{n}H(P|\vee _{k=1}^{n}T^{-k}P)}, which equalshT(P){\textstyle h_{T}(P)}by a proposition.
Discrete-time functions can be interpolated to continuous-time functions. If such interpolationfismeasurable, we may define the continuous-time stationary process accordingly asX~:=f∘X{\displaystyle {\tilde {X}}:=f\circ X}. If the asymptotic equipartition property holds for the discrete-time process, as in the i.i.d. or finite-valued stationary ergodic cases shown above, it automatically holds for the continuous-time stationary process derived from it by some measurable interpolation. i.e.−1nlogp(X~0τ)→H(X){\displaystyle -{\frac {1}{n}}\log p({\tilde {X}}_{0}^{\tau })\to H(X)}wherencorresponds to the degree of freedom in timeτ.nH(X)/τandH(X)are the entropy per unit time and per degree of freedom respectively, defined byShannon.
An important class of such continuous-time stationary process is the bandlimited stationary ergodic process with the sample space being a subset of the continuousL2{\displaystyle {\mathcal {L}}_{2}}functions. The asymptotic equipartition property holds if the process is white, in which case the time samples are i.i.d., or there existsT> 1/2W, whereWis thenominal bandwidth, such that theT-spaced time samples take values in a finite set, in which case we have the discrete-time finite-valued stationary ergodic process.
Anytime-invariantoperations also preserves the asymptotic equipartition property, stationarity and ergodicity and we may easily turn a stationary process to non-stationary without losing the asymptotic equipartition property by nulling out a finite number of time samples in the process.
Acategory theoreticdefinition for the equipartition property is given byGromov.[6]Given a sequence ofCartesian powersPN=P×⋯×P{\displaystyle P^{N}=P\times \cdots \times P}of a measure spaceP, this sequence admits anasymptotically equivalentsequenceHNof homogeneous measure spaces (i.e.all sets have the same measure; all morphisms are invariant under the group of automorphisms, and thus factor as a morphism to theterminal object).
The above requires a definition ofasymptotic equivalence. This is given in terms of a distance function, giving how much aninjective correspondencediffers from anisomorphism. An injective correspondenceπ:P→Q{\displaystyle \pi :P\to Q}is apartially defined mapthat is abijection; that is, it is a bijection between a subsetP′⊂P{\displaystyle P'\subset P}andQ′⊂Q{\displaystyle Q'\subset Q}. Then define|P−Q|π=|P∖P′|+|Q∖Q′|,{\displaystyle |P-Q|_{\pi }=|P\setminus P'|+|Q\setminus Q'|,}where |S| denotes the measure of a setS. In what follows, the measure ofPandQare taken to be 1, so that the measure spaces are probability spaces. This distance|P−Q|π{\displaystyle |P-Q|_{\pi }}is commonly known as theearth mover's distanceorWasserstein metric.
Similarly, define|logP:Q|π=supp∈P′|logp−logπ(p)|logmin(|set(P′)|,|set(Q′)|).{\displaystyle |\log P:Q|_{\pi }={\frac {\sup _{p\in P'}|\log p-\log \pi (p)|}{\log \min \left(|\operatorname {set} (P')|,|\operatorname {set} (Q')|\right)}}.}with|set(P)|{\displaystyle |\operatorname {set} (P)|}taken to be the counting measure onP. Thus, this definition requires thatPbe a finite measure space. Finally, letdistπ(P,Q)=|P−Q|π+|logP:Q|π.{\displaystyle {\text{dist}}_{\pi }(P,Q)=|P-Q|_{\pi }+|\log P:Q|_{\pi }.}
A sequence of injective correspondencesπN:PN→QN{\displaystyle \pi _{N}:P_{N}\to Q_{N}}are thenasymptotically equivalentwhendistπN(PN,QN)→0asN→∞.{\displaystyle {\text{dist}}_{\pi _{N}}(P_{N},Q_{N})\to 0\quad {\text{ as }}\quad N\to \infty .}
Given a homogenous space sequenceHNthat is asymptotically equivalent toPN, the entropyH(P) ofPmay be taken asH(P)=limN→∞1N|set(HN)|.{\displaystyle H(P)=\lim _{N\to \infty }{\frac {1}{N}}|\operatorname {set} (H_{N})|.}
|
https://en.wikipedia.org/wiki/Asymptotic_equipartition_property
|
Ininformation theory,Fano's inequality(also known as theFano converseand theFano lemma) relates the average information lost in a noisy channel to the probability of the categorization error. It was derived byRobert Fanoin the early 1950s while teaching aPh.D.seminar in information theory atMIT, and later recorded in his 1961 textbook.
It is used to find a lower bound on the error probability of any decoder as well as the lower bounds forminimax risksindensity estimation.
Let the discreterandom variablesX{\displaystyle X}andY{\displaystyle Y}represent input and output messages with ajoint probabilityP(x,y){\displaystyle P(x,y)}. Lete{\displaystyle e}represent an occurrence of error; i.e., thatX≠X~{\displaystyle X\neq {\tilde {X}}}, withX~=f(Y){\displaystyle {\tilde {X}}=f(Y)}being an approximate version ofX{\displaystyle X}. Fano's inequality is
whereX{\displaystyle {\mathcal {X}}}denotes the support ofX{\displaystyle X},|X|{\displaystyle |{\mathcal {X}}|}denotes thecardinalityof (number of elements in)X{\displaystyle {\mathcal {X}}},
is theconditional entropy,
is the probability of the communication error, and
is the correspondingbinary entropy.
Define an indicator random variableE{\displaystyle E}, that indicates the event that our estimateX~=f(Y){\displaystyle {\tilde {X}}=f(Y)}is in error,
ConsiderH(E,X|X~){\displaystyle H(E,X|{\tilde {X}})}. We can use thechain rule for entropiesto expand this in two different ways
Equating the two
Expanding the right most term,H(X∣E,X~){\displaystyle H(X\mid E,{\tilde {X}})}
SinceE=0{\displaystyle E=0}meansX=X~{\displaystyle X={\tilde {X}}}; being given the value ofX~{\displaystyle {\tilde {X}}}allows us to know the value ofX{\displaystyle X}with certainty. This makes the termH(X∣E=0,X~)=0{\displaystyle H(X\mid E=0,{\tilde {X}})=0}.
On the other hand,E=1{\displaystyle E=1}means thatX~≠X{\displaystyle {\tilde {X}}\neq X}, hence given the value ofX~{\displaystyle {\tilde {X}}}, we can narrow downX{\displaystyle X}to one of|X|−1{\displaystyle |{\mathcal {X}}|-1}different values, allowing us to upper bound the conditional entropyH(X∣E=1,X~)≤log(|X|−1){\displaystyle H(X\mid E=1,{\tilde {X}})\leq \log(|{\mathcal {X}}|-1)}. Hence
The other term,H(E∣X~)≤H(E){\displaystyle H(E\mid {\tilde {X}})\leq H(E)}, because conditioning reduces entropy. Because of the wayE{\displaystyle E}is defined,H(E)=Hb(e){\displaystyle H(E)=H_{b}(e)}, meaning thatH(E∣X~)≤Hb(e){\displaystyle H(E\mid {\tilde {X}})\leq H_{b}(e)}. Putting it all together,
BecauseX→Y→X~{\displaystyle X\rightarrow Y\rightarrow {\tilde {X}}}is a Markov chain, we haveI(X;X~)≤I(X;Y){\displaystyle I(X;{\tilde {X}})\leq I(X;Y)}by thedata processing inequality, and henceH(X∣X~)≥H(X∣Y){\displaystyle H(X\mid {\tilde {X}})\geq H(X\mid Y)}, giving us
Fano's inequalitycan be interpreted as a way of dividing the uncertainty of a conditional distribution into two questions given an arbitrary predictor. The first question, corresponding to the termHb(e){\displaystyle H_{b}(e)}, relates to the uncertainty of the predictor. If the prediction is correct, there is no more uncertainty remaining. If the prediction is incorrect, the uncertainty of any discrete distribution has an upper bound of the entropy of the uniform distribution over all choices besides the incorrect prediction. This has entropylog(|X|−1){\displaystyle \log(|{\mathcal {X}}|-1)}. Looking at extreme cases, if the predictor is always correct the first and second terms of the inequality are 0, and the existence of a perfect predictor impliesX{\displaystyle X}is totally determined byY{\displaystyle Y}, and soH(X|Y)=0{\displaystyle H(X|Y)=0}. If the predictor is always wrong, then the first term is 0, andH(X∣Y){\displaystyle H(X\mid Y)}can only be upper bounded with a uniform distribution over the remaining choices.
LetX{\displaystyle X}be arandom variablewithdensityequal to one ofr+1{\displaystyle r+1}possible densitiesf1,…,fr+1{\displaystyle f_{1},\ldots ,f_{r+1}}. Furthermore, theKullback–Leibler divergencebetween any pair of densities cannot be too large,
Letψ(X)∈{1,…,r+1}{\displaystyle \psi (X)\in \{1,\ldots ,r+1\}}be an estimate of the index. Then
wherePi{\displaystyle P_{i}}is theprobabilityinduced byfi{\displaystyle f_{i}}.
The following generalization is due to Ibragimov and Khasminskii (1979), Assouad and Birge (1983).
LetFbe a class of densities with a subclass ofr+ 1 densitiesƒθsuch that for anyθ≠θ′
Then in the worst case theexpected valueof error of estimation is bound from below,
whereƒnis anydensity estimatorbased on asampleof sizen.
|
https://en.wikipedia.org/wiki/Fano%27s_inequality
|
Rate–distortion theoryis a major branch ofinformation theorywhich provides the theoretical foundations forlossy data compression; it addresses the problem of determining the minimal number of bits per symbol, as measured by the rateR, that should be communicated over a channel, so that the source (input signal) can be approximately reconstructed at the receiver (output signal) without exceeding an expected distortionD.
Rate–distortion theory gives an analytical expression for how much compression can be achieved using lossy compression methods. Many of the existing audio, speech, image, and video compression techniques have transforms, quantization, and bit-rate allocation procedures that capitalize on the general shape of rate–distortion functions.
Rate–distortion theory was created byClaude Shannonin his foundational work on information theory.
In rate–distortion theory, therateis usually understood as the number ofbitsper data sample to be stored or transmitted. The notion ofdistortionis a subject of on-going discussion.[1]In the most simple case (which is actually used in most cases), the distortion is defined as the expected value of the square of the difference between input and output signal (i.e., themean squared error). However, since we know that mostlossy compressiontechniques operate on data that will be perceived by human consumers (listening tomusic, watching pictures and video) the distortion measure should preferably be modeled on humanperceptionand perhapsaesthetics: much like the use ofprobabilityinlossless compression, distortion measures can ultimately be identified withloss functionsas used in Bayesianestimationanddecision theory. In audio compression, perceptual models (and therefore perceptual distortion measures) are relatively well developed and routinely used in compression techniques such asMP3orVorbis, but are often not easy to include in rate–distortion theory. In image and video compression, the human perception models are less well developed and inclusion is mostly limited to theJPEGandMPEGweighting (quantization,normalization) matrix.
Distortion functions measure the cost of representing a symbolx{\displaystyle x}by an approximated symbolx^{\displaystyle {\hat {x}}}. Typical distortion functions are the Hamming distortion and the Squared-error distortion.
The functions that relate the rate and distortion are found as the solution of the following minimization problem:
HereQY∣X(y∣x){\displaystyle Q_{Y\mid X}(y\mid x)}, sometimes called a test channel, is theconditionalprobability density function(PDF) of the communication channel output (compressed signal)Y{\displaystyle Y}for a given input (original signal)X{\displaystyle X}, andIQ(Y;X){\displaystyle I_{Q}(Y;X)}is themutual informationbetweenY{\displaystyle Y}andX{\displaystyle X}defined as
whereH(Y){\displaystyle H(Y)}andH(Y∣X){\displaystyle H(Y\mid X)}are the entropy of the output signalYand theconditional entropyof the output signal given the input signal, respectively:
The problem can also be formulated as a distortion–rate function, where we find theinfimumover achievable distortions for given rate constraint. The relevant expression is:
The two formulations lead to functions which are inverses of each other.
The mutual information can be understood as a measure for 'prior' uncertainty the receiver has about the sender's signal (H(Y)), diminished by the uncertainty that is left after receiving information about the sender's signal (H(Y∣X){\displaystyle H(Y\mid X)}). Of course the decrease in uncertainty is due to the communicated amount of information, which isI(Y;X){\displaystyle I\left(Y;X\right)}.
As an example, in case there isnocommunication at all, thenH(Y∣X)=H(Y){\displaystyle H(Y\mid X)=H(Y)}andI(Y;X)=0{\displaystyle I(Y;X)=0}. Alternatively, if the communication channel is perfect and the received signalY{\displaystyle Y}is identical to the signalX{\displaystyle X}at the sender, thenH(Y∣X)=0{\displaystyle H(Y\mid X)=0}andI(Y;X)=H(X)=H(Y){\displaystyle I(Y;X)=H(X)=H(Y)}.
In the definition of the rate–distortion function,DQ{\displaystyle D_{Q}}andD∗{\displaystyle D^{*}}are the distortion betweenX{\displaystyle X}andY{\displaystyle Y}for a givenQY∣X(y∣x){\displaystyle Q_{Y\mid X}(y\mid x)}and the prescribed maximum distortion, respectively. When we use themean squared erroras distortion measure, we have (foramplitude-continuous signals):
As the above equations show, calculating a rate–distortion function requires the stochastic description of the inputX{\displaystyle X}in terms of the PDFPX(x){\displaystyle P_{X}(x)}, and then aims at finding the conditional PDFQY∣X(y∣x){\displaystyle Q_{Y\mid X}(y\mid x)}that minimize rate for a given distortionD∗{\displaystyle D^{*}}. These definitions can be formulated measure-theoretically to account for discrete and mixed random variables as well.
Ananalyticalsolution to thisminimization problemis often difficult to obtain except in some instances for which we next offer two of the best known examples. The rate–distortion function of any source is known to obey several fundamental properties, the most important ones being that it is acontinuous,monotonically decreasingconvex(U)functionand thus the shape for the function in the examples is typical (even measured rate–distortion functions in real life tend to have very similar forms).
Although analytical solutions to this problem are scarce, there are upper and lower bounds to these functions including the famousShannon lower bound(SLB), which in the case of squared error and memoryless sources, states that for arbitrary sources with finite differential entropy,
whereh(D) is the differential entropy of a Gaussian random variable with variance D. This lower bound is extensible to sources with memory and other distortion measures. One important feature of the SLB is that it is asymptotically tight in the low distortion regime for a wide class of sources and in some occasions, it actually coincides with the rate–distortion function. Shannon Lower Bounds can generally be found if the distortion between any two numbers can be expressed as a function of the difference between the value of these two numbers.
TheBlahut–Arimoto algorithm, co-invented byRichard Blahut, is an elegant iterative technique for numerically obtaining rate–distortion functions of arbitrary finite input/output alphabet sources and much work has been done to extend it to more general problem instances.
The computation of the rate-distortion function requires knowledge of the underlying distribution, which is often unavailable in contemporary applications in data-science and machine learning. However, this challenge can be addressed using deep learning-based estimators of the rate-distortion function.[2]These estimators are typically referred to as 'neural estimators', involving the optimization of a parametrized variational form of the rate distortion objective.
When working with stationary sources with memory, it is necessary to modify the definition of the rate distortion function and it must be understood in the sense of a limit taken over sequences of increasing lengths.
where
and
where superscripts denote a complete sequence up to that time and the subscript 0 indicates initial state.
If we assume thatX{\displaystyle X}is aGaussianrandom variable withvarianceσ2{\displaystyle \sigma ^{2}}, and if we assume that successive samples of the signalX{\displaystyle X}arestochastically independent(or equivalently, the source ismemoryless, or the signal isuncorrelated), we find the followinganalytical expressionfor the rate–distortion function:
The following figure shows what this function looks like:
Rate–distortion theory tell us that 'no compression system exists that performs outside the gray area'. The closer a practical compression system is to the red (lower) bound, the better it performs. As a general rule, this bound can only be attained by increasing the coding block length parameter. Nevertheless, even at unit blocklengths one can often find good (scalar)quantizersthat operate at distances from the rate–distortion function that are practically relevant.[4]
This rate–distortion function holds only for Gaussian memoryless sources. It is known that the Gaussian source is the most "difficult" source to encode: for a given mean square error, it requires the greatest number of bits. The performance of a practical compression system working on—say—images, may well be below theR(D){\displaystyle R\left(D\right)}lower bound shown.
The rate-distortion function of aBernoulli random variablewith Hamming distortion is given by:
whereHb{\displaystyle H_{b}}denotes thebinary entropy function.
Plot of the rate-distortion function forp=0.5{\displaystyle p=0.5}:
Suppose we want to transmit information about a source to the user with a distortion not exceedingD. Rate–distortion theory tells us that at leastR(D){\displaystyle R(D)}bits/symbol of information from the source must reach the user. We also know from Shannon's channel coding theorem that if the source entropy isHbits/symbol, and thechannel capacityisC(whereC<H{\displaystyle C<H}), thenH−C{\displaystyle H-C}bits/symbol will be lost when transmitting this information over the given channel. For the user to have any hope of reconstructing with a maximum distortionD, we must impose the requirement that the information lost in transmission does not exceed the maximum tolerable loss ofH−R(D){\displaystyle H-R(D)}bits/symbol. This means that the channel capacity must be at least as large asR(D){\displaystyle R(D)}.[5]
|
https://en.wikipedia.org/wiki/Rate%E2%80%93distortion_theory
|
Ininformation theory,Shannon's source coding theorem(ornoiseless coding theorem) establishes the statistical limits to possibledata compressionfor data whose source is anindependent identically-distributed random variable, and the operational meaning of theShannon entropy.
Named afterClaude Shannon, thesource coding theoremshows that, in the limit, as the length of a stream ofindependent and identically-distributed random variable (i.i.d.)data tends to infinity, it is impossible to compress such data such that the code rate (average number of bits per symbol) is less than the Shannon entropy of the source, without it being virtually certain that information will be lost. However it is possible to get the code rate arbitrarily close to the Shannon entropy, with negligible probability of loss.
Thesource coding theorem for symbol codesplaces an upper and a lower bound on the minimal possible expected length of codewords as a function of theentropyof the input word (which is viewed as arandom variable) and of the size of the target alphabet.
Note that, for data that exhibits more dependencies (whose source is not an i.i.d. random variable), theKolmogorov complexity, which quantifies the minimal description length of an object, is more suitable to describe the limits of data compression. Shannon entropy takes into account only frequency regularities while Kolmogorov complexity takes into account all algorithmic regularities, so in general the latter is smaller. On the other hand, if an object is generated by a random process in such a way that it has only frequency regularities, entropy is close to complexity with high probability (Shen et al. 2017).[1]
Source codingis a mapping from (a sequence of) symbols from an informationsourceto a sequence of alphabet symbols (usually bits) such that the source symbols can be exactly recovered from the binary bits (lossless source coding) or recovered within some distortion (lossy source coding). This is one approach todata compression.
In information theory, the source coding theorem (Shannon 1948)[2]informally states that (MacKay 2003, pg. 81,[3]Cover 2006, Chapter 5[4]):
Ni.i.d.random variables each with entropyH(X)can be compressed into more thanN H(X)bitswith negligible risk of information loss, asN→ ∞; but conversely, if they are compressed into fewer thanN H(X)bits it is virtually certain that information will be lost.
TheNH(X){\displaystyle NH(X)}coded sequence represents the compressed message in a biunivocal way, under the assumption that the decoder knows the source. From a practical point of view, this hypothesis is not always true. Consequently, when the entropy encoding is applied the transmitted message isNH(X)+(inf.source){\displaystyle NH(X)+(inf.source)}. Usually, the information that characterizes the source is inserted at the beginning of the transmitted message.
LetΣ1, Σ2denote two finite alphabets and letΣ∗1andΣ∗2denote theset of all finite wordsfrom those alphabets (respectively).
Suppose thatXis a random variable taking values inΣ1and letfbe auniquely decodablecode fromΣ∗1toΣ∗2where|Σ2| =a. LetSdenote the random variable given by the length of codewordf(X).
Iffis optimal in the sense that it has the minimal expected word length forX, then (Shannon 1948):
WhereE{\displaystyle \mathbb {E} }denotes theexpected valueoperator.
GivenXis ani.i.d.source, itstime seriesX1, ...,Xnis i.i.d. withentropyH(X)in the discrete-valued case anddifferential entropyin the continuous-valued case. The Source coding theorem states that for anyε> 0, i.e. for anyrateH(X) +εlarger than theentropyof the source, there is large enoughnand an encoder that takesni.i.d. repetition of the source,X1:n, and maps it ton(H(X) +ε)binary bits such that the source symbolsX1:nare recoverable from the binary bits with probability of at least1 −ε.
Proof of Achievability.Fix someε> 0, and let
Thetypical set,Aεn, is defined as follows:
Theasymptotic equipartition property(AEP) shows that for large enoughn, the probability that a sequence generated by the source lies in the typical set,Aεn, as defined approaches one. In particular, for sufficiently largen,P((X1,X2,⋯,Xn)∈Anε){\displaystyle P((X_{1},X_{2},\cdots ,X_{n})\in A_{n}^{\varepsilon })}can be made arbitrarily close to 1, and specifically, greater than1−ε{\displaystyle 1-\varepsilon }(SeeAEPfor a proof).
The definition of typical sets implies that those sequences that lie in the typical set satisfy:
Since|Anε|≤2n(H(X)+ε),n(H(X)+ε){\displaystyle \left|A_{n}^{\varepsilon }\right|\leq 2^{n(H(X)+\varepsilon )},n(H(X)+\varepsilon )}bits are enough to point to any string in this set.
The encoding algorithm: the encoder checks if the input sequence lies within the typical set; if yes, it outputs the index of the input sequence within the typical set; if not, the encoder outputs an arbitraryn(H(X) +ε)digit number. As long as the input sequence lies within the typical set (with probability at least1 −ε), the encoder does not make any error. So, the probability of error of the encoder is bounded above byε.
Proof of converse: the converse is proved by showing that any set of size smaller thanAεn(in the sense of exponent) would cover a set of probability bounded away from1.
For1 ≤i≤nletsidenote the word length of each possiblexi. Defineqi=a−si/C{\displaystyle q_{i}=a^{-s_{i}}/C}, whereCis chosen so thatq1+ ... +qn= 1. Then
where the second line follows fromGibbs' inequalityand the fifth line follows fromKraft's inequality:
sologC≤ 0.
For the second inequality we may set
so that
and so
and
and so by Kraft's inequality there exists a prefix-free code having those word lengths. Thus the minimalSsatisfies
Define typical setAεnas:
Then, for givenδ> 0, fornlarge enough,Pr(Aεn) > 1 −δ. Now we just encode the sequences in the typical set, and usual methods in source coding show that the cardinality of this set is smaller than2n(Hn¯(X)+ε){\displaystyle 2^{n({\overline {H_{n}}}(X)+\varepsilon )}}. Thus, on an average,Hn(X) +εbits suffice for encoding with probability greater than1 −δ, whereεandδcan be made arbitrarily small, by makingnlarger.
|
https://en.wikipedia.org/wiki/Shannon%27s_source_coding_theorem
|
Ininformation theory, theShannon–Hartley theoremtells the maximum rate at which information can be transmitted over a communications channel of a specifiedbandwidthin the presence ofnoise. It is an application of thenoisy-channel coding theoremto the archetypal case of acontinuous-timeanalogcommunications channelsubject toGaussian noise. The theorem establishes Shannon'schannel capacityfor such a communication link, a bound on the maximum amount of error-freeinformationper time unit that can be transmitted with a specifiedbandwidthin the presence of the noise interference, assuming that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density. The law is named afterClaude ShannonandRalph Hartley.
The Shannon–Hartley theorem states thechannel capacityC{\displaystyle C}, meaning the theoreticaltightestupper bound on theinformation rateof data that can be communicated at an arbitrarily lowerror rateusing an average received signal powerS{\displaystyle S}through an analog communication channel subject toadditive white Gaussian noise(AWGN) of powerN{\displaystyle N}:
where
During the late 1920s,Harry NyquistandRalph Hartleydeveloped a handful of fundamental ideas related to the transmission of information, particularly in the context of thetelegraphas a communications system. At the time, these concepts were powerful breakthroughs individually, but they were not part of a comprehensive theory. In the 1940s,Claude Shannondeveloped the concept of channel capacity, based in part on the ideas of Nyquist and Hartley, and then formulated a complete theory of information and its transmission.
In 1927, Nyquist determined that the number of independent pulses that could be put through a telegraph channel per unit time is limited to twice the one-sidedbandwidthof the channel. In symbolic notation,
wherefp{\displaystyle f_{p}}is the pulse frequency (in pulses per second) andB{\displaystyle B}is the one-sided bandwidth[clarification needed](in hertz). The quantity2B{\displaystyle 2B}later came to be called theNyquist rate, and transmitting at the limiting pulse rate of2B{\displaystyle 2B}pulses per second assignalling at the Nyquist rate. Nyquist published his results in 1928 as part of his paper "Certain topics in Telegraph Transmission Theory".[1]
During 1928, Hartley formulated a way to quantify information and itsline rate(also known asdata signalling rateRbits per second).[2]This method, later known as Hartley's law, became an important precursor for Shannon's more sophisticated notion of channel capacity.
Hartley argued that the maximum number of distinguishable pulse levels that can be transmitted and received reliably over a communications channel is limited by the dynamic range of the signal amplitude and the precision with which the receiver can distinguish amplitude levels. Specifically, if the amplitude of the transmitted signal is restricted to the range of [−A... +A] volts, and the precision of the receiver is ±ΔVvolts, then the maximum number of distinct pulsesMis given by
By taking information per pulse in bit/pulse to be the base-2-logarithmof the number of distinct messagesMthat could be sent, Hartley[3]constructed a measure of the line rateRas:
wherefp{\displaystyle f_{p}}is the pulse rate, also known as the symbol rate, in symbols/second orbaud.
Hartley then combined the above quantification with Nyquist's observation that the number of independent pulses that could be put through a channel of one-sided bandwidthB{\displaystyle B}hertzwas2B{\displaystyle 2B}pulses per second, to arrive at his quantitative measure for achievable line rate.
Hartley's law is sometimes quoted as just a proportionality between theanalog bandwidth,B{\displaystyle B}, in Hertz and what today is called thedigital bandwidth,R{\displaystyle R}, in bit/s.[4]Other times it is quoted in this more quantitative form, as an achievable line rate ofR{\displaystyle R}bits per second:[5]
Hartley did not work out exactly how the numberMshould depend on the noise statistics of the channel, or how the communication could be made reliable even when individual symbol pulses could not be reliably distinguished toMlevels; with Gaussian noise statistics, system designers had to choose a very conservative value ofM{\displaystyle M}to achieve a low error rate.
The concept of an error-free capacity awaited Claude Shannon, who built on Hartley's observations about a logarithmic measure of information and Nyquist's observations about the effect of bandwidth limitations.
Hartley's rate result can be viewed as the capacity of an errorlessM-ary channel of2B{\displaystyle 2B}symbols per second. Some authors refer to it as a capacity. But such an errorless channel is an idealization, and if M is chosen small enough to make the noisy channel nearly errorless, the result is necessarily less than the Shannon capacity of the noisy channel of bandwidthB{\displaystyle B}, which is the Hartley–Shannon result that followed later.
Claude Shannon's development ofinformation theoryduring World War II provided the next big step in understanding how much information could be reliably communicated through noisy channels. Building on Hartley's foundation, Shannon'snoisy channel coding theorem(1948) describes the maximum possible efficiency oferror-correcting methodsversus levels of noise interference and data corruption.[6][7]The proof of the theorem shows that a randomly constructed error-correcting code is essentially as good as the best possible code; the theorem is proved through the statistics of such random codes.
Shannon's theorem shows how to compute achannel capacityfrom a statistical description of a channel, and establishes that given a noisy channel with capacityC{\displaystyle C}and information transmitted at a line rateR{\displaystyle R}, then if
there exists a coding technique which allows the probability of error at the receiver to be made arbitrarily small. This means that theoretically, it is possible to transmit information nearly without error up to nearly a limit ofC{\displaystyle C}bits per second.
The converse is also important. If
the probability of error at the receiver increases without bound as the rate is increased, so no useful information can be transmitted beyond the channel capacity. The theorem does not address the rare situation in which rate and capacity are equal.
The Shannon–Hartley theorem establishes what that channel capacity is for a finite-bandwidthcontinuous-timechannel subject to Gaussian noise. It connects Hartley's result with Shannon's channel capacity theorem in a form that is equivalent to specifying theMin Hartley's line rate formula in terms of a signal-to-noise ratio, but achieving reliability through error-correction coding rather than through reliably distinguishable pulse levels.
If there were such a thing as a noise-free analog channel, one could transmit unlimited amounts of error-free data over it per unit of time (Note that an infinite-bandwidth analog channel could not transmit unlimited amounts of error-free data absent infinite signal power). Real channels, however, are subject to limitations imposed by both finite bandwidth and nonzero noise.
Bandwidth and noise affect the rate at which information can be transmitted over an analog channel. Bandwidth limitations alone do not impose a cap on the maximum information rate because it is still possible for the signal to take on an indefinitely large number of different voltage levels on each symbol pulse, with each slightly different level being assigned a different meaning or bit sequence. Taking into account both noise and bandwidth limitations, however, there is a limit to the amount of information that can be transferred by a signal of a bounded power, even when sophisticated multi-level encoding techniques are used.
In the channel considered by the Shannon–Hartley theorem, noise and signal are combined by addition. That is, the receiver measures a signal that is equal to the sum of the signal encoding the desired information and a continuous random variable that represents the noise. This addition creates uncertainty as to the original signal's value. If the receiver has some information about the random process that generates the noise, one can in principle recover the information in the original signal by considering all possible states of the noise process. In the case of the Shannon–Hartley theorem, the noise is assumed to be generated by a Gaussian process with a known variance. Since the variance of a Gaussian process is equivalent to its power, it is conventional to call this variance the noise power.
Such a channel is called the Additive White Gaussian Noise channel, because Gaussian noise is added to the signal; "white" means equal amounts of noise at all frequencies within the channel bandwidth. Such noise can arise both from random sources of energy and also from coding and measurement error at the sender and receiver respectively. Since sums of independent Gaussian random variables are themselves Gaussian random variables, this conveniently simplifies analysis, if one assumes that such error sources are also Gaussian and independent.
Comparing the channel capacity to the information rate from Hartley's law, we can find the effective number of distinguishable levelsM:[8]
The square root effectively converts the power ratio back to a voltage ratio, so the number of levels is approximately proportional to the ratio of signalRMS amplitudeto noise standard deviation.
This similarity in form between Shannon's capacity and Hartley's law should not be interpreted to mean thatM{\displaystyle M}pulse levels can be literally sent without any confusion. More levels are needed to allow for redundant coding and error correction, but the net data rate that can be approached with coding is equivalent to using thatM{\displaystyle M}in Hartley's law.
In the simple version above, the signal and noise are fully uncorrelated, in which caseS+N{\displaystyle S+N}is the total power of the received signal and noise together. A generalization of the above equation for the case where the additive noise is not white (or that theS/N{\displaystyle S/N}is not constant with frequency over the bandwidth) is obtained by treating the channel as many narrow, independent Gaussian channels in parallel:
where
Note: the theorem only applies to Gaussianstationary processnoise. This formula's way of introducing frequency-dependent noise cannot describe all continuous-time noise processes. For example, consider a noise process consisting of adding a random wave whose amplitude is 1 or −1 at any point in time, and a channel that adds such a wave to the source signal. Such a wave's frequency components are highly dependent. Though such a noise may have a high power, it is fairly easy to transmit a continuous signal with much less power than one would need if the underlying noise was a sum of independent noises in each frequency band.
For large or small and constant signal-to-noise ratios, the capacity formula can be approximated:
When the SNR is large (S/N≫ 1), the logarithm is approximated by
in which case the capacity is logarithmic in power and approximately linear in bandwidth (not quite linear, sinceNincreases with bandwidth, imparting a logarithmic effect). This is called thebandwidth-limited regime.
where
Similarly, when the SNR is small (ifS/N≪1{\displaystyle S/N\ll 1}), applying the approximation to the logarithm:
then the capacity is linear in power. This is called thepower-limited regime.
In this low-SNR approximation, capacity is independent of bandwidth if the noise is white, ofspectral densityN0{\displaystyle N_{0}}watts per hertz, in which case the total noise power isN=B⋅N0{\displaystyle N=B\cdot N_{0}}.
|
https://en.wikipedia.org/wiki/Shannon%E2%80%93Hartley_theorem
|
Ininformation theory,turbo codesare a class of high-performanceforward error correction(FEC) codes developed around 1990–91, but first published in 1993. They were the first practical codes to closely approach the maximum channel capacity orShannon limit, a theoretical maximum for thecode rateat which reliable communication is still possible given a specific noise level. Turbo codes are used in3G/4Gmobile communications (e.g., inUMTSandLTE) and in (deep space)satellitecommunicationsas well as other applications where designers seek to achieve reliable information transfer over bandwidth- or latency-constrained communication links in the presence of data-corrupting noise. Turbo codes compete withlow-density parity-check(LDPC) codes, which provide similar performance. Until the patent for turbo codes expired,[1]the patent-free status of LDPC codes was an important factor in LDPC's continued relevance.[2]
The name "turbo code" arose from the feedback loop used during normal turbo code decoding, which was analogized to the exhaust feedback used for engineturbocharging.Hagenauerhas argued the term turbo code is a misnomer since there is no feedback involved in the encoding process.[3]
The fundamental patent application for turbo codes was filed on 23 April 1991. The patent application listsClaude Berrouas the sole inventor of turbo codes. The patent filing resulted in several patents includingUS Patent 5,446,747, which expired 29 August 2013.
The first public paper on turbo codes was "Near Shannon Limit Error-correcting Coding and Decoding: Turbo-codes".[4]This paper was published 1993 in the Proceedings of IEEE International Communications Conference. The 1993 paper was formed from three separate submissions that were combined due to space constraints. The merger caused the paper to list three authors: Berrou,Glavieux, andThitimajshima(from Télécom Bretagne, formerENST Bretagne, France). However, it is clear from the original patent filing that Berrou is the sole inventor of turbo codes and that the other authors of the paper contributed material other than the core concepts.[improper synthesis]
Turbo codes were so revolutionary at the time of their introduction that many experts in the field of coding did not believe the reported results. When the performance was confirmed a small revolution in the world of coding took place that led to the investigation of many other types of iterative signal processing.[5]
The first class of turbo code was the parallel concatenated convolutional code (PCCC). Since the introduction of the original parallel turbo codes in 1993, many other classes of turbo code have been discovered, includingserial concatenated convolutional codesandrepeat-accumulate codes. Iterative turbo decoding methods have also been applied to more conventional FEC systems, including Reed–Solomon corrected convolutional codes, although these systems are too complex for practical implementations of iterative decoders. Turbo equalization also flowed from the concept of turbo coding.
In addition to turbo codes, Berrou also invented recursive systematic convolutional (RSC) codes, which are used in the example implementation of turbo codes described in the patent. Turbo codes that use RSC codes seem to perform better than turbo codes that do not use RSC codes.
Prior to turbo codes, the best constructions were serialconcatenated codesbased on an outerReed–Solomon error correctioncode combined with an innerViterbi-decodedshort constraint lengthconvolutional code, also known as RSV codes.
In a later paper, Berrou gave credit to the intuition of "G. Battail,J. Hagenauerand P. Hoeher, who, in the late 80s, highlighted the interest of probabilistic processing." He adds "R. Gallagerand M. Tanner had already imagined coding and decoding techniques whose general principles are closely related," although the necessary calculations were impractical at that time.[6]
There are many different instances of turbo codes, using different component encoders, input/output ratios, interleavers, andpuncturing patterns. This example encoder implementation describes a classic turbo encoder, and demonstrates the general design of parallel turbo codes.
This encoder implementation sends three sub-blocks of bits. The first sub-block is them-bit block of payload data. The second sub-block isn/2parity bits for the payload data, computed using a recursive systematicconvolutional code(RSC code). The third sub-block isn/2parity bits for a knownpermutationof the payload data, again computed using an RSC code. Thus, two redundant but different sub-blocks of parity bits are sent with the payload. The complete block hasm+nbits of data with a code rate ofm/(m+n). Thepermutationof the payload data is carried out by a device called aninterleaver.
Hardware-wise, this turbo code encoder consists of two identical RSC coders,C1andC2, as depicted in the figure, which are connected to each other using a concatenation scheme, calledparallel concatenation:
In the figure,Mis a memory register. The delay line and interleaver force input bits dkto appear in different sequences.
At first iteration, the input sequencedkappears at both outputs of the encoder,xkandy1kory2kdue to the encoder's systematic nature. If the encodersC1andC2are used inn1andn2iterations, their rates are respectively equal to
The decoder is built in a similar way to the above encoder. Two elementary decoders are interconnected to each other, but in series, not in parallel. TheDEC1{\displaystyle \textstyle DEC_{1}}decoder operates on lower speed (i.e.,R1{\displaystyle \textstyle R_{1}}), thus, it is intended for theC1{\displaystyle \textstyle C_{1}}encoder, andDEC2{\displaystyle \textstyle DEC_{2}}is forC2{\displaystyle \textstyle C_{2}}correspondingly.DEC1{\displaystyle \textstyle DEC_{1}}yields asoft decisionwhich causesL1{\displaystyle \textstyle L_{1}}delay. The same delay is caused by the delay line in the encoder. TheDEC2{\displaystyle \textstyle DEC_{2}}'s operation causesL2{\displaystyle \textstyle L_{2}}delay.
An interleaver installed between the two decoders is used here to scatter error bursts coming fromDEC1{\displaystyle \textstyle DEC_{1}}output.DIblock is a demultiplexing and insertion module. It works as a switch, redirecting input bits toDEC1{\displaystyle \textstyle DEC_{1}}at one moment and toDEC2{\displaystyle \textstyle DEC_{2}}at another. In OFF state, it feeds bothy1k{\displaystyle \textstyle y_{1k}}andy2k{\displaystyle \textstyle y_{2k}}inputs with padding bits (zeros).
Consider a memorylessAWGNchannel, and assume that atk-th iteration, the decoder receives a pair of random variables:
whereak{\displaystyle \textstyle a_{k}}andbk{\displaystyle \textstyle b_{k}}are independent noise components having the same varianceσ2{\displaystyle \textstyle \sigma ^{2}}.Yk{\displaystyle \textstyle Y_{k}}is ak-th bit fromyk{\displaystyle \textstyle y_{k}}encoder output.
Redundant information is demultiplexed and sent throughDItoDEC1{\displaystyle \textstyle DEC_{1}}(whenyk=y1k{\displaystyle \textstyle y_{k}=y_{1k}}) and toDEC2{\displaystyle \textstyle DEC_{2}}(whenyk=y2k{\displaystyle \textstyle y_{k}=y_{2k}}).
DEC1{\displaystyle \textstyle DEC_{1}}yields a soft decision; i.e.:
and delivers it toDEC2{\displaystyle \textstyle DEC_{2}}.Λ(dk){\displaystyle \textstyle \Lambda (d_{k})}is called thelogarithm of the likelihood ratio(LLR).p(dk=i),i∈{0,1}{\displaystyle \textstyle p(d_{k}=i),\,i\in \{0,1\}}is thea posteriori probability(APP) of thedk{\displaystyle \textstyle d_{k}}data bit which shows the probability of interpreting a receiveddk{\displaystyle \textstyle d_{k}}bit asi{\displaystyle \textstyle i}. Taking theLLRinto account,DEC2{\displaystyle \textstyle DEC_{2}}yields a hard decision; i.e., a decoded bit.
It is known that theViterbi algorithmis unable to calculate APP, thus it cannot be used inDEC1{\displaystyle \textstyle DEC_{1}}. Instead of that, a modifiedBCJR algorithmis used. ForDEC2{\displaystyle \textstyle DEC_{2}}, theViterbi algorithmis an appropriate one.
However, the depicted structure is not an optimal one, becauseDEC1{\displaystyle \textstyle DEC_{1}}uses only a proper fraction of the available redundant information. In order to improve the structure, a feedback loop is used (see the dotted line on the figure).
The decoder front-end produces an integer for each bit in the data stream. This integer is a measure of how likely it is that the bit is a 0 or 1 and is also calledsoft bit. The integer could be drawn from the range [−127, 127], where:
This introduces a probabilistic aspect to the data-stream from the front end, but it conveys more information about each bit than just 0 or 1.
For example, for each bit, the front end of a traditional wireless-receiver has to decide if an internal analog voltage is above or below a given threshold voltage level. For a turbo code decoder, the front end would provide an integer measure of how far the internal voltage is from the given threshold.
To decode them+n-bit block of data, the decoder front-end creates a block of likelihood measures, with one likelihood measure for each bit in the data stream. There are two parallel decoders, one for each of then⁄2-bit parity sub-blocks. Both decoders use the sub-block ofmlikelihoods for the payload data. The decoder working on the second parity sub-block knows the permutation that the coder used for this sub-block.
The key innovation of turbo codes is how they use the likelihood data to reconcile differences between the two decoders. Each of the two convolutional decoders generates a hypothesis (with derived likelihoods) for the pattern ofmbits in the payload sub-block. The hypothesis bit-patterns are compared, and if they differ, the decoders exchange the derived likelihoods they have for each bit in the hypotheses. Each decoder incorporates the derived likelihood estimates from the other decoder to generate a new hypothesis for the bits in the payload. Then they compare these new hypotheses. This iterative process continues until the two decoders come up with the same hypothesis for them-bit pattern of the payload, typically in 15 to 18 cycles.
An analogy can be drawn between this process and that of solving cross-reference puzzles likecrosswordorsudoku. Consider a partially completed, possibly garbled crossword puzzle. Two puzzle solvers (decoders) are trying to solve it: one possessing only the "down" clues (parity bits), and the other possessing only the "across" clues. To start, both solvers guess the answers (hypotheses) to their own clues, noting down how confident they are in each letter (payload bit). Then, they compare notes, by exchanging answers and confidence ratings with each other, noticing where and how they differ. Based on this new knowledge, they both come up with updated answers and confidence ratings, repeating the whole process until they converge to the same solution.
Turbo codes perform well due to the attractive combination of the code's random appearance on the channel together with the physically realisable decoding structure. Turbo codes are affected by anerror floor.
Telecommunications:
From anartificial intelligenceviewpoint, turbo codes can be considered as an instance of loopybelief propagationinBayesian networks.[8]
|
https://en.wikipedia.org/wiki/Turbo_code
|
The following is a list ofmobile telecommunicationsnetworks usingthird-generationUniversal Mobile Telecommunications System(UMTS) technology. This list does not aim to cover all networks, but instead focuses on networks deployed on frequencies other than 2100 MHz which is commonly deployed around the globe and on Multiband deployments.
Networks in Europe, the Middle East and Africa are exclusively deployed on 2100 MHz (Band 1) and/or 900 MHz (Band 8).
Networks in this region are commonly deployed on 850 MHz (Band 5) and/or 1900 MHz (Band 2) unless denoted otherwise.
Networks in Asia are commonly deployed on 2100 MHz (Band 1) unless denoted otherwise.
|
https://en.wikipedia.org/wiki/List_of_UMTS_networks
|
CDMA2000++(also known asC2KorIMT Multi‑Carrier(IMT‑MC)) is a family of3G[1]mobile technology standards for sending voice, data, andsignalingdata betweenmobile phonesandcell sites. It is developed by3GPP2as a backwards-compatible successor tosecond-generationcdmaOne(IS-95) set of standards and used especially in North America and South Korea.
CDMA2000 compares toUMTS, a competing set of3Gstandards, which is developed by3GPPand used in Europe, Japan, China, and Singapore.
The name CDMA2000 denotes a family of standards that represent the successive, evolutionary stages of the underlying technology. These are:
All are approved radio interfaces for theITU'sIMT-2000. In the United States,CDMA2000is a registered trademark of theTelecommunications Industry Association(TIA-USA).[2]
CDMA2000 1X (IS-2000), also known as1xand1xRTT, is the core CDMA2000 wireless air interface standard. The designation "1x", meaning1 times radio transmission technology, indicates the sameradio frequency(RF) bandwidth asIS-95: aduplexpair of 1.25 MHz radio channels. 1xRTT almost doubles the capacity of IS-95 by adding 64 more traffic channels to theforward link,orthogonalto (inquadraturewith) the original set of 64. The 1X standard supports packet data speeds of up to 153kbit/swith real world data transmission averaging 80–100 kbit/s in most commercial applications.[3]IMT-2000 also made changes to thedata link layerfor greater use of data services, including medium and link access control protocols andquality of service(QoS). The IS-95 data link layer only providedbest-effort deliveryfor data and circuit switched channel for voice (i.e., a voice frame once every 20 ms).
CDMA2000 1xEV-DO (Evolution-Data Optimized), often abbreviated asEV-DOorEV, is atelecommunicationsstandard for thewirelesstransmission of data throughradiosignals, typically forbroadband Internet access. It usesmultiplexingtechniques includingcode-division multiple access(CDMA) as well astime-division multiple accessto maximize both individual user's throughput and the overall system throughput. It is standardized (IS-856) by3rd Generation Partnership Project 2(3GPP2) as part of the CDMA2000 family of standards and has been adopted by manymobile phoneservice providers around the world – particularly those previously employing CDMA networks.
1X Advanced (Rev.E)[4][5]is the evolution of CDMA2000 1X. It provides up to four times the capacity and 70% more coverage compared to 1X.[6]
The CDMA Development Group states that, as of April 2014, there are 314operatorsin 118 countries offering CDMA2000 1X and/or 1xEV-DO service.[7][needs update]
CDMA2000 technology was developed byQualcommin the late 1990s as an enhancement to the CDMA standard.
The intended4Gsuccessor to CDMA2000 wasUMB (Ultra Mobile Broadband); however, in November 2008,Qualcommannounced it was ending development of the technology, favoringLTEinstead.[8]
In 2007, Qualcomm provided a global patent license for CDMA2000 to the Chinese company Teleepoch.[9]
|
https://en.wikipedia.org/wiki/CDMA2000
|
Wi-Fi calling, also calledVoWiFi,[1]refers tomobile phonevoice calls and data that are made overIPnetworks usingWi-Fi, instead of thecell towersprovided bycellular networks.[2]Using this feature, compatible handsets are able to route regular cellular calls through a wireless LAN (Wi-Fi) network withbroadband Internet, while seamlessly changing connections between the two where necessary.[3]This feature makes use of theGeneric Access Network(GAN) protocol, also known asUnlicensed Mobile Access(UMA).[4][5]
Voice over wireless LAN(VoWLAN), alsovoice over Wi‑Fi(VoWiFi[6]), is the use of awirelessbroadband network according to theIEEE 802.11standards for the purpose of vocal conversation. In essence, it isvoice over IP(VoIP) over aWi-Finetwork.
Essentially, GAN/UMA allows cell phone packets to be forwarded to a network access point over the internet, rather than over-the-air usingGSM/GPRS,UMTSor similar. A separate device known as a "GAN Controller" (GANC)[5]receives this data from the Internet and feeds it into the phone network as if it were coming from an antenna on a tower. Calls can be placed from or received to the handset as if it were connected over-the-air directly to the GANC'spoint of presence, making the call invisible to the network as a whole.[7]This can be useful in locations with poor cell coverage where some other form ofinternet accessis available,[2]especially at the home or office. The system offers seamlesshandoff, so the user can move from cell to Wi-Fi and back again with the same invisibility that the cell network offers when moving from tower to tower.[3]
Since the GAN system works over the internet, a UMA-capable handset can connect to its service provider from any location with internet access. This is particularly useful for travelers, who can connect to their provider's GANC and make calls into their home service area from anywhere in the world.[citation needed]This is subject to the quality of the internet connection, however, and may not work well over limited bandwidth or long-latency connection. To improvequality of service(QoS) in the home or office, some providers also supply a specially programmedwireless access pointthat prioritizes UMA packets.[8]Another benefit of Wi-Fi calling is that mobile calls can be made through the internet using the same native calling client; it does not require third-partyVoice over IP(VoIP) closed services likeWhatsApporSkype, relying instead on the mobile cellular operator.[9]
The GAN protocol that extends mobile voice, data and multimedia (IP Multimedia Subsystem/Session Initiation Protocol(IMS/SIP)) applications over IP networks. The latest generation system is named orVoWiFiby a number of handset manufacturers, includingAppleandSamsung, a move that is being mirrored by carriers likeT-Mobile USandVodafone.[citation needed]The service is dependent on IMS, IPsec,IWLANandePDG.
The original Release 6 GAN specification supported a 2G (A/Gb) connection from the GANC into the mobile core network (MSC/GSN). Today[when?]all commercial GAN dual-mode handset deployments are based on a 2G connection and all GAN enabled devices are dual-mode 2G/Wi-Fi. The specification, though, defined support for multimode handset operation. Therefore, 3G/2G/Wi-Fi handsets are supported in the standard. The first 3G/UMA devices were announced in the second half of 2008.
A typical UMA/GAN handset will have four modes of operation:
In all cases, the handset scans for GSM cells when it first turns on, to determine its location area. This allows the carrier to route the call to the nearest GANC, set the correct rate plan, and comply with existing roaming agreements.
At the end of 2007, the GAN specification was enhanced to support 3G (Iu) interfaces from the GANC to the mobile core network (MSC/GSN). This native 3G interface can be used for dual-mode handset as well as 3Gfemtocellservice delivery. The GAN release 8 documentation describes these new capabilities.
While UMA is nearly always associated with dual-mode GSM/Wi-Fi services, it is actually a ‘generic’ access network technology that provides a generic method for extending the services and applications in an operator's mobile core (voice, data, IMS) over IP and the public Internet.
GAN defines a secure, managed connection from the mobile core (GANC) to different devices/access points over IP.
A Wi-Fi network that supports voice telephony must be carefully designed in a way that maximizes performance and is able to support the applicable call density.[12]A voice network includes call gateways in addition to the Wi-Fi access points. The gateways provide call handling among wireless IP phones and connections to traditional telephone systems. The Wi-Fi network supporting voice applications must provide much stronger signal coverage than what's needed for most data-only applications. In addition, the Wi-Fi network must provide seamless roaming between access points.
UMA was developed by a group of operator and vendor companies.[13]The initial specifications were published on 2 September 2004. The companies then contributed the specifications to the3rd Generation Partnership Project(3GPP) as part of 3GPP work item "Generic Access to A/Gb interfaces". On 8 April 2005, 3GPP approved specifications for Generic Access to A/Gb interfaces for 3GPP Release 6 and renamed the system to GAN.[14][15]But the termGANis little known outside the 3GPP community, and the termUMAis more common in marketing.[citation needed]
For carriers:
For subscribers:
The first service launch was BT withBT Fusionin the autumn of 2005. The service is based on pre-3GPP GAN standard technology. Initially, BT Fusion used UMA over Bluetooth with phones fromMotorola. From January 2007, it used UMA over 802.11 with phones from Nokia, Motorola and Samsung[18]and was branded as a "Wi-Fi mobile service". BT has since discontinued the service.
On August 28, 2006,TeliaSonerawas the first to launch an 802.11 based UMA service called "Home Free".[19]The service started in Denmark but is no longer offered.
On September 25, 2006Orangeannounced its "Unik service", also known as Signal Boost in the UK.[20][21]However this service is no longer available to new customers in the UK.[22]The announcement, the largest to date, covers more than 60m of Orange's mobile subscribers in the UK, France, Poland, Spain and the Netherlands.
Cincinnati Bellannounced the first UMA deployment in the United States.[23]The service, originally called CB Home Run, allows users to transfer seamlessly from the Cincinnati Bell cellular network to a home wireless network or to Cincinnati Bell's WiFi HotSpots. It has since been rebranded as Fusion WiFi.
This was followed shortly byT-Mobile USon June 27, 2007.[24]T-Mobile's service, originally named "Hotspot Calling", and rebranded to "Wi-Fi Calling" in 2009, allows users to seamlessly transfer from the T-Mobile cellular network to an 802.11x wireless network or T-Mobile HotSpot in the United States.
In Canada, bothFidoandRogers Wirelesslaunched UMA plans under the names UNO and Rogers Home Calling Zone (later rebranded Talkspot, and subsequently rebranded again as Wi-Fi Calling), respectively, on May 6, 2008.[25]
In Australia, GAN has been implemented by Vodafone, Optus and Telstra.[26]
Since 10 April 2015, Wi-Fi Calling has been available for customers ofEEin the UK initially on theNokia Lumia 640andSamsung Galaxy S6andSamsung Galaxy S6 Edgehandsets.[27]
In March 2016,Vodafone Netherlandslaunched Wi-Fi Calling support along withVoLTE.[28]
Since the Autumn of 2016, Wifi Calling / Voice over Wifi has been available for customers of Telenor Denmark, including the ability to do handover to and from the 4G (VoLTE) network. This is available for several Samsung and Apple handsets.
AT&T[29]andVerizon[30]are going to launch Wi-Fi calling in 2015.
Industry organisationUMA Todaytracks all operator activities and handset development.
In September 2015, South African cellular network Cell C launched WiFi Calling on its South African network.[31]
In November 2024, Belgian cellular network Voo launched WiFi Calling on its Belgian network.[32]
GAN/UMA is not the first system to allow the use of unlicensed spectrum to connect handsets to a GSM network. TheGIP/IWPstandard forDECTprovides similar functionality, but requires a more direct connection to the GSM network from the base station. While dual-mode DECT/GSM phones have appeared, these have generally been functionally cordless phones with a GSM handset built-in (or vice versa, depending on your point of view), rather than phones implementing DECT/GIP, due to the lack of suitable infrastructure to hook DECT base-stations supporting GIP to GSM networks on an ad-hoc basis.[33]
GAN/UMA's ability to use the Internet to provide the "last mile" connection to the GSM network solves the major issue that DECT/GIP has faced. Had GIP emerged as a practical standard, the low power usage of DECT technology when idle would have been an advantage compared to GAN.[citation needed]
There is nothing preventing an operator from deploying micro- and pico-cells that use towers that connect with the home network over the Internet. Several companies have developed femtocell systems that do precisely that, broadcasting a "real" GSM or UMTS signal, bypassing the need for special handsets that require 802.11 technology. In theory, such systems are more universal, and again require lower power than 802.11, but their legality will vary depending on the jurisdiction, and will require the cooperation of the operator. Further, users may be charged at higher cell phone rates, even though they are paying for the DSL or other network that ultimately carries their traffic; in contrast, GAN/UMA providers charge reduced rates when making calls off the providers cellular phone network.[citation needed]
|
https://en.wikipedia.org/wiki/Generic_Access_Network
|
Opportunity-Driven Multiple Access (ODMA)is aUMTScommunications relaying protocol standard first introduced by the European Telecommunication Standards Institute (ETSI) in 1996. ODMA has been adopted by the 3rd-Generation Partnership Project,3GPPto improve the efficiency of UMTS networks using theTDDmode. One of the objectives of ODMA is to enhance the capacity and the coverage of radio transmissions towards the boundaries of the cell. While mobile stations under the cell coverage area can communicate directly with the base station, mobile stations outside the cell boundary can still access the network and communicating with the base station viamultihoptransmission. Mobile stations with high data rate inside the cell are used as multihop relays.
The initial concept of Opportunity Driven Multiple Access (ODMA) was conceived and patented in South Africa by David Larsen and James Larsen of SRD Pty Ltd in 1978[1]
The ODMA standard was tabled by the 3GPP committee in 1999 due to complexity issues. The technology continues to be developed and enhanced by IWICS who holds the key patents describing the methods employed in ODMA to effect opportunity driven communications.
With the explosion of cellular phone use and Internet multi-media services, wireless networks are becoming increasingly congested. The increased demand has raised our expectations, while creating capacity problems and a need for greater bandwidth. However, if the transmitted power of wireless units is significantly reduced, then there is a potential solution. This implies a signal-to-noise ratio improvement: the ratio is affected by numerous parameters, including radio frequency and path. Opportunity Driven Multiple Access (ODMA) continually determines optimal points along that path to support each transmission.
Adaptation
ODMA uses many adaptation techniques to optimize communications, but one of the most powerful is path diversity. From origin to destination, ODMA stations relay the transmissions in an intelligent and efficient manner.
The available optimal paths will increase as subscribers join the network, supporting a fundamental aspect of the ODMA philosophy: Communications are dynamic and local, best controlled at the station level, rather than from some centralized source. Each ODMA-network station is an intelligent burst-mode radio, which can use all the available bandwidth some of the time. However, as with any technology, weather or general network conditions can affect transmissions.
Like cellular networks, the ODMA-network stations operate in the same wide frequency band, but frequency hopping, at lower data rates, introduces sub-bands. Because transmission is packet based and connectionless, stations relay packets from neighbor stations. For each packet, a station optimizes the transmission by adapting the route, power, data rate, packet length, frequency, time window and data quality over a wide range. Each station has responsibility and much autonomy for routing and service-enhancing adaptation to the current environment. For security, stations accept the authority of a network supervisor.
|
https://en.wikipedia.org/wiki/Opportunity-Driven_Multiple_Access
|
Aduplexcommunication systemis apoint-to-pointsystem composed of two or more connected parties or devices that can communicate with one another in both directions. Duplex systems are employed in many communications networks, either to allow for simultaneous communication in both directions between two connected parties or to provide a reverse path for the monitoring and remote adjustment of equipment in the field. There are two types of duplex communication systems: full-duplex (FDX) and half-duplex (HDX).
In afull-duplexsystem, both parties can communicate with each other simultaneously. An example of a full-duplex device isplain old telephone service; the parties at both ends of a call can speak and be heard by the other party simultaneously. The earphone reproduces the speech of the remote party as the microphone transmits the speech of the local party. There is a two-way communication channel between them, or more strictly speaking, there are two communication channels between them.
In ahalf-duplexorsemiduplexsystem, both parties can communicate with each other, but not simultaneously; the communication is one direction at a time. An example of a half-duplex device is awalkie-talkie, atwo-way radiothat has apush-to-talkbutton. When the local user wants to speak to the remote person, they push this button, which turns on the transmitter and turns off the receiver, preventing them from hearing the remote person while talking. To listen to the remote person, they release the button, which turns on the receiver and turns off the transmitter. This terminology is not completely standardized, and some sources define this mode assimplex.[1][2]
Systems that do not need duplex capability may instead usesimplex communication, in which one device transmits and the others can only listen. Examples arebroadcastradio and television,garage door openers,baby monitors,wireless microphones, andsurveillance cameras. In these devices, the communication is only in one direction.
Simplex communicationis acommunication channelthat sends information in one direction only.[3]
TheInternational Telecommunication Uniondefinition is a communications channel that operates in one direction at a time, but that may be reversible; this is termedhalf duplexin other contexts.
For example, in TV and radiobroadcasting, information flows only from the transmitter site to multiple receivers. A pair ofwalkie-talkietwo-way radiosprovide a simplex circuit in the ITU sense; only one party at a time can talk, while the other listens until it can hear an opportunity to transmit. The transmission medium (the radio signal over the air) can carry information in only one direction.
TheWestern Unioncompany used the termsimplexwhen describing the half-duplex and simplex capacity of their newtransatlantic telegraph cablecompleted betweenNewfoundlandand theAzoresin 1928.[4]The same definition for a simplex radio channel was used by theNational Fire Protection Associationin 2002.[5]
Ahalf-duplex(HDX) system provides communication in both directions, but only one direction at a time, not simultaneously in both directions.[6][7][8]This terminology is not completely standardized between defining organizations, and in radio communication some sources classify this mode assimplex.[2][1][9]Typically, once one party begins a transmission, the other party on the channel must wait for the transmission to complete, before replying.[10]
An example of a half-duplex system is a two-party system such as awalkie-talkie, wherein one must say "over" or another previously designated keyword to indicate the end of transmission, to ensure that only one party transmits at a time. A good analogy for a half-duplex system would be a one-lane road that allows two-way traffic, traffic can only flow in one direction at a time.
Half-duplex systems are usually used to conservebandwidth, at the cost of reducing the overall bidirectional throughput, since only a singlecommunication channelis needed and is shared alternately between the two directions. For example, a walkie-talkie or a DECT phone or so-called TDD 4G or 5G phones requires only a singlefrequencyfor bidirectional communication, while acell phonein the so-called FDD mode is a full-duplex device, and generally requires two frequencies to carry the two simultaneous voice channels, one in each direction.
In automatic communications systems such as two-way data-links,time-division multiplexingcan be used for time allocations for communications in a half-duplex system. For example, station A on one end of the data link could be allowed to transmit for exactly one second, then station B on the other end could be allowed to transmit for exactly one second, and then the cycle repeats. In this scheme, the channel is never left idle.
In half-duplex systems, if more than one party transmits at the same time, acollisionoccurs, resulting in lost or distorted messages.
Afull-duplex(FDX) system allows communication in both directions, and, unlike half-duplex, allows this to happen simultaneously.[6][7][8]Land-linetelephonenetworks are full-duplex since they allow both callers to speak and be heard at the same time. Full-duplex operation is achieved on atwo-wire circuitthrough the use of ahybrid coilin atelephone hybrid. Modern cell phones are also full-duplex.[11]
There is a technical distinction between full-duplex communication, which uses a single physical communication channel for both directions simultaneously, anddual-simplexcommunication which uses two distinct channels, one for each direction. From the user perspective, the technical difference does not matter and both variants are commonly referred to asfull duplex.
ManyEthernetconnections achieve full-duplex operation by making simultaneous use of two physicaltwisted pairsinside the same jacket, or two optical fibers which are directly connected to each networked device: one pair or fiber is for receiving packets, while the other is for sending packets. Other Ethernet variants, such as1000BASE-Tuse the same channels in each direction simultaneously. In any case, with full-duplex operation, the cable itself becomes a collision-free environment and doubles the maximum total transmission capacity supported by each Ethernet connection.
Full-duplex has also several benefits over the use of half-duplex. Since there is only one transmitter on each twisted pair there is no contention and no collisions so time is not wasted by having to wait or retransmit frames. Full transmission capacity is available in both directions because the send and receive functions are separate.
Some computer-based systems of the 1960s and 1970s required full-duplex facilities, even for half-duplex operation, since their poll-and-response schemes could not tolerate the slight delays in reversing the direction of transmission in a half-duplex line.[citation needed]
Full-duplex audio systems like telephones can create echo, which is distracting to users and impedes the performance of modems. Echo occurs when the sound originating from the far end comes out of the speaker at the near end and re-enters the microphone[a]there and is then sent back to the far end. The sound then reappears at the original source end but delayed.
Echo cancellationis a signal-processing operation that subtracts the far-end signal from the microphone signal before it is sent back over the network. Echo cancellation is important technology allowingmodemsto achieve good full-duplex performance. TheV.32,V.34,V.56, andV.90modem standards require echo cancellation.[12]Echo cancelers are available as both software and hardware implementations. They can be independent components in a communications system or integrated into the communication system'scentral processing unit.
Wherechannel access methodsare used inpoint-to-multipointnetworks (such ascellular networks) for dividing forward and reverse communication channels on the same physical communications medium, they are known as duplexing methods.[13]
Time-division duplexing(TDD) is the application oftime-division multiplexingto separate outward and return signals. It emulates full-duplex communication over a half-duplex communication link.
Time-division duplexing is flexible in the case where there isasymmetryof theuplinkanddownlinkdata rates or utilization. As the amount of uplink data increases, more communication capacity can be dynamically allocated, and as the traffic load becomes lighter, capacity can be taken away. The same applies in the downlink direction.
Thetransmit/receive transition gap(TTG) is the gap (time) between a downlink burst and the subsequent uplink burst. Similarly, thereceive/transmit transition gap(RTG) is the gap between an uplink burst and the subsequent downlink burst.[14]
Examples of time-division duplexing systems include:
Frequency-division duplexing(FDD) means that thetransmitterandreceiveroperate using differentcarrier frequencies.
The method is frequently used inham radiooperation, where an operator is attempting to use arepeaterstation. The repeater station must be able to send and receive a transmission at the same time and does so by slightly altering the frequency at which it sends and receives. This mode of operation is referred to asduplex modeoroffset mode. Uplink and downlink sub-bands are said to be separated by thefrequency offset.
Frequency-division duplex systems can extend their range by using sets of simple repeater stations because the communications transmitted on any single frequency always travel in the same direction.
Frequency-division duplexing can be efficient in the case of symmetric traffic. In this case, time-division duplexing tends to waste bandwidth during the switch-over from transmitting to receiving, has greater inherentlatency, and may require more complexcircuitry.
Another advantage of frequency-division duplexing is that it makes radio planning easier and more efficient since base stations do notheareach other (as they transmit and receive in different sub-bands) and therefore will normally not interfere with each other. Conversely, with time-division duplexing systems, care must be taken to keep guard times between neighboring base stations (which decreasesspectral efficiency) or to synchronize base stations, so that they will transmit and receive at the same time (which increases network complexity and therefore cost, and reduces bandwidth allocation flexibility as all base stations and sectors will be forced to use the same uplink/downlink ratio).
Examples of frequency-division duplexing systems include:
|
https://en.wikipedia.org/wiki/Time-division_duplex
|
Awireless ad hoc network[1](WANET) ormobile ad hoc network(MANET) is a decentralized type ofwireless network. The network isad hocbecause it does not rely on a pre-existing infrastructure, such asroutersorwireless access points. Instead, eachnodeparticipates in routing byforwardingdata for other nodes. The determination of which nodes forward data is made dynamically on the basis of network connectivity and therouting algorithmin use.[2]
Such wireless networks lack the complexities of infrastructure setup and administration, enabling devices to create and join networks "on the fly".[3]
Each device in a MANET is free to move independently in any direction, and will therefore change its links to other devices frequently. Each must forward traffic unrelated to its own use, and therefore be arouter. The primary challenge in building a MANET is equipping each device to continuously maintain the information required to properly route traffic. This becomes harder as the scale of the MANET increases due to 1) the desire to routepacketsto/through every other node, 2) the percentage of overhead traffic needed to maintain real-time routing status, 3) each node has its owngoodputto route independent and unaware of others needs, and 4) all must share limited communicationbandwidth, such as a slice of radio spectrum.
Such networks may operate by themselves or may be connected to the largerInternet. They may contain one or multiple and differenttransceiversbetween nodes. This results in a highly dynamic, autonomous topology. MANETs usually have a routable networking environment on top of alink layerad hoc network.
The earliest wireless data network was calledPRNET, thepacket radionetwork, and was sponsored byDefense Advanced Research Projects Agency(DARPA) in the early 1970s.Bolt, Beranek and Newman Inc.(BBN) and SRI International designed, built, and experimented with these earliest systems. Experimenters includedRobert Kahn,[4]Jerry Burchfiel, andRay Tomlinson.[5]Similar experiments took place in the amateur radio community with the x25 protocol. These early packet radio systems predated the Internet, and indeed were part of the motivation of the original Internet Protocol suite. Later DARPA experiments included the Survivable Radio Network (SURAN) project,[6]which took place in the 1980s. A successor to these systems was fielded in the mid-1990s for the US Army, and later other nations, as theNear-term digital radio.
Another third wave of academic and research activity started in the mid-1990s with the advent of inexpensive802.11radio cards forpersonal computers. Current wireless ad hoc networks are designed primarily for military utility.[7]Problems with packet radios are: (1) bulky elements, (2) slow data rate, (3) unable to maintain links if mobility is high. The project did not proceed much further until the early 1990s when wireless ad hoc networks were born.
The growth oflaptopsand802.11/Wi-Fiwireless networking have made MANETs a popular research topic since the mid-1990s. Many academic papers evaluateprotocolsand their abilities, assuming varying degrees of mobility within a bounded space, usually with all nodes within a fewhopsof each other. Different protocols are then evaluated based on measures such as the packet drop rate, the overhead introduced by the routing protocol, end-to-end packet delays, network throughput, ability to scale, etc.
In the early 1990s, Charles Perkins from SUN Microsystems USA, andChai Keong Tohfrom Cambridge University separately started to work on a different Internet, that of a wireless ad hoc network. Perkins was working on the dynamic addressing issues. Toh worked on a new routing protocol, which was known as ABR –associativity-based routing.[8]Perkins eventually proposed DSDV – Destination Sequence Distance Vector routing, which was based on distributed distance vector routing. Toh's proposal was an on-demand based routing, i.e. routes are discovered on-the-fly in real-time as and when needed. ABR[9]was submitted toIETFas RFCs. ABR was implemented successfully into Linux OS on Lucent WaveLAN 802.11a enabled laptops and a practical ad hoc mobile network was therefore proven[3][10][11]to be possible in 1999. Another routing protocol known as AODV was subsequently introduced and later proven and implemented in 2005.[12]In 2007, David Johnson and Dave Maltz proposed DSR –Dynamic Source Routing.[13]
The decentralized nature of wireless ad hoc networks makes them suitable for a variety of applications where central nodes can't be relied on and may improve the scalability of networks compared to wireless managed networks, though theoretical and practical limits to the overall capacity of such networks have been identified.[citation needed]Minimal configuration and quick deployment make ad hoc networks suitable for emergency situations like natural disasters or military conflicts. The presence of dynamic and adaptive routing protocols enables ad hoc networks to be formed quickly.
A mobile ad hoc network (MANET) is a continuously self-configuring, self-organizing, infrastructure-less[14]network of mobile devices connected without wires. It is sometimes known as "on-the-fly" networks or "spontaneous networks".[15]
VANETsare used for communication between vehicles and roadside equipment.[16]Intelligent vehicular ad hoc networks(InVANETs) are a kind of artificial intelligence that helps vehicles to behave in intelligent manners during vehicle-to-vehicle collisions, accidents. Vehicles are using radio waves to communicate with each other, creating communication networks instantly on-the-fly while vehicles move along roads. VANET needs to be secured with lightweight protocols.[17]
ASPANleverages existing hardware (primarilyWi-FiandBluetooth) and software (protocols) in commercially available smartphones to create peer-to-peer networks without relying on cellular carrier networks, wireless access points, or traditional network infrastructure. SPANs differ from traditionalhub and spokenetworks, such asWi-Fi Direct, in that they support multi-hop relays and there is no notion of a group leader so peers can join and leave at will without destroying the network. Apple'siPhonewith iOS version 7.0 and higher is capable of multi-peer ad hoc mesh networking.[18]
Mesh networks take their name from the topology of the resultant network. In a fully connected mesh, each node is connected to every other node, forming a "mesh". A partial mesh, by contrast, has a topology in which some nodes are not connected to others, although this term is seldom in use. Wireless ad hoc networks can take the form of a mesh networks or others. A wireless ad hoc network does not have fixed topology, and its connectivity among nodes is totally dependent on the behavior of the devices, their mobility patterns, distance with each other, etc. Hence, wireless mesh networks are a particular type of wireless ad hoc networks, with special emphasis on the resultant network topology. While some wireless mesh networks (particularly those within a home) have relatively infrequent mobility and thus infrequent link breaks, other more mobile mesh networks require frequent routing adjustments to account for lost links.[19]
Military or tactical MANETs are used by military units with emphasis on data rate, real-time requirement, fast re-routing during mobility, data security, radio range, and integration with existing systems.[20]Common radio waveforms include the US Army's JTRSSRW, Silvus Technologies MN-MIMO Waveform (Mobile Networked MIMO),[21][22][23][24]and Codan DTC MeshUltra Waveform.[25][26][27]Ad hoc mobile communications come in well to fulfill this need, especially its infrastructureless nature, fast deployment and operation. Military MANETs are used by military units with an emphasis on rapid deployment, infrastructureless, all-wireless networks (no fixed radio towers), robustness (link breaks are no problem), security, range, and instant operation.
Flying ad hoc networks (FANETs) are composed ofunmanned aerial vehicles, allowing great mobility and providing connectivity to remote areas.[28]
Unmanned aerial vehicle, is an aircraft with no pilot on board. UAVs can be remotely controlled (i.e., flown by a pilot at a ground control station) or can fly autonomously based on pre-programmed flight plans. Civilian usage of UAV include modeling 3D terrains, package delivery (Logistics), etc.[29]
UAVs have also been used by US Air Force[30]for data collection and
situation sensing, without risking the pilot in a foreign unfriendly environment.
With wireless ad hoc network technology embedded into the UAVs, multiple UAVs can communicate with each other and work as a team, collaboratively to complete a task and mission. If a UAV is destroyed by an enemy, its data can be quickly offloaded wirelessly to other neighboring UAVs.
The UAV ad hoc communication network is also sometimes referred to UAV instant sky network. More generally, aerial MANET in UAVs are now (as of 2021) successfully implemented and operational as mini tactical reconnaissance ISR UAVs like theBRAMOR C4EYEfrom Slovenia.
Navy ships traditionally use satellite communications and other maritime radios to communicate with each other or with ground station back on land. However, such communications are restricted by delays and limited bandwidth. Wireless ad hoc networks enable ship-area-networks to be formed while at sea, enabling high-speed wireless communications among ships, enhancing their sharing of imaging and multimedia data, and better co-ordination in battlefield operations.[31]Some defense companies (such as Rockwell Collins, Silvus Technologies and Rohde & Schwartz) have produced products that enhance ship-to-ship and ship-to-shore communications.[32]
Sensors are useful devices that collect information related to a specific parameter, such as noise, temperature, humidity, pressure, etc. Sensors are increasingly connected via wireless to allow large-scale collection of sensor data. With a large sample of sensor data, analytics processing can be used to make sense out of these data. The connectivity ofwireless sensor networksrely on the principles behind wireless ad hoc networks, since sensors can now be deploy without any fixed radio towers, and they can now form networks on-the-fly. "Smart Dust" was one of the early projects done at U C Berkeley, where tiny radios were used to interconnect smart dust.[33]More recently,mobile wireless sensor networks (MWSNs)have also become an area of academic interest.
Efforts have been made to co-ordinate and control a group of robots to undertake collaborative work to complete a task. Centralized control is often based on a "star" approach, where robots take turns to talk to the controller station. However, with wireless ad hoc networks, robots can form a communication network on-the-fly, i.e., robots can now "talk" to each other and collaborate in a distributed fashion.[34]With a network of robots, the robots can communicate among themselves, share local information, and distributively decide how to resolve a task in the most effective and efficient way.[35]
Another civilian use of wireless ad hoc network is for public safety. At times of disasters (floods, storms, earthquakes, fires, etc.), a quick and instant wireless communication network is necessary. Especially at times of earthquakes when radio towers had collapsed or were destroyed, wireless ad hoc networks can be formed independently. Firefighters and rescue workers can use ad hoc networks to communicate and rescue those injured. Commercial radios with such capability are available on the market.[31][36]
Wireless ad hoc networks allow sensors, videos, instruments, and other devices to be deployed and interconnected wirelessly for clinic and hospital patient monitoring, doctor and nurses alert notification, and also making senses of such data quickly at fusion points, so that lives can be saved.[37][38]
MANETS can be used for facilitating the collection ofsensordata fordata miningfor a variety of applications such asair pollutionmonitoring and different types of architectures can be used for such applications.[39]A key characteristic of such applications is that nearby sensor nodes monitoring an environmental feature typically register similar values. This kind ofdata redundancydue to thespatial correlationbetween sensor observations inspires the techniques for in-network data aggregation and mining. By measuring the spatial correlation between data sampled by different sensors, a wide class of specialized algorithms can be developed to develop more efficient spatial data mining algorithms as well as more efficient routing strategies.[40]Also, researchers have developed performance models for MANET to applyqueueing theory.[41][42]
Several books[43][1]and works have revealed the technical and research challenges[44][45]facing wireless ad hoc networks or MANETs. The advantages for users, the technical difficulties in implementation, and the side effect onradio spectrum pollutioncan be briefly summarized below:
The obvious appeal of MANETs is that the network is decentralised and nodes/devices are mobile, that is to say there is no fixed infrastructure which provides the possibility for numerous applications in different areas such asenvironmental monitoring, disaster relief and military communications. Since the early 2000s, interest in MANETs has greatly increased which, in part, is due to the fact mobility can improve network capacity, shown by Grossglauser and Tse along with the introduction of new technologies.[46]
One main advantage to a decentralised network is that they are typically more robust than centralised networks due to the multi-hop fashion in which information is relayed. For example, in the cellular network setting, a drop in coverage occurs if a base station stops working, however the chance of a single point of failure in a MANET is reduced significantly since the data can take multiple paths. Since the MANET architecture evolves with time it has the potential to resolve issues such as isolation/disconnection from the network. Further advantages of MANETS over networks with a fixed topology include flexibility (an ad hoc network can be created anywhere with mobile devices), scalability (adding nodes to the network is easy) and lower administration costs (no need to build an infrastructure first).[47][48]
With a time evolving network it is clear we should expect variations in network performance due to no fixed architecture (no fixed connections). Furthermore, since network topology determines interference and thus connectivity, the mobility pattern of devices within the network will impact on network performance, possibly resulting in data having to be resent a lot of times (increased delay) and finally allocation of network resources such as power remains unclear.[46]Finally, finding a model that accurately represents human mobility whilst remaining mathematically tractable remains an open problem due to the large range of factors that influence it.[49]Some typical models used include the random walk, random waypoint and levy flight models.[50][51][52][53]
Wireless ad hoc networks can operate over different types of radios. All radios usemodulationto move information over a certainbandwidthof radio frequencies. Given the need to move large amounts of information quickly over long distances, a MANET radio channel ideally has large bandwidth (e.g. amount of radio spectrum), lower frequencies, and higher power. Given the desire to communicate with many other nodes ideally simultaneously, many channels are needed. Given radio spectrum is shared andregulated, there is less bandwidth available at lower frequencies. Processing many radio channels requires many resources. Given the need for mobility, small size and lower power consumption are very important. Picking a MANET radio and modulation has many trade-offs; many start with the specific frequency and bandwidth they are allowed to use.
Radios can beUHF(300 – 3000 MHz),SHF(3 – 30 GHz), andEHF(30 – 300 GHz).Wi-Fiad hoc uses the unlicensed ISM 2.4 GHz radios. They can also be used on 5.8 GHz radios.
The higher the frequency, such as those of 300 GHz, absorption of the signal will be more predominant. Army tactical radios usually employ a variety of UHF and SHF radios, including those ofVHFto provide a variety of communication modes. At the 800, 900, 1200, 1800 MHz range, cellular radios are predominant. Some cellular radios use ad hoc communications to extend cellular range to areas and devices not reachable by the cellular base station.
Next generation Wi-Fi known as802.11axprovides low delay, high capacity (up to 10 Gbit/s) and low packet loss rate, offering 12 streams – 8 streams at 5 GHz and 4 streams at 2.4 GHz. IEEE 802.11ax uses 8x8 MU-MIMO, OFDMA, and 80 MHz channels. Hence, 802.11ax has the ability to form high capacity Wi-Fi ad hoc networks.
At 60 GHz, there is another form of Wi-Fi known as WiGi – wireless gigabit. This has the ability to offer up to 7 Gbit/s throughput. Currently, WiGi is targeted to work with 5G cellular networks.[54]
Circa 2020, the general consensus finds the 'best' modulation for moving information over higher frequency waves to beorthogonal frequency-division multiplexing, as used in4G LTE,5G, andWi-Fi.
The challenges[43][1]affecting MANETs span from various layers of theOSI protocolstack. The media access layer (MAC) has to be improved to resolve collisions and hidden terminal problems. The network layer routing protocol has to be improved to resolve dynamically changing network topologies and broken routes. The transport layer protocol has to be improved to handle lost or broken connections. The session layer protocol has to deal with discovery of servers and services.
A major limitation with mobile nodes is that they have high mobility, causing links to be frequently broken and reestablished. Moreover, the bandwidth of a wireless channel is also limited, and nodes operate on limited battery power, which will eventually be exhausted. These factors make the design of a mobile ad hoc network challenging.
The cross-layer design deviates from the traditionalnetwork designapproach in which each layer of the stack would be made to operate independently. The modified transmission power will help that node to dynamically vary its propagation range at the physical layer. This is because the propagation distance is always directly proportional to transmission power. This information is passed from the physical layer to the network layer so that it can take optimal decisions in routing protocols. A major advantage of this protocol is that it allows access of information between physical layer and top layers (MAC and network layer).
Some elements of the software stack were developed to allow code updatesin situ, i.e., with the nodes embedded in their physical environment and without needing to bring the nodes back into the lab facility.[55]Such software updating relied on epidemic mode of dissemination of information and had to be done both efficiently (few network transmissions) and fast.
Routing[56]in wireless ad hoc networks or MANETs generally falls into three categories, namely: proactive routing, reactive routing, and hybrid routing.
This type of protocols maintains fresh lists of destinations and their routes by periodically distributing routing tables throughout the network. The main disadvantages of such algorithms are:
Example:Optimized Link State Routing Protocol(OLSR)
As in a fix net nodes maintain routing tables. Distance-vector protocols are based on calculating the direction and distance to any link in a network. "Direction" usually means the next hop address and the exit interface. "Distance" is a measure of the cost to reach a certain node. The least cost route between any two nodes is the route with minimum distance. Each node maintains a vector (table) of minimum distance to every node. The cost of reaching a destination is calculated using various route metrics.RIPuses the hop count of the destination whereasIGRPtakes into account other information such as node delay and available bandwidth.
This type of protocol finds a route based on user and traffic demand by flooding the network with Route Request or Discovery packets. The main disadvantages of such algorithms are:
However, clustering can be used to limit flooding. The latency incurred during route discovery is not significant compared to periodic route update exchanges by all nodes in the network.
Example:Ad hoc On-Demand Distance Vector Routing(AODV)
Is a simple routing algorithm in which every incoming packet is sent through every outgoing link except the one it arrived on. Flooding is used in bridging and in systems such as Usenet andpeer-to-peer file sharingand as part of some routing protocols, includingOSPF,DVMRP, and those used in wireless ad hoc networks.
This type of protocol combines the advantages ofproactiveandreactive routing. The routing is initially established with some proactively prospected routes and then serves the demand from additionally activated nodes through reactive flooding. The choice of one or the other method requires predetermination for typical cases. The main disadvantages of such algorithms are:
Example:Zone Routing Protocol(ZRP)
Position-based routing methods use information on the exact locations of the nodes. This information is obtained for example via aGPSreceiver. Based on the exact location the best path between source and destination nodes can be determined.
Example: "Location-Aided Routing in mobile ad hoc networks" (LAR)
An ad hoc network is made up of multiple "nodes" connected by "links."
Links are influenced by the node's resources (e.g., transmitter power, computing power and memory) and behavioral properties (e.g., reliability), as well as link properties (e.g. length-of-link and signal loss, interference and noise). Since links can be connected or disconnected at any time, a functioning network must be able to cope with this dynamic restructuring, preferably in a way that is timely, efficient, reliable, robust, and scalable.
The network must allow any two nodes to communicate by relaying the information via other nodes. A "path" is a series of links that connects two nodes. Various routing methods use one or two paths between any two nodes; flooding methods use all or most of the available paths.[59]
In most wireless ad hoc networks, the nodes compete for access to shared wireless medium, often resulting incollisions(interference).[60]Collisions can be handled using centralized scheduling or distributed contention access protocols.[60]Usingcooperative wireless communicationsimproves immunity tointerferenceby having the destination node combine self-interference and other-node interference to improve decoding of the desired signals.
One key problem in wireless ad hoc networks is foreseeing the variety of possible situations that can occur. As a result,modeling and simulation(M&S) using extensive parameter sweeping and what-if analysis becomes an extremely important paradigm for use in ad hoc networks. One solution is the use of simulation tools likeOPNET, NetSim orns2. A comparative study of various simulators for VANETs reveal that factors such as constrained road topology, multi-path fading and roadside obstacles, traffic flow models, trip models, varying vehicular speed and mobility, traffic lights, traffic congestion, drivers' behavior, etc., have to be taken into consideration in the simulation process to reflect realistic conditions.[61]
In 2009, theU.S. Army Research Laboratory(ARL) andNaval Research Laboratory(NRL) developed a Mobile Ad-HocNetwork emulationtestbed, where algorithms and applications were subjected to representative wireless network conditions. The testbed was based on a version of the "MANE" (Mobile Ad hoc Network Emulator) software originally developed by NRL.[62]
The traditional model is therandom geometric graph. Early work included simulating ad hoc mobile networks on sparse and densely connected topologies. Nodes are firstly scattered in a constrained physical space randomly. Each node then has a predefined fixed cell size (radio range). A node is said to be connected to another node if this neighbor is within its radio range. Nodes are then moved (migrated away) based on a random model, using random walk or brownian motion. Different mobility and number of nodes present yield different route length and hence different number of multi-hops.
These aregraphsconsisting of a set ofnodesplaced according to apoint processin some usually boundedsubsetof then-dimensional plane, mutuallycoupledaccording to aBooleanprobability mass functionof theirspatial separation(see e.g.unit disk graphs). The connections between nodes may have different weights to model the difference in channel attenuations.[60]One can then study networkobservables(such asconnectivity,[63]centrality[64]or thedegree distribution[65]) from agraph-theoreticperspective. One can further study network protocols and algorithms to improve network throughput and fairness.[60]
Most wireless ad hoc networks do not implement any network access control, leaving these networks vulnerable to resource consumption attacks where a malicious node injects packets into the network with the goal of depleting the resources of the nodes relaying the packets.[66]
To thwart or prevent such attacks, it was necessary to employ authentication mechanisms that ensure that only authorized nodes can inject traffic into the network.[67]Even with authentication, these networks are vulnerable to packet dropping or delaying attacks, whereby an intermediate node drops the packet or delays it, rather than promptly sending it to the next hop.
In a multicast and dynamic environment, establishing temporary 1:1 secure 'sessions' usingPKIwith every other node is not feasible (like is done withHTTPS, mostVPNs, etc. at the transport layer). Instead, a common solution is to use pre-shared keys for symmetric, authenticated encryption at the link layer, for exampleMACsecusingAES-256-GCM. With this method, every properly formatted packet received is authenticated then passed along for decryption or dropped. It also means the key(s) in each node must be changed more often and simultaneously (e.g. to avoid reusing anIV).
Trust establishment and management in MANETs face challenges due to resource constraints and the complex interdependency of networks. Managing trust in a MANET needs to consider the interactions between the composite cognitive, social, information and communication networks, and take into account the resource constraints (e.g., computing power, energy, bandwidth, time), and dynamics (e.g., topology changes, node mobility, node failure, propagation channel conditions).[68]
Researchers of trust management in MANET suggested that such complex interactions require a composite trust metric that captures aspects of communications and social networks, and corresponding trust measurement, trust distribution, and trust management schemes.[68]
Continuous monitoringof every node within a MANET is necessary for trust and reliability but difficult because it by definition is dis-continuous, 2) it requires input from the node itself and 3) from its 'nearby' peers.
|
https://en.wikipedia.org/wiki/Wireless_ad_hoc_network
|
High Speed Packet Access(HSPA)[1]is an amalgamation of twomobileprotocols—High Speed Downlink Packet Access (HSDPA) and High Speed Uplink Packet Access (HSUPA)—that extends and improves the performance of existing3Gmobile telecommunication networks using theWCDMAprotocols. A further-improved3GPPstandard calledEvolved High Speed Packet Access(also known as HSPA+) was released late in 2008, with subsequent worldwide adoption beginning in 2010. The newer standard allowsbit ratesto reach as high as 337 Mbit/s in the downlink and 34 Mbit/s in the uplink; however, these speeds are rarely achieved in practice.[2]
The first HSPA specifications supported increased peak data rates of up to 14 Mbit/s in the downlink and 5.76 Mbit/s in the uplink. They also reduced latency and provided up to five times more system capacity in the downlink and up to twice as much system capacity in the uplink compared with original WCDMA protocol.
High Speed Downlink Packet Access(HSDPA) is an enhanced3G(third-generation)mobilecommunications protocolin the High-Speed Packet Access (HSPA) family. HSDPA is also known as3.5Gand3G+. It allows networks based on theUniversal Mobile Telecommunications System(UMTS) to have higher data speeds and capacity. HSDPA also decreaseslatency, and therefore theround-trip timefor applications.
HSDPA was introduced in3GPPRelease 5. It was accompanied by an improvement to the uplink that provided a new bearer of 384 kbit/s (the previous maximum bearer was 128 kbit/s).Evolved High Speed Packet Access(HSPA+), introduced in 3GPP Release 7, further increased data rates by adding 64QAM modulation,MIMO, andDual-Carrier HSDPAoperation. Under 3GPP Release 11, even higher speeds of up to 337.5 Mbit/s were possible.[3]
The first phase of HSDPA was specified in 3GPP Release 5. This phase introduced new basic functions and was aimed to achieve peak data rates of 14.0 Mbit/s with significantly reduced latency. The improvement in speed and latency reduced the cost per bit and enhanced support for high-performance packet data applications. HSDPA is based on shared channel transmission, and its key features are shared channel and multi-code transmission,higher-order modulation, shortTransmission Time Interval(TTI), fast link adaptation and scheduling, and fasthybrid automatic repeat request(HARQ). Additional new features include the High Speed Downlink Shared Channels (HS-DSCH),quadrature phase-shift keying, 16-quadrature amplitude modulation, and the High Speed Medium Access protocol (MAC-hs) in base stations.
The upgrade to HSDPA is often just a software update for WCDMA networks. In HSDPA, voice calls are usually prioritized over data transfer.
The following table is derived from table 5.1a of the release 11 of 3GPP TS 25.306[4]and shows maximum data rates of different device classes and by what combination of features they are achieved. The per-cell, per-stream data rate is limited by the "maximum number of bits of an HS-DSCH transport block received within an HS-DSCH TTI" and the "minimum inter-TTI interval". The TTI is 2 milliseconds. So, for example, Cat 10 can decode 27,952 bits / 2 ms = 13.976 Mbit/s (and not 14.4 Mbit/s as often claimed incorrectly). Categories 1-4 and 11 have inter-TTI intervals of 2 or 3, which reduces the maximum data rate by that factor. Dual-Cell and MIMO 2x2 each multiply the maximum data rate by 2, because multiple independent transport blocks are transmitted over different carriers or spatial streams, respectively. The data rates given in the table are rounded to one decimal point.
Further UE categories were defined from 3GGP Release 7 onwards asEvolved HSPA(HSPA+) and are listed inEvolved HSDPA UE Categories.
As of 28 August 2009[update], 250 HSDPA networks had commercially launchedmobile broadbandservices in 109 countries. 169 HSDPA networks supported 3.6 Mbit/s peak downlink data throughput, and a growing number delivered 21 Mbit/s peak data downlink.[citation needed]
CDMA2000-EVDOnetworks had the early lead on performance. In particular,Japaneseproviders were highly successful benchmarks for this network standard. However, this later changed in favor of HSDPA, as an increasing number of providers worldwide began adopting it.
In 2007, an increasing number of telcos worldwide began sellingHSDPA USB modemsto provide mobile broadband connections. In addition, the popularity of HSDPA landline replacement boxes grew—these provided HSDPA for data viaEthernetandWi-Fi, as well as ports for connecting traditional landline telephones. Some were marketed with connection speeds of "up to 7.2 Mbit/s"[5]under ideal conditions. However, these services could be slower, such as when in fringe coverage indoors.
High-Speed Uplink Packet Access(HSUPA) is a 3G mobile telephonyprotocolin the HSPA family. It is specified and standardized in 3GPP Release 6 to improve the uplink data rate to 5.76 Mbit/s, extend capacity, and reduce latency. Together with additional improvements, this allows for new features such asVoice over Internet Protocol(VoIP), uploading pictures, and sending large e-mail messages.
HSUPA was the second major step in the UMTS evolution process. It has since been superseded by newer technologies with higher transfer rates, such asLTE(150 Mbit/s for downlink and 50 Mbit/s for uplink) andLTE Advanced(maximum downlink rates of over 1 Gbit/s).
HSUPA adds a new transport channel to WCDMA, called the Enhanced Dedicated Channel (E-DCH). It also features several improvements similar to those of HSDPA, including multi-code transmission, shorter transmission time interval enabling fasterlink adaptation, fast scheduling, and fasthybrid automatic repeat request(HARQ) with incremental redundancy, makingretransmissionsmore effective. Similar to HSDPA, HSUPA uses a "packet scheduler", but it operates on a "request-grant" principle where theuser equipment(UE) requests permission to send data and the scheduler decides when and how many UEs will be allowed to do so. A request for transmission contains data about the state of the transmission buffer and the queue at the UE and its available power margin. However, unlike HSDPA, uplink transmissions are notorthogonalto each other.
In addition to this "scheduled" mode of transmission, the standards allow a self-initiated transmission mode from the UEs, denoted "non-scheduled". The non-scheduled mode can, for example, be used for VoIP services for which even the reduced TTI and theNode Bbased scheduler are unable to provide the necessary short delay time and constant bandwidth.
Each MAC-d flow (i.e., QoS flow) is configured to use either scheduled or non-scheduled modes. The UE adjusts the data rate for scheduled and non-scheduled flows independently. The maximum data rate of each non-scheduled flow is configured at call setup, and typically not frequently changed. The power used by the scheduled flows is controlled dynamically by the Node B through absolute grant (consisting of an actual value) and relative grant (consisting of a single up/down bit) messages.
At thephysical layer, HSUPA introduces the following new channels:
The following table shows uplink speeds for the different categories of HSUPA:
Further UE categories were defined from 3GGP Release 7 onwards as Evolved HSPA (HSPA+) and are listed inEvolved HSUPA UE Categories.
Evolved HSPA(also known as HSPA Evolution, HSPA+) is a wireless broadband standard defined in3GPPrelease 7 of the WCDMA specification. It provides extensions to the existing HSPA definitions and is thereforebackward compatibleall the way to the original Release 99 WCDMA network releases. Evolved HSPA provides data rates between 42.2 and 56 Mbit/s in the downlink and 22 Mbit/s in the uplink (per 5 MHz carrier) with multiple input, multiple output (2x2 MIMO) technologies and higher order modulation (64 QAM). With Dual Cell technology, these can be doubled.
Since 2011, HSPA+ has been widely deployed among WCDMA operators, with nearly 200 commitments.[6]
|
https://en.wikipedia.org/wiki/HSDPA
|
High Speed Packet Access(HSPA)[1]is an amalgamation of twomobileprotocols—High Speed Downlink Packet Access (HSDPA) and High Speed Uplink Packet Access (HSUPA)—that extends and improves the performance of existing3Gmobile telecommunication networks using theWCDMAprotocols. A further-improved3GPPstandard calledEvolved High Speed Packet Access(also known as HSPA+) was released late in 2008, with subsequent worldwide adoption beginning in 2010. The newer standard allowsbit ratesto reach as high as 337 Mbit/s in the downlink and 34 Mbit/s in the uplink; however, these speeds are rarely achieved in practice.[2]
The first HSPA specifications supported increased peak data rates of up to 14 Mbit/s in the downlink and 5.76 Mbit/s in the uplink. They also reduced latency and provided up to five times more system capacity in the downlink and up to twice as much system capacity in the uplink compared with original WCDMA protocol.
High Speed Downlink Packet Access(HSDPA) is an enhanced3G(third-generation)mobilecommunications protocolin the High-Speed Packet Access (HSPA) family. HSDPA is also known as3.5Gand3G+. It allows networks based on theUniversal Mobile Telecommunications System(UMTS) to have higher data speeds and capacity. HSDPA also decreaseslatency, and therefore theround-trip timefor applications.
HSDPA was introduced in3GPPRelease 5. It was accompanied by an improvement to the uplink that provided a new bearer of 384 kbit/s (the previous maximum bearer was 128 kbit/s).Evolved High Speed Packet Access(HSPA+), introduced in 3GPP Release 7, further increased data rates by adding 64QAM modulation,MIMO, andDual-Carrier HSDPAoperation. Under 3GPP Release 11, even higher speeds of up to 337.5 Mbit/s were possible.[3]
The first phase of HSDPA was specified in 3GPP Release 5. This phase introduced new basic functions and was aimed to achieve peak data rates of 14.0 Mbit/s with significantly reduced latency. The improvement in speed and latency reduced the cost per bit and enhanced support for high-performance packet data applications. HSDPA is based on shared channel transmission, and its key features are shared channel and multi-code transmission,higher-order modulation, shortTransmission Time Interval(TTI), fast link adaptation and scheduling, and fasthybrid automatic repeat request(HARQ). Additional new features include the High Speed Downlink Shared Channels (HS-DSCH),quadrature phase-shift keying, 16-quadrature amplitude modulation, and the High Speed Medium Access protocol (MAC-hs) in base stations.
The upgrade to HSDPA is often just a software update for WCDMA networks. In HSDPA, voice calls are usually prioritized over data transfer.
The following table is derived from table 5.1a of the release 11 of 3GPP TS 25.306[4]and shows maximum data rates of different device classes and by what combination of features they are achieved. The per-cell, per-stream data rate is limited by the "maximum number of bits of an HS-DSCH transport block received within an HS-DSCH TTI" and the "minimum inter-TTI interval". The TTI is 2 milliseconds. So, for example, Cat 10 can decode 27,952 bits / 2 ms = 13.976 Mbit/s (and not 14.4 Mbit/s as often claimed incorrectly). Categories 1-4 and 11 have inter-TTI intervals of 2 or 3, which reduces the maximum data rate by that factor. Dual-Cell and MIMO 2x2 each multiply the maximum data rate by 2, because multiple independent transport blocks are transmitted over different carriers or spatial streams, respectively. The data rates given in the table are rounded to one decimal point.
Further UE categories were defined from 3GGP Release 7 onwards asEvolved HSPA(HSPA+) and are listed inEvolved HSDPA UE Categories.
As of 28 August 2009[update], 250 HSDPA networks had commercially launchedmobile broadbandservices in 109 countries. 169 HSDPA networks supported 3.6 Mbit/s peak downlink data throughput, and a growing number delivered 21 Mbit/s peak data downlink.[citation needed]
CDMA2000-EVDOnetworks had the early lead on performance. In particular,Japaneseproviders were highly successful benchmarks for this network standard. However, this later changed in favor of HSDPA, as an increasing number of providers worldwide began adopting it.
In 2007, an increasing number of telcos worldwide began sellingHSDPA USB modemsto provide mobile broadband connections. In addition, the popularity of HSDPA landline replacement boxes grew—these provided HSDPA for data viaEthernetandWi-Fi, as well as ports for connecting traditional landline telephones. Some were marketed with connection speeds of "up to 7.2 Mbit/s"[5]under ideal conditions. However, these services could be slower, such as when in fringe coverage indoors.
High-Speed Uplink Packet Access(HSUPA) is a 3G mobile telephonyprotocolin the HSPA family. It is specified and standardized in 3GPP Release 6 to improve the uplink data rate to 5.76 Mbit/s, extend capacity, and reduce latency. Together with additional improvements, this allows for new features such asVoice over Internet Protocol(VoIP), uploading pictures, and sending large e-mail messages.
HSUPA was the second major step in the UMTS evolution process. It has since been superseded by newer technologies with higher transfer rates, such asLTE(150 Mbit/s for downlink and 50 Mbit/s for uplink) andLTE Advanced(maximum downlink rates of over 1 Gbit/s).
HSUPA adds a new transport channel to WCDMA, called the Enhanced Dedicated Channel (E-DCH). It also features several improvements similar to those of HSDPA, including multi-code transmission, shorter transmission time interval enabling fasterlink adaptation, fast scheduling, and fasthybrid automatic repeat request(HARQ) with incremental redundancy, makingretransmissionsmore effective. Similar to HSDPA, HSUPA uses a "packet scheduler", but it operates on a "request-grant" principle where theuser equipment(UE) requests permission to send data and the scheduler decides when and how many UEs will be allowed to do so. A request for transmission contains data about the state of the transmission buffer and the queue at the UE and its available power margin. However, unlike HSDPA, uplink transmissions are notorthogonalto each other.
In addition to this "scheduled" mode of transmission, the standards allow a self-initiated transmission mode from the UEs, denoted "non-scheduled". The non-scheduled mode can, for example, be used for VoIP services for which even the reduced TTI and theNode Bbased scheduler are unable to provide the necessary short delay time and constant bandwidth.
Each MAC-d flow (i.e., QoS flow) is configured to use either scheduled or non-scheduled modes. The UE adjusts the data rate for scheduled and non-scheduled flows independently. The maximum data rate of each non-scheduled flow is configured at call setup, and typically not frequently changed. The power used by the scheduled flows is controlled dynamically by the Node B through absolute grant (consisting of an actual value) and relative grant (consisting of a single up/down bit) messages.
At thephysical layer, HSUPA introduces the following new channels:
The following table shows uplink speeds for the different categories of HSUPA:
Further UE categories were defined from 3GGP Release 7 onwards as Evolved HSPA (HSPA+) and are listed inEvolved HSUPA UE Categories.
Evolved HSPA(also known as HSPA Evolution, HSPA+) is a wireless broadband standard defined in3GPPrelease 7 of the WCDMA specification. It provides extensions to the existing HSPA definitions and is thereforebackward compatibleall the way to the original Release 99 WCDMA network releases. Evolved HSPA provides data rates between 42.2 and 56 Mbit/s in the downlink and 22 Mbit/s in the uplink (per 5 MHz carrier) with multiple input, multiple output (2x2 MIMO) technologies and higher order modulation (64 QAM). With Dual Cell technology, these can be doubled.
Since 2011, HSPA+ has been widely deployed among WCDMA operators, with nearly 200 commitments.[6]
|
https://en.wikipedia.org/wiki/HSUPA
|
Packet Data Convergence Protocol(PDCP) is specified by3GPPin TS 25.323[1]forUMTS, TS 36.323[2]forLTEand TS 38.323[3]for5G. PDCP is located in the Radio Protocol Stack in the UMTS/LTE/5Gair interfaceon top of theRLClayer.
PDCP provides its services to theRRCand user plane upper layers, e.g.IPat theUEor to the relay at the base station. The following services are provided by PDCP to upper layers:
The header compression technique can be based on either IP header compression (RFC 2507) orRobust Header Compression(RFC 3095). IfPDCPis configured forNo Compressionit will send the IP Packets without compression; otherwise it will compress the packets according to its configuration by upper layer and attach aPDCPheader and send the packet.
Different header formats are defined, dependent on the type of data to be transported. In LTE, there are e.g. header formats for Control Plane PDCP DataPDUwith long PDCP SN (12 bits), for User plane PDCP Data PDU with short PDCP SN (7 bits) and others.
|
https://en.wikipedia.org/wiki/PDCP
|
ASIM cardorSIM(subscriber identity module) is anintegrated circuit(IC) intended to securely store aninternational mobile subscriber identity(IMSI) number and its related key, which are used to identify and authenticate subscribers onmobile telephonedevices (such asmobile phones,tablets, andlaptops). SIMs are also able to storeaddress bookcontacts information,[1]and may be protected using aPIN codeto prevent unauthorized use.
SIMs are always used onGSMphones; forCDMAphones, they are needed only forLTE-capable handsets. SIM cards are also used in varioussatellite phones, smart watches, computers, or cameras.[2]The first SIM cards were the size ofcredit and bank cards; sizes were reduced several times over the years, usually keeping electrical contacts the same, to fit smaller-sized devices.[3]SIMs are transferable between different mobile devices by removing the card itself.
Technically, the actual physical card is known as auniversal integrated circuit card(UICC); thissmart cardis usually made ofPVCwith embedded contacts andsemiconductors, with the SIM as its primary component. In practice the term "SIM card" is still used to refer to the entire unit and not simply the IC. A SIM contains a unique serial number, integrated circuit card identification (ICCID), international mobile subscriber identity (IMSI) number, security authentication and ciphering information, temporary information related to the local network, a list of the services the user has access to, and four passwords: apersonal identification number(PIN) for ordinary use, and apersonal unblocking key(PUK) for PIN unlocking as well as a second pair (called PIN2 and PUK2 respectively) which are used for managingfixed dialing numberand some other functionality.[4][5]In Europe, the serial SIM number (SSN) is also sometimes accompanied by aninternational article number(IAN) or aEuropean article number(EAN) required when registering online for the subscription of a prepaid card.
As of 2020,eSIMis superseding physical SIM cards in some domains, including cellular telephony. eSIM uses a software-based SIM embedded into an irremovableeUICC.
The SIM card is a type ofsmart card,[2]the basis for which is thesiliconintegrated circuit(IC) chip.[6]The idea of incorporating a silicon IC chip onto a plastic card originates from the late 1960s.[6]Smart cards have since usedMOS integrated circuitchips, along withMOS memorytechnologies such asflash memoryandEEPROM(electricallyEPROM).[7]
The SIM was initially specified by theETSIin the specification TS 11.11. This describes the physical and logical behaviour of the SIM. With the development ofUMTS, the specification work was partially transferred to3GPP. 3GPP is now responsible for the further development of applications like SIM (TS 51.011[8]) and USIM (TS 31.102[9]) and ETSI for the further development of the physical cardUICC.
The first SIM card was manufactured in 1991 byMunichsmart-card makerGiesecke+Devrient, who sold the first 300 SIM cards to the Finnishwireless network operatorRadiolinja,[10][11]who launched the world's first commercial2GGSMcell network that year.[12]
Today, SIM cards are considered ubiquitous, allowing over 8 billion devices to connect to cellular networks around the world daily. According to the International Card Manufacturers Association (ICMA), there were 5.4 billion SIM cards manufactured globally in 2016 creating over $6.5 billion in revenue for traditional SIM card vendors.[13]The rise of cellular IoT and 5G networks was predicted by Ericsson to drive the growth of the addressable market for SIM cards to over 20 billion devices by 2020.[14]The introduction ofembedded-SIM(eSIM) andremote SIM provisioning(RSP) from the GSMA[15]may disrupt the traditional SIM card ecosystem with the entrance of new players specializing in "digital" SIM card provisioning and other value-added services for mobile network operators.[7]
There are three operating voltages for SIM cards:5 V,3 Vand1.8 V(ISO/IEC 7816-3 classes A, B and C, respectively). The operating voltage of the majority of SIM cards launched before 1998 was5 V. SIM cards produced subsequently are compatible with3 Vand5 V. Modern cards support5 V,3 Vand1.8 V.[7]
Modern SIM cards allow applications to load when the SIM is in use by the subscriber. These applications communicate with the handset or a server usingSIM Application Toolkit, which was initially specified by3GPPin TS 11.14. (There is an identical ETSI specification with different numbering.) ETSI and 3GPP maintain the SIM specifications. The main specifications are: ETSI TS 102 223 (the toolkit for smart cards), ETSI TS 102 241 (API), ETSI TS 102 588 (application invocation), and ETSI TS 131 111 (toolkit for more SIM-likes). SIM toolkit applications were initially written in native code using proprietary APIs. To provide interoperability of the applications, ETSI choseJava Card.[16]A multi-company collaboration calledGlobalPlatformdefines some extensions on the cards, with additional APIs and features like more cryptographic security andRFIDcontactless use added.[17]
SIM cards store network-specific information used to authenticate and identify subscribers on the network. The most important of these are the ICCID, IMSI,authentication key (Ki), local area identity (LAI) and operator-specific emergency number. The SIM also stores other carrier-specific data such as the SMSC (Short Message service center) number, service provider name (SPN), service dialing numbers (SDN), advice-of-charge parameters and value-added service (VAS) applications. (Refer to GSM 11.11.[18])
SIM cards can come in various data capacities, from8 KBto at least256 KB.[11]All can store a maximum of 250 contacts on the SIM, but while the32 KBhas room for 33Mobile country code(MCCs) ornetwork identifiers, the64 KBversion has room for 80 MNCs.[1]This is used by network operators to store data on preferred networks, mostly used when the SIM is not in its home network but isroaming. The network operator that issued the SIM card can use this to have a phone connect to a preferred network that is more economic for the provider instead of having to pay the network operator that the phone discovered first. This does not mean that a phone containing this SIM card can connect to a maximum of only 33 or 80 networks, instead it means that the SIM card issuer can specify only up to that number of preferred networks. If a SIM is outside these preferred networks, it uses the first or best available network.[14]
Each SIM is internationally identified by itsintegrated circuit card identifier(ICCID). Nowadays ICCID numbers are also used to identify eSIM profiles, not only physical SIM cards. ICCIDs are stored in the SIM cards and are also engraved or printed on the SIM card body during a process called personalisation.
The ICCID is defined by the ITU-T recommendationE.118as theprimary account number.[19]Its layout is based onISO/IEC 7812. According to E.118, the number can be up to 19 digits long, including a single check digit calculated using theLuhn algorithm. However, the GSM Phase 1[20]defined the ICCID length as an opaque data field, 10 octets (20 digits) in length, whose structure is specific to amobile network operator.
The number is composed of three subparts:
Their format is as follows.
Issuer identification number (IIN)
Individual account identification
Check digit
With the GSM Phase 1 specification using 10octetsinto which ICCID is stored as packed BCD[clarification needed], the data field has room for 20 digits with hexadecimal digit "F" being used as filler when necessary. In practice, this means that on GSM cards there are 20-digit (19+1) and 19-digit (18+1) ICCIDs in use, depending upon the issuer. However, a single issuer always uses the same size for its ICCIDs.
As required by E.118, the ITU-T updates a list of all current internationally assigned IIN codes in its Operational Bulletins which are published twice a month (the last as of January 2019 was No. 1163 from 1 January 2019).[22]ITU-T also publishes complete lists: as of August 2023, the list issued on 1 December 2018 was current, having all issuer identifier numbers before 1 December 2018.[23]
SIM cards are identified on their individual operator networks by a uniqueinternational mobile subscriber identity(IMSI).Mobile network operatorsconnect mobile phone calls and communicate with their market SIM cards using their IMSIs. The format is:
The Kiis a 128-bit value used in authenticating the SIMs on aGSMmobile network (for USIM network, the Kiis still needed but other parameters are also needed). Each SIM holds a unique Kiassigned to it by the operator during the personalisation process. The Kiis also stored in a database (termedauthentication centeror AuC) on the carrier's network.
The SIM card is designed to prevent someone from getting the Kiby using thesmart-card interface. Instead, the SIM card provides a function,Run GSM Algorithm, that the phone uses to pass data to the SIM card to be signed with the Ki. This, by design, makes using the SIM card mandatory unless the Kican be extracted from the SIM card, or the carrier is willing to reveal the Ki. In practice, the GSM cryptographic algorithm for computing a signed response (SRES_1/SRES_2: see steps 3 and 4, below) from the Kihas certain vulnerabilities[1]that can allow the extraction of the Kifrom a SIM card and the making of aduplicate SIM card.
Authentication process:
The SIM stores network state information, which is received from thelocation area identity(LAI). Operator networks are divided into location areas, each having a unique LAI number. When the device changes locations, it stores the new LAI to the SIM and sends it back to the operator network with its new location. If the device is power cycled, it takes data off the SIM, and searches for the prior LAI.
Most SIM cards store a number of SMS messages and phone book contacts. It stores the contacts in simple "name and number" pairs. Entries that contain multiple phone numbers and additional phone numbers are usually not stored on the SIM card. When a user tries to copy such entries to a SIM, the handset's software breaks them into multiple entries, discarding information that is not a phone number. The number of contacts and messages stored depends on the SIM; early models stored as few as five messages and 20 contacts, while modern SIM cards can usually store over 250 contacts.[24]
SIM cards have been made smaller over the years; functionality is independent of format. Full-size SIM was followed by mini-SIM, micro-SIM, and nano-SIM. SIM cards are also made to embed in devices.
JEDECDesign Guide 4.8, SON-8GSMA SGP.22 V1.0
All versions of the non-embedded SIM cards share the sameISO/IEC 7816pin arrangement.
Themini-SIMor (2FF , 2nd form factor) card has the same contact arrangement as the full-size SIM card and is normally supplied within a full-size card carrier, attached by a number of linking pieces. This arrangement (defined inISO/IEC 7810asID-1/000) lets such a card be used in a device that requires a full-size card – or in a device that requires a mini-SIM card, after breaking the linking pieces. As the full-size SIM is obsolete, some suppliers refer to the mini-SIM as a "standard SIM" or "regular SIM".
Themicro-SIM(or 3FF) card has the same thickness and contact arrangements, but reduced length and width as shown in the table above.[25]
The micro-SIM was introduced by theEuropean Telecommunications Standards Institute(ETSI) along with SCP,3GPP(UTRAN/GERAN),3GPP2(CDMA2000),ARIB,GSM Association(GSMA SCaG and GSMNA), GlobalPlatform,Liberty Alliance, and theOpen Mobile Alliance(OMA) for the purpose of fitting into devices too small for a mini-SIM card.[21][26]
The form factor was mentioned in the December 1998 3GPP SMG9UMTSWorking Party, which is the standards-setting body for GSM SIM cards,[24]and the form factor was agreed upon in late 2003.[27]
The micro-SIM was designed for backward compatibility. The major issue for backward compatibility was the contact area of the chip. Retaining the same contact area makes the micro-SIM compatible with the prior, larger SIM readers through the use of plastic cutout surrounds. The SIM was also designed to run at the same speed (5 MHz) as the prior version. The same size and positions of pins resulted in numerous "How-to" tutorials and YouTube videos with detailed instructions how to cut a mini-SIM card to micro-SIM size.
The chairman of EP SCP, Klaus Vedder, said[27]
ETSI has responded to a market need from ETSI customers, but additionally there is a strong desire not to invalidate, overnight, the existing interface, nor reduce the performance of the cards.
Micro-SIM cards were introduced by various mobile service providers for the launch of the original iPad, and later for smartphones, from April 2010. TheiPhone 4was the first smartphone to use a micro-SIM card in June 2010, followed by many others.[28]
After a debate in early 2012 between a few designs created by Apple,NokiaandRIM, Apple's design for an even smaller SIM card was accepted by the ETSI.[29][30]Thenano-SIM(or 4FF) card was introduced in June 2012, when mobile service providers in various countries first supplied it for phones that supported the format. The nano-SIM measures 12.3 mm × 8.8 mm × 0.67 mm (0.484 in × 0.346 in × 0.026 in) and reduces the previous format to the contact area while maintaining the existing contact arrangements.[31]A small rim of isolating material is left around the contact area to avoid short circuits with the socket. The nano-SIM can be put into adapters for use with devices designed for 2FF or 3FF SIMs, and is made thinner for that purpose,[32]and telephone companies give due warning about this.[33]4FF is 0.67 mm (0.026 in) thick, compared to the 0.76 mm (0.030 in) of its predecessors.
TheiPhone 5, released in September 2012, was the first device to use a nano-SIM card,[34]followed by other handsets.
In July 2013, Karsten Nohl, a security researcher from SRLabs, described[35][36]vulnerabilities in some SIM cards that supportedDES, which, despite its age, is still used by some operators.[36]The attack could lead to the phone being remotelyclonedor let someone steal payment credentials from the SIM.[36]Further details of the research were provided atBlackHaton 31 July 2013.[36][37]In response, theInternational Telecommunication Unionsaid that the development was "hugely significant" and that it would be contacting its members.[38]
In February 2015,The Interceptreported that theNSAandGCHQhad stolen the encryption keys (Ki's) used byGemalto(now known asThales DIS, manufacturer of 2 billion SIM cards annually)[39]), enabling these intelligence agencies to monitor voice and data communications without the knowledge or approval of cellular network providers or judicial oversight.[40]Having finished its investigation, Gemalto claimed that it has “reasonable grounds” to believe that the NSA and GCHQ carried out an operation to hack its network in 2010 and 2011, but says the number of possibly stolen keys would not have been massive.[41]
In September 2019, Cathal Mc Daid, a security researcher from Adaptive Mobile Security, described[42][43]how vulnerabilities in some SIM cards that contained the S@T Browser library were being actively exploited. This vulnerability was namedSimjacker. Attackers were using the vulnerability to track the location of thousands of mobile phone users in several countries.[44]Further details of the research were provided atVirusBulletinon 3 October 2019.[45][46]
When GSM was already in use, the specifications were further developed and enhanced with functionality such asSMSandGPRS. These development steps are referred as releases by ETSI. Within these development cycles, the SIM specification was enhanced as well: new voltage classes, formats and files were introduced.
In GSM-only times, the SIM consisted of the hardware and the software. With the advent of UMTS, this naming was split: the SIM was now an application and hence only software. The hardware part was called UICC. This split was necessary because UMTS introduced a new application, the universal subscriber identity module (USIM). The USIM brought, among other things, security improvements like mutual authentication and longer encryption keys, and an improved address book.
"SIM cards" in developed countries today are usuallyUICCscontaining at least a SIM application and a USIM application. This configuration is necessary because older GSM only handsets are solely compatible with the SIM application and some UMTS security enhancements rely on the USIM application.
OncdmaOnenetworks, the equivalent of the SIM card is theR-UIMand the equivalent of the SIM application is theCSIM.
Avirtual SIMis a mobile phone number provided by amobile network operatorthat does not require a SIM card to connect phone calls to a user's mobile phone.
An embedded SIM (eSIM) is a form of programmable SIM that is embedded directly into a device.[47]The surface mount format provides the same electrical interface as the full size, 2FF and 3FF SIM cards, but is soldered to a circuit board as part of the manufacturing process. In M2M applications where there is no requirement[15]to change the SIM card, this avoids the requirement for a connector, improving reliability and security.[citation needed]An eSIM can beprovisioned remotely; end-users can add or remove operators without the need to physically swap a SIM from the device or use multiple eSIM profiles at the same time.[48][49]
The eSIM standard, initially introduced in 2016, has progressively supplanted traditional physical SIM cards across various sectors, notably in cellular telephony.[50][51][52]In September 2017, Apple introduced the Apple Watch Series 3 featuring eSIM.[53]In October 2018, Apple introduced theiPad Pro (3rd generation),[54]which was the first iPad to support eSIM. In September 2022, Apple introduced the iPhone 14 series which was the first eSIM exclusive iPhone in the United States.[55]
An integrated SIM (iSIM) is a form of SIM directly integrated into the modem chip or main processor of the device itself. As a consequence they are smaller, cheaper and more reliable than eSIMs, they can improve security and ease the logistics and production of small devices i.e. forIoTapplications. In 2021,Deutsche Telekomintroduced thenuSIM, an "Integrated SIM for IoT".[56][57][58]
The use of SIM cards is mandatory inGSMdevices.[59][60]
Thesatellite phonenetworksIridium,ThurayaandInmarsat'sBGANalso use SIM cards. Sometimes, these SIM cards work in regular GSM phones and also allow GSM customers to roam in satellite networks by using their own SIM cards in a satellite phone.
Japan's 2GPDCsystem (which was shut down in 2012;SoftBank Mobileshut down PDC from 31 March 2010) also specified a SIM, but this has never been implemented commercially. The specification of the interface between the Mobile Equipment and the SIM is given in theRCRSTD-27 annexe 4. The Subscriber Identity Module Expert Group was a committee of specialists assembled by the European Telecommunications Standards Institute (ETSI) to draw up the specifications (GSM11.11) for interfacing between smart cards and mobile telephones. In 1994, the name SIMEG was changed to SMG9.
Japan's current and next-generation cellular systems are based on W-CDMA (UMTS) andCDMA2000and all use SIM cards. However, Japanese CDMA2000-based phones are locked to the R-UIM they are associated with and thus, the cards are not interchangeable with other Japanese CDMA2000 handsets (though they may be inserted into GSM/WCDMA handsets for roaming purposes outside Japan).
CDMA-based devices originally did not use a removable card, and the service for these phones is bound to a unique identifier contained in the handset itself. This is most prevalent in operators in the Americas. The first publication of the TIA-820 standard (also known as 3GPP2 C.S0023) in 2000 defined the Removable User Identity Module (R-UIM). Card-based CDMA devices are most prevalent in Asia.
The equivalent of a SIM inUMTSis called the universal integrated circuit card (UICC), which runs a USIM application. The UICC is still colloquially called aSIM card.[61]
The SIM card introduced a new and significant business opportunity forMVNOswho lease capacity from one of the network operators rather than owning or operating a cellular telecoms network and only provide a SIM card to their customers. MVNOs first appeared in Denmark, Hong Kong, Finland and the UK. By 2011 they existed in over 50 countries, including most of Europe, the United States, Canada, Mexico, Australia and parts of Asia, and accounted for approximately 10% of all mobile phone subscribers around the world.[62]
On some networks, the mobile phone islocked to its carrier SIM card, meaning that the phone only works with SIM cards from the specific carrier. This is more common in markets where mobile phones are heavily subsidised by the carriers, and the business model depends on the customer staying with the service provider for a minimum term (typically 12, 18 or 24 months). SIM cards that are issued by providers with an associated contract, but where the carrier does not provide a mobile device (such as a mobile phone) are calledSIM-onlydeals. Common examples are the GSM networks in the United States, Canada, Australia, and Poland. UK mobile networks ended SIM lock practices in December 2021. Many businesses offer the ability to remove the SIM lock from a phone, effectively making it possible to then use the phone on any network by inserting a different SIM card. Mostly, GSM and 3G mobile handsets can easily be unlocked and used on any suitable network with any SIM card.
In countries where the phones are not subsidised, e.g., India, Israel and Belgium, all phones are unlocked. Where the phone is not locked to its SIM card, the users can easily switch networks by simply replacing the SIM card of one network with that of another while using only one phone. This is typical, for example, among users who may want to optimise their carrier's traffic by different tariffs to different friends on different networks, or when travelling internationally.
In 2016, carriers started using the concept of automatic SIM reactivation[63]whereby they let users reuse expired SIM cards instead of purchasing new ones when they wish to re-subscribe to that operator. This is particularly useful in countries whereprepaid callsdominate and where competition drives highchurn rates, as users had to return to a carrier shop to purchase a new SIM each time they wanted to churn back to an operator.
Commonly sold as a product by mobiletelecommunicationscompanies, "SIM-only" refers to a type oflegally liabilitycontract between a mobile network provider and a customer. The contract itself takes the form of a credit agreement and is subject to a credit check.
SIM-only contracts can bepre-pay- where the subscriber buyscreditbefore use (often called pay as you go, abbreviated to PAYG), orpost-pay, where the subscriber pays in arrears, typically monthly.
Within a SIM-only contract, the mobile network provider supplies their customer with just one piece of hardware, a SIM card, which includes an agreed amount of network usage in exchange for a monthly payment. Network usage within a SIM-only contract can be measured in minutes, text, data or any combination of these. The duration of a SIM-only contract varies depending on the deal selected by the customer, but in the UK they are typically available over 1, 3, 6, 12 or 24-month periods.
SIM-only contracts differ from mobile phone contracts in that they do not include any hardware other than a SIM card. In terms of network usage, SIM-only is typically more cost-effective than other contracts because the provider does not charge more to offset the cost of a mobile device over the contract period. The short contract length is one of the key features of SIM-only – made possible by the absence of a mobile device.
SIM-only is increasing in popularity very quickly.[64]In 2010 pay monthly based mobile phone subscriptions grew from 41 percent to 49 percent of all UK mobile phone subscriptions.[65]According to German research companyGfK, 250,000 SIM-only mobile contracts were taken up in the UK during July 2012 alone, the highest figure since GfK began keeping records.
Increasing smartphone penetration combined with financial concerns is leading customers to save money by moving onto a SIM-only when their initial contract term is over.
Dual SIMdevices have two SIM card slots for the use of two SIM cards, from one or multiple carriers. Multiple SIM devices are commonplace in developing markets such as inAfrica,East Asia,South AsiaandSoutheast Asia, where variable billing rates, network coverage and speed make it desirable for consumers to use multiple SIMs from competing networks. Dual-SIM phones are also useful to separate one's personal phone number from a business phone number, without having to carry multiple devices. Some popular devices, such as theBlackBerry KeyOne, have dual-SIM variants; however, dual-SIM devices were not common in the US or Europe due to lack of demand. This has changed with mainline products from Apple and Google featuring either two SIM slots or a combination of a physical SIM slot and an eSIM.
In September 2018,AppleintroducediPhone XS,iPhone XS Max, andiPhone XRfeaturing Dual SIM (nano-SIM andeSIM) andApple Watch Series 4featuring DualeSIM.
Athin SIM(oroverlay SIMorSIM overlay) is a very thin device shaped like a SIM card, approximately 120 microns (1⁄200inch) thick. It has contacts on its front and back. It is used by placing it on top of a regular SIM card. It provides its own functionality while passing through the functionality of the SIM card underneath. It can be used to bypass the mobile operating network and run custom applications, particularly on non-programmable cell phones.[66]
Its top surface is a connector that connects to the phone in place of the normal SIM. Its bottom surface is a connector that connects to the SIM in place of the phone. With electronics, it can modify signals in either direction, thus presenting a modified SIM to the phone, and/or presenting a modified phone to the SIM. (It is a similar concept to theGame Genie, which connects between a game console and a game cartridge, creating a modified game). Similar devices have also been developed for iPhones to circumvent SIM card restrictions on carrier-locked models.[67]
In 2014,Equitel, an MVNO operated by Kenya'sEquity Bank, announced its intention to begin issuing thin SIMs to customers, raising security concerns by competition, particularly concerning the safety of mobile money accounts. However, after months of security testing and legal hearings before the country's Parliamentary Committee on Energy, Information and Communications, theCommunications Authority of Kenya(CAK) gave the bank the green light to roll out its thin SIM cards.[68]
|
https://en.wikipedia.org/wiki/Subscriber_Identity_Module
|
TheUniversal Mobile Telecommunications System(UMTS) is a3Gmobile cellular system for networks based on theGSMstandard.[1]UMTS useswideband code-division multiple access(W-CDMA) radio access technology to offer greater spectral efficiency and bandwidth tomobile network operatorscompared to previous2Gsystems likeGPRSandCSD.[2]UMTS on its provides a peak theoretical data rate of 2Mbit/s.[3]
Developed and maintained by the3GPP(3rd Generation Partnership Project), UMTS is a component of theInternational Telecommunication UnionIMT-2000standard set and compares with theCDMA2000standard set for networks based on the competingcdmaOnetechnology. The technology described in UMTS is sometimes also referred to asFreedom of Mobile Multimedia Access(FOMA)[4]or 3GSM.
UMTS specifies a complete network system, which includes theradio access network(UMTS Terrestrial Radio Access Network, or UTRAN), thecore network(Mobile Application Part, or MAP) and the authentication of users via SIM (subscriber identity module) cards. UnlikeEDGE(IMT Single-Carrier, based on GSM) and CDMA2000 (IMT Multi-Carrier), UMTS requires new base stations and new frequency allocations. UMTS has since been enhanced asHigh Speed Packet Access(HSPA).[5]
UMTS supports theoretical maximum datatransfer ratesof 42Mbit/swhenEvolved HSPA(HSPA+) is implemented in the network.[6]Users in deployed networks can expect a transfer rate of up to 384 kbit/s for Release '99 (R99) handsets (the original UMTS release), and 7.2 Mbit/s forHigh-Speed Downlink Packet Access(HSDPA) handsets in the downlink connection. These speeds are significantly faster than the 9.6 kbit/s of a single GSM error-corrected circuit switched data channel, multiple 9.6 kbit/s channels inHigh-Speed Circuit-Switched Data(HSCSD) and 14.4 kbit/s for CDMAOne channels.
Since 2006, UMTS networks in many countries have been or are in the process of being upgraded with High-Speed Downlink Packet Access (HSDPA), sometimes known as3.5G. Currently, HSDPA enablesdownlinktransfer speeds of up to 21 Mbit/s. Work is also progressing on improving the uplink transfer speed with theHigh-Speed Uplink Packet Access(HSUPA). The 3GPPLTEstandard succeeds UMTS and initially provided 4G speeds of 100 Mbit/s down and 50 Mbit/s up, with scalability up to 3 Gbps, using a next generation air interface technology based uponorthogonal frequency-division multiplexing.
The first national consumer UMTS networks launched in 2002 with a heavy emphasis on telco-provided mobile applications such as mobile TV andvideo calling. The high data speeds of UMTS are now most often utilised for Internet access: experience in Japan and elsewhere has shown that user demand for video calls is not high, and telco-provided audio/video content has declined in popularity in favour of high-speed access to the World Wide Web – either directly on a handset or connected to a computer viaWi-Fi,BluetoothorUSB.[citation needed]
UMTS combines three different terrestrialair interfaces,GSM'sMobile Application Part(MAP) core, and the GSM family ofspeech codecs.
The air interfaces are called UMTS Terrestrial Radio Access (UTRA).[7]All air interface options are part ofITU'sIMT-2000. In the currently most popular variant for cellular mobile telephones, W-CDMA (IMT Direct Spread) is used. It is also called "Uu interface", as it links User Equipment to the UMTS Terrestrial Radio Access Network.
Please note that the termsW-CDMA,TD-CDMAandTD-SCDMAare misleading. While they suggest covering just achannel access method(namely a variant ofCDMA), they are actually the common names for the whole air interface standards.[8]
W-CDMA (WCDMA; WidebandCode-Division Multiple Access), along with UMTS-FDD, UTRA-FDD, or IMT-2000 CDMA Direct Spread is an air interface standard found in3Gmobile telecommunicationsnetworks. It supports conventional cellular voice, text andMMSservices, but can also carry data at high speeds, allowing mobile operators to deliver higher bandwidth applications including streaming and broadband Internet access.[9]
W-CDMA uses theDS-CDMAchannel access method with a pair of 5 MHz wide channels. In contrast, the competingCDMA2000system uses one or more available 1.25 MHz channels for each direction of communication. W-CDMA systems are widely criticized for their large spectrum usage, which delayed deployment in countries that acted relatively slowly in allocating new frequencies specifically for 3G services (such as the United States).
The specificfrequency bandsoriginally defined by the UMTS standard are 1885–2025 MHz for the mobile-to-base (uplink) and 2110–2200 MHz for the base-to-mobile (downlink). In the US, 1710–1755 MHz and 2110–2155 MHz are used instead, as the 1900 MHz band was already used.[10]While UMTS2100 is the most widely deployed UMTS band, some countries' UMTS operators use the 850 MHz (900 MHz in Europe) and/or 1900 MHz bands (independently, meaning uplink and downlink are within the same band), notably in the US byAT&T Mobility, New Zealand byTelecom New Zealandon theXT Mobile Networkand in Australia byTelstraon theNext Gnetwork. Some carriers such asT-Mobileuse band numbers to identify the UMTS frequencies. For example, Band I (2100 MHz), Band IV (1700/2100 MHz), and Band V (850 MHz).
UMTS-FDD is an acronym for Universal Mobile Telecommunications System (UMTS) –frequency-division duplexing(FDD) and a3GPPstandardizedversion of UMTS networks that makes use of frequency-division duplexing forduplexingover an UMTS Terrestrial Radio Access (UTRA) air interface.[11]
W-CDMA is the basis of Japan'sNTT DoCoMo'sFOMAservice and the most-commonly used member of the Universal Mobile Telecommunications System (UMTS) family and sometimes used as a synonym for UMTS.[12]It uses the DS-CDMA channel access method and the FDD duplexing method to achieve higher speeds and support more users compared to most previously usedtime-division multiple access(TDMA) andtime-division duplex(TDD) schemes.
While not an evolutionary upgrade on the airside, it uses the samecore networkas the2GGSM networks deployed worldwide, allowingdual-mode mobileoperation along with GSM/EDGE; a feature it shares with other members of the UMTS family.
In the late 1990s, W-CDMA was developed by NTT DoCoMo as the air interface for their 3G networkFOMA. Later NTT DoCoMo submitted the specification to theInternational Telecommunication Union(ITU) as a candidate for the international 3G standard known as IMT-2000. The ITU eventually accepted W-CDMA as part of the IMT-2000 family of 3G standards, as an alternative to CDMA2000, EDGE, and the short rangeDECTsystem. Later, W-CDMA was selected as an air interface forUMTS.
As NTT DoCoMo did not wait for the finalisation of the 3G Release 99 specification, their network was initially incompatible with UMTS.[13]However, this has been resolved by NTT DoCoMo updating their network.
Code-Division Multiple Access communication networks have been developed by a number of companies over the years, but development of cell-phone networks based on CDMA (prior to W-CDMA) was dominated byQualcomm, the first company to succeed in developing a practical and cost-effective CDMA implementation for consumer cell phones and its earlyIS-95air interface standard has evolved into the current CDMA2000 (IS-856/IS-2000) standard. Qualcomm created an experimental wideband CDMA system called CDMA2000 3x which unified the W-CDMA (3GPP) and CDMA2000 (3GPP2) network technologies into a single design for a worldwide standard air interface. Compatibility with CDMA2000 would have beneficially enabled roaming on existing networks beyond Japan, since Qualcomm CDMA2000 networks are widely deployed, especially in the Americas, with coverage in 58 countries as of 2006[update]. However, divergent requirements resulted in the W-CDMA standard being retained and deployed globally. W-CDMA has then become the dominant technology with 457 commercial networks in 178 countries as of April 2012.[14]Several CDMA2000 operators have even converted their networks to W-CDMA for international roaming compatibility and smooth upgrade path toLTE.
Despite incompatibility with existing air-interface standards, late introduction and the high upgrade cost of deploying an all-new transmitter technology, W-CDMA has become the dominant standard.
W-CDMA transmits on a pair of 5 MHz-wide radio channels, while CDMA2000 transmits on one or several pairs of 1.25 MHz radio channels. Though W-CDMA does use adirect-sequenceCDMA transmission technique like CDMA2000, W-CDMA is not simply a wideband version of CDMA2000 and differs in many aspects from CDMA2000. From an engineering point of view, W-CDMA provides a different balance of trade-offs between cost, capacity, performance, and density[citation needed]; it also promises to achieve a benefit of reduced cost for video phone handsets. W-CDMA may also be better suited for deployment in the very dense cities of Europe and Asia. However, hurdles remain, andcross-licensingofpatentsbetween Qualcomm and W-CDMA vendors has not eliminated possible patent issues due to the features of W-CDMA which remain covered by Qualcomm patents.[15]
W-CDMA has been developed into a complete set of specifications, a detailed protocol that defines how a mobile phone communicates with the tower, how signals are modulated, how datagrams are structured, and system interfaces are specified allowing free competition on technology elements.
The world's first commercial W-CDMA service, FOMA, was launched by NTT DoCoMo inJapanin 2001.
Elsewhere, W-CDMA deployments are usually marketed under the UMTS brand.
W-CDMA has also been adapted for use in satellite communications on the U.S.Mobile User Objective Systemusing geosynchronous satellites in place of cell towers.
J-PhoneJapan (onceVodafoneand nowSoftBank Mobile) soon followed by launching their own W-CDMA based service, originally branded "Vodafone Global Standard" and claiming UMTS compatibility. The name of the service was changed to "Vodafone 3G" (now "SoftBank 3G") in December 2004.
Beginning in 2003,Hutchison Whampoagradually launched their upstart UMTS networks.
Most countries have, since the ITU approved of the 3G mobile service, either "auctioned" the radio frequencies to the company willing to pay the most, or conducted a "beauty contest" – asking the various companies to present what they intend to commit to if awarded the licences. This strategy has been criticised for aiming to drain the cash of operators to the brink of bankruptcy in order to honour their bids or proposals. Most of them have a time constraint for the rollout of the service – where a certain "coverage" must be achieved within a given date or the licence will be revoked.
Vodafone launched several UMTS networks in Europe in February 2004.MobileOneofSingaporecommercially launched its 3G (W-CDMA) services in February 2005.New Zealandin August 2005 andAustraliain October 2005.
AT&T Mobilityutilized a UMTS network, with HSPA+, from 2005 until its shutdown in February 2022.
Rogers inCanadaMarch 2007 has launched HSDPA in the Toronto Golden Horseshoe district on W-CDMA at 850/1900 MHz and plan the launch the service commercial in the top 25 cities October, 2007.
TeliaSoneraopened W-CDMA service inFinlandOctober 13, 2004, with speeds up to 384 kbit/s. Availability only in main cities. Pricing is approx. €2/MB.[citation needed]
SK TelecomandKTF, two largest mobile phone service providers inSouth Korea, have each started offering W-CDMA service in December 2003. Due to poor coverage and lack of choice in handhelds, the W-CDMA service has barely made a dent in the Korean market which was dominated by CDMA2000. By October 2006 both companies are covering more than 90 cities whileSK Telecomhas announced that it will provide nationwide coverage for its WCDMA network in order for it to offer SBSM (Single Band Single Mode) handsets by the first half of 2007.KT Freecelwill thus cut funding to its CDMA2000 network development to the minimum.
InNorway,Telenorintroduced W-CDMA in major cities by the end of 2004, while their competitor,NetCom, followed suit a few months later. Both operators have 98% national coverage on EDGE, but Telenor has parallel WLAN roaming networks on GSM, where the UMTS service is competing with this. For this reason Telenor is dropping support of their WLAN service in Austria (2006).
Maxis CommunicationsandCelcom, two mobile phone service providers inMalaysia, started offering W-CDMA services in 2005.
InSweden,Teliaintroduced W-CDMA in March 2004.
UMTS-TDD, an acronym for Universal Mobile Telecommunications System (UMTS) – time-division duplexing (TDD), is a 3GPP standardized version of UMTS networks that use UTRA-TDD.[11]UTRA-TDD is a UTRA that usestime-division duplexingfor duplexing.[11]While a full implementation of UMTS, it is mainly used to provide Internet access in circumstances similar to those whereWiMAXmight be used.[citation needed]UMTS-TDD is not directly compatible with UMTS-FDD: a device designed to use one standard cannot, unless specifically designed to, work on the other, because of the difference in air interface technologies and frequencies used.[citation needed]It is more formally as IMT-2000 CDMA-TDD or IMT 2000 Time-Division (IMT-TD).[16][17]
The two UMTS air interfaces (UTRAs) for UMTS-TDD are TD-CDMA and TD-SCDMA. Both air interfaces use a combination of two channel access methods,code-division multiple access(CDMA) and time-division multiple access (TDMA): the frequency band is divided into time slots (TDMA), which are further divided into channels using CDMA spreading codes. These air interfaces are classified as TDD, because time slots can be allocated to either uplink or downlink traffic.
TD-CDMA, an acronym for Time-Division-Code-Division Multiple Access, is a channel-access method based on usingspread-spectrummultiple-access (CDMA) across multiple time slots (TDMA). TD-CDMA is the channel access method for UTRA-TDD HCR, which is an acronym for UMTS Terrestrial Radio Access-Time Division Duplex High Chip Rate.[16]
UMTS-TDD's air interfaces that use the TD-CDMA channel access technique are standardized as UTRA-TDD HCR, which uses increments of 5MHzof spectrum, each slice divided into 10 ms frames containing fifteen time slots (1500 per second).[16]The time slots (TS) are allocated in fixed percentage for downlink and uplink. TD-CDMA is used to multiplex streams from or to multiple transceivers. Unlike W-CDMA, it does not need separate frequency bands for up- and downstream, allowing deployment in tightfrequency bands.[18]
TD-CDMA is a part of IMT-2000, defined as IMT-TD Time-Division (IMT CDMA TDD), and is one of the three UMTS air interfaces (UTRAs), as standardized by the 3GPP in UTRA-TDD HCR. UTRA-TDD HCR is closely related to W-CDMA, and provides the same types of channels where possible. UMTS's HSDPA/HSUPA enhancements are also implemented under TD-CDMA.[19]
In the United States, the technology has been used for public safety and government use in theNew York Cityand a few other areas.[needs update][20]In Japan, IPMobile planned to provide TD-CDMA service in year 2006, but it was delayed, changed to TD-SCDMA, and bankrupt before the service officially started.
Time-Division Synchronous Code-Division Multiple Access(TD-SCDMA) or UTRA TDD 1.28Mcpslow chip rate (UTRA-TDD LCR)[17][8]is an air interface[17]found in UMTS mobile telecommunications networks in China as an alternative to W-CDMA.
TD-SCDMA uses the TDMA channel access method combined with an adaptivesynchronous CDMAcomponent[17]on 1.6 MHz slices of spectrum, allowing deployment in even tighter frequency bands than TD-CDMA. It is standardized by the 3GPP and also referred to as "UTRA-TDD LCR". However, the main incentive for development of this Chinese-developed standard was avoiding or reducing the license fees that have to be paid to non-Chinese patent owners. Unlike the other air interfaces, TD-SCDMA was not part of UMTS from the beginning but has been added in Release 4 of the specification.
Like TD-CDMA, TD-SCDMA is known as IMT CDMA TDD within IMT-2000.
The term "TD-SCDMA" is misleading. While it suggests covering only a channel access method, it is actually the common name for the whole air interface specification.[8]
TD-SCDMA / UMTS-TDD (LCR) networks are incompatible with W-CDMA / UMTS-FDD and TD-CDMA / UMTS-TDD (HCR) networks.
TD-SCDMA was developed in the People's Republic of China by the Chinese Academy of Telecommunications Technology (CATT),Datang TelecomandSiemensin an attempt to avoid dependence on Western technology. This is likely primarily for practical reasons, since other 3G formats require the payment of patent fees to a large number of Western patent holders.
TD-SCDMA proponents also claim it is better suited for densely populated areas.[17]Further, it is supposed to cover all usage scenarios, whereas W-CDMA is optimised for symmetric traffic and macro cells, while TD-CDMA is best used in low mobility scenarios within micro or pico cells.[17]
TD-SCDMA is based on spread-spectrum technology which makes it unlikely that it will be able to completely escape the payment of license fees to western patent holders. The launch of a national TD-SCDMA network was initially projected by 2005[21]but only reached large scale commercial trials with 60,000 users across eight cities in 2008.[22]
On January 7, 2009, China granted a TD-SCDMA 3G licence toChina Mobile.[23]
On September 21, 2009, China Mobile officially announced that it had 1,327,000 TD-SCDMA subscribers as of the end of August, 2009.
TD-SCDMA is not commonly used outside of China.[24]
TD-SCDMA uses TDD, in contrast to the FDD scheme used byW-CDMA. By dynamically adjusting the number of timeslots used for downlink anduplink, the system can more easily accommodate asymmetric traffic with different data rate requirements on downlink and uplink than FDD schemes. Since it does not require paired spectrum for downlink and uplink, spectrum allocation flexibility is also increased. Using the same carrier frequency for uplink and downlink also means that the channel condition is the same on both directions, and thebase stationcan deduce the downlink channel information from uplink channel estimates, which is helpful to the application ofbeamformingtechniques.
TD-SCDMA also uses TDMA in addition to the CDMA used in WCDMA. This reduces the number of users in each timeslot, which reduces the implementation complexity ofmultiuser detectionand beamforming schemes, but the non-continuous transmission also reducescoverage(because of the higher peak power needed), mobility (because of lowerpower controlfrequency) and complicatesradio resource managementalgorithms.
The "S" in TD-SCDMA stands for "synchronous", which means that uplink signals are synchronized at the base station receiver, achieved by continuous timing adjustments. This reduces theinterferencebetween users of the same timeslot using different codes by improving theorthogonalitybetween the codes, therefore increasing system capacity, at the cost of some hardware complexity in achieving uplink synchronization.
On January 20, 2006,Ministry of Information Industryof the People's Republic of China formally announced that TD-SCDMA is the country's standard of 3G mobile telecommunication. On February 15, 2006, a timeline for deployment of the network in China was announced, stating pre-commercial trials would take place starting after completion of a number of test networks in select cities. These trials ran from March to October, 2006, but the results were apparently unsatisfactory. In early 2007, the Chinese government instructed the dominant cellular carrier, China Mobile, to build commercial trial networks in eight cities, and the two fixed-line carriers,China TelecomandChina Netcom, to build one each in two other cities. Construction of these trial networks was scheduled to finish during the fourth quarter of 2007, but delays meant that construction was not complete until early 2008.
The standard has been adopted by 3GPP since Rel-4, known as "UTRA TDD 1.28 Mcps Option".[17]
On March 28, 2008, China Mobile Group announced TD-SCDMA "commercial trials" for 60,000 test users in eight cities from April 1, 2008. Networks using other 3G standards (WCDMA and CDMA2000 EV/DO) had still not been launched in China, as these were delayed until TD-SCDMA was ready for commercial launch.
In January 2009, theMinistry of Industry and Information Technology(MIIT) in China took the unusual step of assigning licences for 3 different third-generation mobile phone standards to three carriers in a long-awaited step that is expected to prompt $41 billion in spending on new equipment. The Chinese-developed standard, TD-SCDMA, was assigned to China Mobile, the world's biggest phone carrier by subscribers. That appeared to be an effort to make sure the new system has the financial and technical backing to succeed. Licences for two existing 3G standards, W-CDMA andCDMA2000 1xEV-DO, were assigned toChina Unicomand China Telecom, respectively. Third-generation, or 3G, technology supports Web surfing, wireless video and other services and the start of service is expected to spur new revenue growth.
The technical split by MIIT has hampered the performance of China Mobile in the 3G market, with users and China Mobile engineers alike pointing to the lack of suitable handsets to use on the network.[25]Deployment of base stations has also been slow, resulting in lack of improvement of service for users.[26]The network connection itself has consistently been slower than that from the other two carriers, leading to a sharp decline in market share. By 2011 China Mobile has already moved its focus onto TD-LTE.[27][28]Gradual closures of TD-SCDMA stations started in 2016.[29][30]
The following is a list ofmobile telecommunicationsnetworks using third-generation TD-SCDMA / UMTS-TDD (LCR) technology.
In Europe,CEPTallocated the 2010–2020 MHz range for a variant of UMTS-TDD designed for unlicensed, self-provided use.[33]Some telecom groups and jurisdictions have proposed withdrawing this service in favour of licensed UMTS-TDD,[34]due to lack of demand, and lack of development of a UMTS TDD air interface technology suitable for deployment in this band.
Ordinary UMTS uses UTRA-FDD as an air interface and is known asUMTS-FDD. UMTS-FDD uses W-CDMA for multiple access andfrequency-division duplexfor duplexing, meaning that the up-link and down-link transmit on different frequencies. UMTS is usually transmitted on frequencies assigned for1G,2G, or 3G mobile telephone service in the countries of operation.
UMTS-TDD uses time-division duplexing, allowing the up-link and down-link to share the same spectrum. This allows the operator to more flexibly divide the usage of available spectrum according to traffic patterns. For ordinary phone service, you would expect the up-link and down-link to carry approximately equal amounts of data (because every phone call needs a voice transmission in either direction), but Internet-oriented traffic is more frequently one-way. For example, when browsing a website, the user will send commands, which are short, to the server, but the server will send whole files, that are generally larger than those commands, in response.
UMTS-TDD tends to be allocated frequency intended for mobile/wireless Internet services rather than used on existing cellular frequencies. This is, in part, because TDD duplexing is not normally allowed oncellular,PCS/PCN, and 3G frequencies. TDD technologies open up the usage of left-over unpaired spectrum.
Europe-wide, several bands are provided either specifically for UMTS-TDD or for similar technologies. These are 1900 MHz and 1920 MHz and between 2010 MHz and 2025 MHz. In several countries the 2500–2690 MHz band (also known as MMDS in the USA) have been used for UMTS-TDD deployments. Additionally, spectrum around the 3.5 GHz range has been allocated in some countries, notably Britain, in a technology-neutral environment. In the Czech Republic UTMS-TDD is also used in a frequency range around 872 MHz.[35]
UMTS-TDD has been deployed for public and/or private networks in at least nineteen countries around the world, with live systems in, amongst other countries, Australia, Czech Republic, France, Germany, Japan, New Zealand, Botswana, South Africa, the UK, and the USA.
Deployments in the US thus far have been limited. It has been selected for a public safety support network used by emergency responders in New York,[36]but outside of some experimental systems, notably one fromNextel, thus far the WiMAX standard appears to have gained greater traction as a general mobile Internet access system.
A variety of Internet-access systems exist which provide broadband speed access to the net. These include WiMAX andHIPERMAN. UMTS-TDD has the advantages of being able to use an operator's existing UMTS/GSM infrastructure, should it have one, and that it includes UMTS modes optimized for circuit switching should, for example, the operator want to offer telephone service. UMTS-TDD's performance is also more consistent. However, UMTS-TDD deployers often have regulatory problems with taking advantage of some of the services UMTS compatibility provides. For example, the UMTS-TDD spectrum in the UK cannot be used to provide telephone service, though the regulatorOFCOMis discussing the possibility of allowing it at some point in the future. Few operators considering UMTS-TDD have existing UMTS/GSM infrastructure.
Additionally, the WiMAX and HIPERMAN systems provide significantly larger bandwidths when the mobile station is near the tower.
Like most mobile Internet access systems, many users who might otherwise choose UMTS-TDD will find their needs covered by the ad hoc collection of unconnected Wi-Fi access points at many restaurants and transportation hubs, and/or by Internet access already provided by their mobile phone operator. By comparison, UMTS-TDD (and systems like WiMAX) offers mobile, and more consistent, access than the former, and generally faster access than the latter.
UMTS also specifies the Universal Terrestrial Radio Access Network (UTRAN), which is composed of multiple base stations, possibly using different terrestrial air interface standards and frequency bands.
UMTS and GSM/EDGE can share a Core Network (CN), making UTRAN an alternative radio access network toGERAN(GSM/EDGE RAN), and allowing (mostly) transparent switching between the RANs according to available coverage and service needs. Because of that, UMTS's and GSM/EDGE's radio access networks are sometimes collectively referred to as UTRAN/GERAN.
UMTS networks are often combined with GSM/EDGE, the latter of which is also a part of IMT-2000.
The UE (User Equipment) interface of theRAN(Radio Access Network) primarily consists ofRRC(Radio Resource Control),PDCP(Packet Data Convergence Protocol),RLC(Radio Link Control) and MAC (Media Access Control) protocols. RRC protocol handles connection establishment, measurements, radio bearer services, security and handover decisions. RLC protocol primarily divides into three Modes – Transparent Mode (TM), Unacknowledge Mode (UM), Acknowledge Mode (AM). The functionality of AM entity resembles TCP operation whereas UM operation resembles UDP operation. In TM mode, data will be sent to lower layers without adding any header toSDUof higher layers. MAC handles the scheduling of data on air interface depending on higher layer (RRC) configured parameters.
The set of properties related to data transmission is called Radio Bearer (RB). This set of properties decides the maximum allowed data in a TTI (Transmission Time Interval). RB includes RLC information and RB mapping. RB mapping decides the mapping between RB<->logical channel<->transport channel. Signaling messages are sent on Signaling Radio Bearers (SRBs) and data packets (either CS or PS) are sent on data RBs. RRC andNASmessages go on SRBs.
Security includes two procedures: integrity and ciphering. Integrity validates the resource of messages and also makes sure that no one (third/unknown party) on the radio interface has modified the messages. Ciphering ensures that no one listens to your data on the air interface. Both integrity and ciphering are applied for SRBs whereas only ciphering is applied for data RBs.
With Mobile Application Part, UMTS uses the same core network standard as GSM/EDGE. This allows a simple migration for existing GSM operators. However, the migration path to UMTS is still costly: while much of the core infrastructure is shared with GSM, the cost of obtaining new spectrum licenses and overlaying UMTS at existing towers is high.
The CN can be connected to variousbackbone networks, such as theInternetor anIntegrated Services Digital Network(ISDN) telephone network. UMTS (and GERAN) include the three lowest layers ofOSI model. The network layer (OSI 3) includes theRadio Resource Managementprotocol (RRM) that manages the bearer channels between the mobile terminals and the fixed network, including the handovers.
A UARFCN (abbreviationfor UTRA Absolute Radio Frequency Channel Number, where UTRA stands for UMTS Terrestrial Radio Access) is used to identify a frequency in theUMTS frequency bands.
Typically channel number is derived from the frequency in MHz through the formula Channel Number = Frequency * 5. However, this is only able to represent channels that are centered on a multiple of 200 kHz, which do not align with licensing in North America. 3GPP added several special values for the common North American channels.
Over 130 licenses had been awarded to operators worldwide, as of December 2004, specifying W-CDMA radio access technology that builds on GSM. In Europe, the license process occurred at the tail end of the technology bubble, and the auction mechanisms for allocation set up in some countries resulted in some extremely high prices being paid for the original 2100 MHz licenses, notably in the UK and Germany. InGermany, bidders paid a total €50.8 billion for six licenses, two of which were subsequently abandoned and written off by their purchasers (Mobilcom and theSonera/Telefónicaconsortium). It has been suggested that these huge license fees have the character of a very large tax paid on future income expected many years down the road. In any event, the high prices paid put some European telecom operators close to bankruptcy (most notablyKPN). Over the last few years some operators have written off some or all of the license costs. Between 2007 and 2009, all three Finnish carriers began to use 900 MHz UMTS in a shared arrangement with its surrounding 2G GSM base stations for rural area coverage, a trend that is expected to expand over Europe in the next 1–3 years.[needs update]
The 2100 MHz band (downlink around 2100 MHz and uplink around 1900 MHz) allocated for UMTS in Europe and most of Asia is already used in North America. The 1900 MHz range is used for 2G (PCS) services, and 2100 MHz range is used for satellite communications. Regulators have, however, freed up some of the 2100 MHz range for 3G services, together with a different range around 1700 MHz for the uplink.[needs update]
AT&T Wireless launched UMTS services in the United States by the end of 2004 strictly using the existing 1900 MHz spectrum allocated for 2G PCS services. Cingular acquired AT&T Wireless in 2004 and has since then launched UMTS in select US cities. Cingular renamed itself AT&T Mobility and rolled out[37]some cities with a UMTS network at 850 MHz to enhance its existing UMTS network at 1900 MHz and now offers subscribers a number of dual-band UMTS 850/1900 phones.
T-Mobile's rollout of UMTS in the US was originally focused on the 1700 MHz band. However, T-Mobile has been moving users from 1700 MHz to 1900 MHz (PCS) in order to reallocate the spectrum to 4GLTEservices.[38]
In Canada, UMTS coverage is being provided on the 850 MHz and 1900 MHz bands on the Rogers and Bell-Telus networks. Bell and Telus share the network. Recently, new providersWind Mobile,MobilicityandVideotronhave begun operations in the 1700 MHz band.
In 2008, Australian telco Telstra replaced its existing CDMA network with a national UMTS-based 3G network, branded asNextG, operating in the 850 MHz band. Telstra currently provides UMTS service on this network, and also on the 2100 MHz UMTS network, through a co-ownership of the owning and administrating company 3GIS. This company is also co-owned byHutchison 3G Australia, and this is the primary network used by their customers.Optusis currently rolling out a 3G network operating on the 2100 MHz band in cities and most large towns, and the 900 MHz band in regional areas.Vodafoneis also building a 3G network using the 900 MHz band.
In India,BSNLhas started its 3G services since October 2009, beginning with the larger cities and then expanding over to smaller cities. The 850 MHz and 900 MHz bands provide greater coverage compared to equivalent 1700/1900/2100 MHz networks, and are best suited to regional areas where greater distances separate base station and subscriber.
Carriers in South America are now also rolling out 850 MHz networks.
UMTS phones (and data cards) are highly portable – they have been designed to roam easily onto other UMTS networks (if the providers have roaming agreements in place). In addition, almost all UMTS phones are UMTS/GSM dual-mode devices, so if a UMTS phone travels outside of UMTS coverage during a call the call may be transparently handed off to available GSM coverage. Roaming charges are usually significantly higher than regular usage charges.
Most UMTS licensees consider ubiquitous, transparent globalroamingan important issue. To enable a high degree of interoperability, UMTS phones usually support several different frequencies in addition to their GSM fallback. Different countries support different UMTS frequency bands – Europe initially used 2100 MHz while the most carriers in the USA use 850 MHz and 1900 MHz. T-Mobile has launched a network in the US operating at 1700 MHz (uplink) /2100 MHz (downlink), and these bands also have been adopted elsewhere in the US and in Canada and Latin America. A UMTS phone and network must support a common frequency to work together. Because of the frequencies used, early models of UMTS phones designated for the United States will likely not be operable elsewhere and vice versa. There are now 11 different frequency combinations used around the world – including frequencies formerly used solely for 2G services.
UMTS phones can use aUniversal Subscriber Identity Module, USIM (based on GSM'sSIM card) and also work (including UMTS services) with GSM SIM cards. This is a global standard of identification, and enables a network to identify and authenticate the (U)SIM in the phone. Roaming agreements between networks allow for calls to a customer to be redirected to them while roaming and determine the services (and prices) available to the user. In addition to user subscriber information and authentication information, the (U)SIM provides storage space for phone book contact. Handsets can store their data on their own memory or on the (U)SIM card (which is usually more limited in its phone book contact information). A (U)SIM can be moved to another UMTS or GSM phone, and the phone will take on the user details of the (U)SIM, meaning it is the (U)SIM (not the phone) which determines the phone number of the phone and the billing for calls made from the phone.
Japan was the first country to adopt 3G technologies, and since they had not used GSM previously they had no need to build GSM compatibility into their handsets and their 3G handsets were smaller than those available elsewhere. In 2002, NTT DoCoMo's FOMA 3G network was the first commercial UMTS network – using a pre-release specification,[39]it was initially incompatible with the UMTS standard at the radio level but used standard USIM cards, meaning USIM card based roaming was possible (transferring the USIM card into a UMTS or GSM phone when travelling). Both NTT DoCoMo and SoftBank Mobile (which launched 3G in December 2002) now use standard UMTS.
All of the major 2G phone manufacturers (that are still in business) are now manufacturers of 3G phones. The early 3G handsets and modems were specific to the frequencies required in their country, which meant they could only roam to other countries on the same 3G frequency (though they can fall back to the older GSM standard). Canada and USA have a common share of frequencies, as do most European countries. The article UMTS frequency bands is an overview of UMTS network frequencies around the world.
Using acellular router, PCMCIA or USB card, customers are able to access 3G broadband services, regardless of their choice of computer (such as atablet PCor aPDA). Some softwareinstalls itselffrom the modem, so that in some cases absolutely no knowledge of technology is required to getonlinein moments. Using a phone that supports 3G and Bluetooth 2.0, multiple Bluetooth-capable laptops can be connected to the Internet. Some smartphones can also act as a mobileWLAN access point.
There are very few 3G phones or modems available supporting all 3G frequencies (UMTS850/900/1700/1900/2100 MHz). In 2010, Nokia released a range of phones withPentaband3G coverage, including theN8andE7. Many other phones are offering more than one band which still enables extensive roaming. For example, Apple'siPhone 4contains a quadband chipset operating on 850/900/1900/2100 MHz, allowing usage in the majority of countries where UMTS-FDD is deployed.
The main competitor to UMTS is CDMA2000 (IMT-MC), which is developed by the3GPP2. Unlike UMTS, CDMA2000 is an evolutionary upgrade to an existing 2G standard, cdmaOne, and is able to operate within the same frequency allocations. This and CDMA2000's narrower bandwidth requirements make it easier to deploy in existing spectra. In some, but not all, cases, existing GSM operators only have enough spectrum to implement either UMTS or GSM, not both. For example, in the US D, E, and F PCS spectrum blocks, the amount of spectrum available is 5 MHz in each direction. A standard UMTS system would saturate that spectrum. Where CDMA2000 is deployed, it usually co-exists with UMTS. In many markets however, the co-existence issue is of little relevance, as legislative hurdles exist to co-deploying two standards in the same licensed slice of spectrum.
Another competitor to UMTS isEDGE(IMT-SC), which is an evolutionary upgrade to the 2G GSM system, leveraging existing GSM spectrums. It is also much easier, quicker, and considerably cheaper for wireless carriers to "bolt-on" EDGE functionality by upgrading their existing GSM transmission hardware to support EDGE rather than having to install almost all brand-new equipment to deliver UMTS. However, being developed by 3GPP just as UMTS, EDGE is not a true competitor. Instead, it is used as a temporary solution preceding UMTS roll-out or as a complement for rural areas. This is facilitated by the fact that GSM/EDGE and UMTS specifications are jointly developed and rely on the same core network, allowing dual-mode operation includingvertical handovers.
China'sTD-SCDMAstandard is often seen as a competitor, too. TD-SCDMA has been added to UMTS' Release 4 as UTRA-TDD 1.28 Mcps Low Chip Rate (UTRA-TDD LCR). UnlikeTD-CDMA(UTRA-TDD 3.84 Mcps High Chip Rate, UTRA-TDD HCR) which complements W-CDMA (UTRA-FDD), it is suitable for both micro and macrocells. However, the lack of vendors' support is preventing it from being a real competitor.
While DECT is technically capable of competing with UMTS and other cellular networks in densely populated, urban areas, it has only been deployed for domestic cordless phones and private in-house networks.
All of these competitors have been accepted by ITU as part of the IMT-2000 family of 3G standards, along with UMTS-FDD.
On the Internet access side, competing systems include WiMAX andFlash-OFDM.
From a GSM/GPRS network, the following network elements can be reused:
From a GSM/GPRS communication radio network, the following elements cannot be reused:
They can remain in the network and be used in dual network operation where 2G and 3G networks co-exist while network migration and new 3G terminals become available for use in the network.
The UMTS network introduces new network elements that function as specified by 3GPP:
The functionality of MSC changes when going to UMTS. In a GSM system the MSC handles all the circuit switched operations like connecting A- and B-subscriber through the network. In UMTS the Media gateway (MGW) takes care of data transfer in circuit switched networks. MSC controls MGW operations.
Some countries, including the United States, have allocated spectrum differently from theITUrecommendations, so that the standard bands most commonly used for UMTS (UMTS-2100) have not been available.[citation needed]In those countries, alternative bands are used, preventing the interoperability of existing UMTS-2100 equipment, and requiring the design and manufacture of different equipment for the use in these markets. As is the case with GSM900 today[when?], standard UMTS 2100 MHz equipment will not work in those markets. However, it appears as though UMTS is not suffering as much from handset band compatibility issues as GSM did, as many UMTS handsets are multi-band in both UMTS and GSM modes. Penta-band (850, 900, 1700, 2100, and 1900 MHz bands), quad-band GSM (850, 900, 1800, and 1900 MHz bands) and tri-band UMTS (850, 1900, and 2100 MHz bands) handsets are becoming more commonplace.[40]
In its early days[when?], UMTS had problems in many countries: Overweight handsets with poor battery life were first to arrive on a market highly sensitive to weight and form factor.[citation needed]The Motorola A830, a debut handset on Hutchison's 3 network, weighed more than 200 grams and even featured a detachable camera to reduce handset weight. Another significant issue involved call reliability, related to problems with handover from UMTS to GSM. Customers found their connections being dropped as handovers were possible only in one direction (UMTS → GSM), with the handset only changing back to UMTS after hanging up. In most networks around the world this is no longer an issue.[citation needed]
Compared to GSM, UMTS networks initially required a higherbase stationdensity. For fully-fledged UMTS incorporatingvideo on demandfeatures, one base station needed to be set up every 1–1.5 km (0.62–0.93 mi). This was the case when only the 2100 MHz band was being used, however with the growing use of lower-frequency bands (such as 850 and 900 MHz) this is no longer so. This has led to increasing rollout of the lower-band networks by operators since 2006.[citation needed]
Even with current technologies and low-band UMTS, telephony and data over UMTS requires more power than on comparable GSM networks.Apple Inc.cited[41]UMTS power consumption as the reason that the first generationiPhoneonly supported EDGE. Their release of the iPhone 3G quotes talk time on UMTS as half that available when the handset is set to use GSM. Other manufacturers indicate different battery lifetime for UMTS mode compared to GSM mode as well. As battery and network technology improve, this issue is diminishing.
As early as 2008, it was known that carrier networks can be used to surreptitiously gather user location information.[42]In August 2014, theWashington Postreported on widespread marketing of surveillance systems usingSignalling System No. 7(SS7) protocols to locate callers anywhere in the world.[42]
In December 2014, news broke that SS7's very own functions can be repurposed for surveillance, because of its relaxed security, in order to listen to calls in real time or to record encrypted calls and texts for later decryption, or to defraud users and cellular carriers.[43]
Deutsche Telekomand Vodafone declared the same day that they had fixed gaps in their networks, but that the problem is global and can only be fixed with a telecommunication system-wide solution.[44]
The evolution of UMTS progresses according to planned releases. Each release is designed to introduce new features and improve upon existing ones.
|
https://en.wikipedia.org/wiki/UMTS-TDD
|
TheUMTS frequency bandsareradio frequenciesused bythird generation (3G)wirelessUniversal Mobile Telecommunications Systemnetworks. They were allocated by delegates to theWorld Administrative Radio Conference(WARC-92) held in Málaga-Torremolinos, Spain between 3 February 1992 and 3 March 1992.[1]Resolution 212 (Rev.WRC-97), adopted at the World Radiocommunication Conference held in Geneva, Switzerland in 1997, endorsed the bands specifically for the International Mobile Telecommunications-2000 (IMT-2000) specification by referring to S5.388, which states "The bands 1,885-2,025 MHz and 2,110-2,200 MHz are intended for use, on a worldwide basis, by administrations wishing to implement International Mobile Telecommunications 2000 (IMT-2000). Such use does not preclude the use of these bands by other services to which they are allocated. The bands should be made available for IMT-2000 in accordance with Resolution 212 (Rev. WRC-97)." To accommodate the reality that these initially defined bands were already in use in various regions of the world, the initial allocation has been amended multiple times to include other radio frequency bands.[2][3]
From Tables 5.0 "UTRA FDD frequency bands" of the latest published version of the 3GPP TS 25.101,[4]the following table lists the specified frequency bands ofUMTS (FDD):
The following table shows the standardized UMTS bands and their regional use. The main UMTS bands are inboldprint.
UMTS-TDDtechnology is standardized for usage in the following bands:[5]
|
https://en.wikipedia.org/wiki/UMTS_frequency_bands
|
TheUMTS channelsare communication channels used bythird generation (3G)wirelessUniversal Mobile Telecommunications System(UMTS) networks.[1][2][3]UMTS channels can be divided into three levels:
|
https://en.wikipedia.org/wiki/UMTS_channels
|
TheUniversal Mobile Telecommunications System(UMTS) is a3Gmobile cellular system for networks based on theGSMstandard.[1]UMTS useswideband code-division multiple access(W-CDMA) radio access technology to offer greater spectral efficiency and bandwidth tomobile network operatorscompared to previous2Gsystems likeGPRSandCSD.[2]UMTS on its provides a peak theoretical data rate of 2Mbit/s.[3]
Developed and maintained by the3GPP(3rd Generation Partnership Project), UMTS is a component of theInternational Telecommunication UnionIMT-2000standard set and compares with theCDMA2000standard set for networks based on the competingcdmaOnetechnology. The technology described in UMTS is sometimes also referred to asFreedom of Mobile Multimedia Access(FOMA)[4]or 3GSM.
UMTS specifies a complete network system, which includes theradio access network(UMTS Terrestrial Radio Access Network, or UTRAN), thecore network(Mobile Application Part, or MAP) and the authentication of users via SIM (subscriber identity module) cards. UnlikeEDGE(IMT Single-Carrier, based on GSM) and CDMA2000 (IMT Multi-Carrier), UMTS requires new base stations and new frequency allocations. UMTS has since been enhanced asHigh Speed Packet Access(HSPA).[5]
UMTS supports theoretical maximum datatransfer ratesof 42Mbit/swhenEvolved HSPA(HSPA+) is implemented in the network.[6]Users in deployed networks can expect a transfer rate of up to 384 kbit/s for Release '99 (R99) handsets (the original UMTS release), and 7.2 Mbit/s forHigh-Speed Downlink Packet Access(HSDPA) handsets in the downlink connection. These speeds are significantly faster than the 9.6 kbit/s of a single GSM error-corrected circuit switched data channel, multiple 9.6 kbit/s channels inHigh-Speed Circuit-Switched Data(HSCSD) and 14.4 kbit/s for CDMAOne channels.
Since 2006, UMTS networks in many countries have been or are in the process of being upgraded with High-Speed Downlink Packet Access (HSDPA), sometimes known as3.5G. Currently, HSDPA enablesdownlinktransfer speeds of up to 21 Mbit/s. Work is also progressing on improving the uplink transfer speed with theHigh-Speed Uplink Packet Access(HSUPA). The 3GPPLTEstandard succeeds UMTS and initially provided 4G speeds of 100 Mbit/s down and 50 Mbit/s up, with scalability up to 3 Gbps, using a next generation air interface technology based uponorthogonal frequency-division multiplexing.
The first national consumer UMTS networks launched in 2002 with a heavy emphasis on telco-provided mobile applications such as mobile TV andvideo calling. The high data speeds of UMTS are now most often utilised for Internet access: experience in Japan and elsewhere has shown that user demand for video calls is not high, and telco-provided audio/video content has declined in popularity in favour of high-speed access to the World Wide Web – either directly on a handset or connected to a computer viaWi-Fi,BluetoothorUSB.[citation needed]
UMTS combines three different terrestrialair interfaces,GSM'sMobile Application Part(MAP) core, and the GSM family ofspeech codecs.
The air interfaces are called UMTS Terrestrial Radio Access (UTRA).[7]All air interface options are part ofITU'sIMT-2000. In the currently most popular variant for cellular mobile telephones, W-CDMA (IMT Direct Spread) is used. It is also called "Uu interface", as it links User Equipment to the UMTS Terrestrial Radio Access Network.
Please note that the termsW-CDMA,TD-CDMAandTD-SCDMAare misleading. While they suggest covering just achannel access method(namely a variant ofCDMA), they are actually the common names for the whole air interface standards.[8]
W-CDMA (WCDMA; WidebandCode-Division Multiple Access), along with UMTS-FDD, UTRA-FDD, or IMT-2000 CDMA Direct Spread is an air interface standard found in3Gmobile telecommunicationsnetworks. It supports conventional cellular voice, text andMMSservices, but can also carry data at high speeds, allowing mobile operators to deliver higher bandwidth applications including streaming and broadband Internet access.[9]
W-CDMA uses theDS-CDMAchannel access method with a pair of 5 MHz wide channels. In contrast, the competingCDMA2000system uses one or more available 1.25 MHz channels for each direction of communication. W-CDMA systems are widely criticized for their large spectrum usage, which delayed deployment in countries that acted relatively slowly in allocating new frequencies specifically for 3G services (such as the United States).
The specificfrequency bandsoriginally defined by the UMTS standard are 1885–2025 MHz for the mobile-to-base (uplink) and 2110–2200 MHz for the base-to-mobile (downlink). In the US, 1710–1755 MHz and 2110–2155 MHz are used instead, as the 1900 MHz band was already used.[10]While UMTS2100 is the most widely deployed UMTS band, some countries' UMTS operators use the 850 MHz (900 MHz in Europe) and/or 1900 MHz bands (independently, meaning uplink and downlink are within the same band), notably in the US byAT&T Mobility, New Zealand byTelecom New Zealandon theXT Mobile Networkand in Australia byTelstraon theNext Gnetwork. Some carriers such asT-Mobileuse band numbers to identify the UMTS frequencies. For example, Band I (2100 MHz), Band IV (1700/2100 MHz), and Band V (850 MHz).
UMTS-FDD is an acronym for Universal Mobile Telecommunications System (UMTS) –frequency-division duplexing(FDD) and a3GPPstandardizedversion of UMTS networks that makes use of frequency-division duplexing forduplexingover an UMTS Terrestrial Radio Access (UTRA) air interface.[11]
W-CDMA is the basis of Japan'sNTT DoCoMo'sFOMAservice and the most-commonly used member of the Universal Mobile Telecommunications System (UMTS) family and sometimes used as a synonym for UMTS.[12]It uses the DS-CDMA channel access method and the FDD duplexing method to achieve higher speeds and support more users compared to most previously usedtime-division multiple access(TDMA) andtime-division duplex(TDD) schemes.
While not an evolutionary upgrade on the airside, it uses the samecore networkas the2GGSM networks deployed worldwide, allowingdual-mode mobileoperation along with GSM/EDGE; a feature it shares with other members of the UMTS family.
In the late 1990s, W-CDMA was developed by NTT DoCoMo as the air interface for their 3G networkFOMA. Later NTT DoCoMo submitted the specification to theInternational Telecommunication Union(ITU) as a candidate for the international 3G standard known as IMT-2000. The ITU eventually accepted W-CDMA as part of the IMT-2000 family of 3G standards, as an alternative to CDMA2000, EDGE, and the short rangeDECTsystem. Later, W-CDMA was selected as an air interface forUMTS.
As NTT DoCoMo did not wait for the finalisation of the 3G Release 99 specification, their network was initially incompatible with UMTS.[13]However, this has been resolved by NTT DoCoMo updating their network.
Code-Division Multiple Access communication networks have been developed by a number of companies over the years, but development of cell-phone networks based on CDMA (prior to W-CDMA) was dominated byQualcomm, the first company to succeed in developing a practical and cost-effective CDMA implementation for consumer cell phones and its earlyIS-95air interface standard has evolved into the current CDMA2000 (IS-856/IS-2000) standard. Qualcomm created an experimental wideband CDMA system called CDMA2000 3x which unified the W-CDMA (3GPP) and CDMA2000 (3GPP2) network technologies into a single design for a worldwide standard air interface. Compatibility with CDMA2000 would have beneficially enabled roaming on existing networks beyond Japan, since Qualcomm CDMA2000 networks are widely deployed, especially in the Americas, with coverage in 58 countries as of 2006[update]. However, divergent requirements resulted in the W-CDMA standard being retained and deployed globally. W-CDMA has then become the dominant technology with 457 commercial networks in 178 countries as of April 2012.[14]Several CDMA2000 operators have even converted their networks to W-CDMA for international roaming compatibility and smooth upgrade path toLTE.
Despite incompatibility with existing air-interface standards, late introduction and the high upgrade cost of deploying an all-new transmitter technology, W-CDMA has become the dominant standard.
W-CDMA transmits on a pair of 5 MHz-wide radio channels, while CDMA2000 transmits on one or several pairs of 1.25 MHz radio channels. Though W-CDMA does use adirect-sequenceCDMA transmission technique like CDMA2000, W-CDMA is not simply a wideband version of CDMA2000 and differs in many aspects from CDMA2000. From an engineering point of view, W-CDMA provides a different balance of trade-offs between cost, capacity, performance, and density[citation needed]; it also promises to achieve a benefit of reduced cost for video phone handsets. W-CDMA may also be better suited for deployment in the very dense cities of Europe and Asia. However, hurdles remain, andcross-licensingofpatentsbetween Qualcomm and W-CDMA vendors has not eliminated possible patent issues due to the features of W-CDMA which remain covered by Qualcomm patents.[15]
W-CDMA has been developed into a complete set of specifications, a detailed protocol that defines how a mobile phone communicates with the tower, how signals are modulated, how datagrams are structured, and system interfaces are specified allowing free competition on technology elements.
The world's first commercial W-CDMA service, FOMA, was launched by NTT DoCoMo inJapanin 2001.
Elsewhere, W-CDMA deployments are usually marketed under the UMTS brand.
W-CDMA has also been adapted for use in satellite communications on the U.S.Mobile User Objective Systemusing geosynchronous satellites in place of cell towers.
J-PhoneJapan (onceVodafoneand nowSoftBank Mobile) soon followed by launching their own W-CDMA based service, originally branded "Vodafone Global Standard" and claiming UMTS compatibility. The name of the service was changed to "Vodafone 3G" (now "SoftBank 3G") in December 2004.
Beginning in 2003,Hutchison Whampoagradually launched their upstart UMTS networks.
Most countries have, since the ITU approved of the 3G mobile service, either "auctioned" the radio frequencies to the company willing to pay the most, or conducted a "beauty contest" – asking the various companies to present what they intend to commit to if awarded the licences. This strategy has been criticised for aiming to drain the cash of operators to the brink of bankruptcy in order to honour their bids or proposals. Most of them have a time constraint for the rollout of the service – where a certain "coverage" must be achieved within a given date or the licence will be revoked.
Vodafone launched several UMTS networks in Europe in February 2004.MobileOneofSingaporecommercially launched its 3G (W-CDMA) services in February 2005.New Zealandin August 2005 andAustraliain October 2005.
AT&T Mobilityutilized a UMTS network, with HSPA+, from 2005 until its shutdown in February 2022.
Rogers inCanadaMarch 2007 has launched HSDPA in the Toronto Golden Horseshoe district on W-CDMA at 850/1900 MHz and plan the launch the service commercial in the top 25 cities October, 2007.
TeliaSoneraopened W-CDMA service inFinlandOctober 13, 2004, with speeds up to 384 kbit/s. Availability only in main cities. Pricing is approx. €2/MB.[citation needed]
SK TelecomandKTF, two largest mobile phone service providers inSouth Korea, have each started offering W-CDMA service in December 2003. Due to poor coverage and lack of choice in handhelds, the W-CDMA service has barely made a dent in the Korean market which was dominated by CDMA2000. By October 2006 both companies are covering more than 90 cities whileSK Telecomhas announced that it will provide nationwide coverage for its WCDMA network in order for it to offer SBSM (Single Band Single Mode) handsets by the first half of 2007.KT Freecelwill thus cut funding to its CDMA2000 network development to the minimum.
InNorway,Telenorintroduced W-CDMA in major cities by the end of 2004, while their competitor,NetCom, followed suit a few months later. Both operators have 98% national coverage on EDGE, but Telenor has parallel WLAN roaming networks on GSM, where the UMTS service is competing with this. For this reason Telenor is dropping support of their WLAN service in Austria (2006).
Maxis CommunicationsandCelcom, two mobile phone service providers inMalaysia, started offering W-CDMA services in 2005.
InSweden,Teliaintroduced W-CDMA in March 2004.
UMTS-TDD, an acronym for Universal Mobile Telecommunications System (UMTS) – time-division duplexing (TDD), is a 3GPP standardized version of UMTS networks that use UTRA-TDD.[11]UTRA-TDD is a UTRA that usestime-division duplexingfor duplexing.[11]While a full implementation of UMTS, it is mainly used to provide Internet access in circumstances similar to those whereWiMAXmight be used.[citation needed]UMTS-TDD is not directly compatible with UMTS-FDD: a device designed to use one standard cannot, unless specifically designed to, work on the other, because of the difference in air interface technologies and frequencies used.[citation needed]It is more formally as IMT-2000 CDMA-TDD or IMT 2000 Time-Division (IMT-TD).[16][17]
The two UMTS air interfaces (UTRAs) for UMTS-TDD are TD-CDMA and TD-SCDMA. Both air interfaces use a combination of two channel access methods,code-division multiple access(CDMA) and time-division multiple access (TDMA): the frequency band is divided into time slots (TDMA), which are further divided into channels using CDMA spreading codes. These air interfaces are classified as TDD, because time slots can be allocated to either uplink or downlink traffic.
TD-CDMA, an acronym for Time-Division-Code-Division Multiple Access, is a channel-access method based on usingspread-spectrummultiple-access (CDMA) across multiple time slots (TDMA). TD-CDMA is the channel access method for UTRA-TDD HCR, which is an acronym for UMTS Terrestrial Radio Access-Time Division Duplex High Chip Rate.[16]
UMTS-TDD's air interfaces that use the TD-CDMA channel access technique are standardized as UTRA-TDD HCR, which uses increments of 5MHzof spectrum, each slice divided into 10 ms frames containing fifteen time slots (1500 per second).[16]The time slots (TS) are allocated in fixed percentage for downlink and uplink. TD-CDMA is used to multiplex streams from or to multiple transceivers. Unlike W-CDMA, it does not need separate frequency bands for up- and downstream, allowing deployment in tightfrequency bands.[18]
TD-CDMA is a part of IMT-2000, defined as IMT-TD Time-Division (IMT CDMA TDD), and is one of the three UMTS air interfaces (UTRAs), as standardized by the 3GPP in UTRA-TDD HCR. UTRA-TDD HCR is closely related to W-CDMA, and provides the same types of channels where possible. UMTS's HSDPA/HSUPA enhancements are also implemented under TD-CDMA.[19]
In the United States, the technology has been used for public safety and government use in theNew York Cityand a few other areas.[needs update][20]In Japan, IPMobile planned to provide TD-CDMA service in year 2006, but it was delayed, changed to TD-SCDMA, and bankrupt before the service officially started.
Time-Division Synchronous Code-Division Multiple Access(TD-SCDMA) or UTRA TDD 1.28Mcpslow chip rate (UTRA-TDD LCR)[17][8]is an air interface[17]found in UMTS mobile telecommunications networks in China as an alternative to W-CDMA.
TD-SCDMA uses the TDMA channel access method combined with an adaptivesynchronous CDMAcomponent[17]on 1.6 MHz slices of spectrum, allowing deployment in even tighter frequency bands than TD-CDMA. It is standardized by the 3GPP and also referred to as "UTRA-TDD LCR". However, the main incentive for development of this Chinese-developed standard was avoiding or reducing the license fees that have to be paid to non-Chinese patent owners. Unlike the other air interfaces, TD-SCDMA was not part of UMTS from the beginning but has been added in Release 4 of the specification.
Like TD-CDMA, TD-SCDMA is known as IMT CDMA TDD within IMT-2000.
The term "TD-SCDMA" is misleading. While it suggests covering only a channel access method, it is actually the common name for the whole air interface specification.[8]
TD-SCDMA / UMTS-TDD (LCR) networks are incompatible with W-CDMA / UMTS-FDD and TD-CDMA / UMTS-TDD (HCR) networks.
TD-SCDMA was developed in the People's Republic of China by the Chinese Academy of Telecommunications Technology (CATT),Datang TelecomandSiemensin an attempt to avoid dependence on Western technology. This is likely primarily for practical reasons, since other 3G formats require the payment of patent fees to a large number of Western patent holders.
TD-SCDMA proponents also claim it is better suited for densely populated areas.[17]Further, it is supposed to cover all usage scenarios, whereas W-CDMA is optimised for symmetric traffic and macro cells, while TD-CDMA is best used in low mobility scenarios within micro or pico cells.[17]
TD-SCDMA is based on spread-spectrum technology which makes it unlikely that it will be able to completely escape the payment of license fees to western patent holders. The launch of a national TD-SCDMA network was initially projected by 2005[21]but only reached large scale commercial trials with 60,000 users across eight cities in 2008.[22]
On January 7, 2009, China granted a TD-SCDMA 3G licence toChina Mobile.[23]
On September 21, 2009, China Mobile officially announced that it had 1,327,000 TD-SCDMA subscribers as of the end of August, 2009.
TD-SCDMA is not commonly used outside of China.[24]
TD-SCDMA uses TDD, in contrast to the FDD scheme used byW-CDMA. By dynamically adjusting the number of timeslots used for downlink anduplink, the system can more easily accommodate asymmetric traffic with different data rate requirements on downlink and uplink than FDD schemes. Since it does not require paired spectrum for downlink and uplink, spectrum allocation flexibility is also increased. Using the same carrier frequency for uplink and downlink also means that the channel condition is the same on both directions, and thebase stationcan deduce the downlink channel information from uplink channel estimates, which is helpful to the application ofbeamformingtechniques.
TD-SCDMA also uses TDMA in addition to the CDMA used in WCDMA. This reduces the number of users in each timeslot, which reduces the implementation complexity ofmultiuser detectionand beamforming schemes, but the non-continuous transmission also reducescoverage(because of the higher peak power needed), mobility (because of lowerpower controlfrequency) and complicatesradio resource managementalgorithms.
The "S" in TD-SCDMA stands for "synchronous", which means that uplink signals are synchronized at the base station receiver, achieved by continuous timing adjustments. This reduces theinterferencebetween users of the same timeslot using different codes by improving theorthogonalitybetween the codes, therefore increasing system capacity, at the cost of some hardware complexity in achieving uplink synchronization.
On January 20, 2006,Ministry of Information Industryof the People's Republic of China formally announced that TD-SCDMA is the country's standard of 3G mobile telecommunication. On February 15, 2006, a timeline for deployment of the network in China was announced, stating pre-commercial trials would take place starting after completion of a number of test networks in select cities. These trials ran from March to October, 2006, but the results were apparently unsatisfactory. In early 2007, the Chinese government instructed the dominant cellular carrier, China Mobile, to build commercial trial networks in eight cities, and the two fixed-line carriers,China TelecomandChina Netcom, to build one each in two other cities. Construction of these trial networks was scheduled to finish during the fourth quarter of 2007, but delays meant that construction was not complete until early 2008.
The standard has been adopted by 3GPP since Rel-4, known as "UTRA TDD 1.28 Mcps Option".[17]
On March 28, 2008, China Mobile Group announced TD-SCDMA "commercial trials" for 60,000 test users in eight cities from April 1, 2008. Networks using other 3G standards (WCDMA and CDMA2000 EV/DO) had still not been launched in China, as these were delayed until TD-SCDMA was ready for commercial launch.
In January 2009, theMinistry of Industry and Information Technology(MIIT) in China took the unusual step of assigning licences for 3 different third-generation mobile phone standards to three carriers in a long-awaited step that is expected to prompt $41 billion in spending on new equipment. The Chinese-developed standard, TD-SCDMA, was assigned to China Mobile, the world's biggest phone carrier by subscribers. That appeared to be an effort to make sure the new system has the financial and technical backing to succeed. Licences for two existing 3G standards, W-CDMA andCDMA2000 1xEV-DO, were assigned toChina Unicomand China Telecom, respectively. Third-generation, or 3G, technology supports Web surfing, wireless video and other services and the start of service is expected to spur new revenue growth.
The technical split by MIIT has hampered the performance of China Mobile in the 3G market, with users and China Mobile engineers alike pointing to the lack of suitable handsets to use on the network.[25]Deployment of base stations has also been slow, resulting in lack of improvement of service for users.[26]The network connection itself has consistently been slower than that from the other two carriers, leading to a sharp decline in market share. By 2011 China Mobile has already moved its focus onto TD-LTE.[27][28]Gradual closures of TD-SCDMA stations started in 2016.[29][30]
The following is a list ofmobile telecommunicationsnetworks using third-generation TD-SCDMA / UMTS-TDD (LCR) technology.
In Europe,CEPTallocated the 2010–2020 MHz range for a variant of UMTS-TDD designed for unlicensed, self-provided use.[33]Some telecom groups and jurisdictions have proposed withdrawing this service in favour of licensed UMTS-TDD,[34]due to lack of demand, and lack of development of a UMTS TDD air interface technology suitable for deployment in this band.
Ordinary UMTS uses UTRA-FDD as an air interface and is known asUMTS-FDD. UMTS-FDD uses W-CDMA for multiple access andfrequency-division duplexfor duplexing, meaning that the up-link and down-link transmit on different frequencies. UMTS is usually transmitted on frequencies assigned for1G,2G, or 3G mobile telephone service in the countries of operation.
UMTS-TDD uses time-division duplexing, allowing the up-link and down-link to share the same spectrum. This allows the operator to more flexibly divide the usage of available spectrum according to traffic patterns. For ordinary phone service, you would expect the up-link and down-link to carry approximately equal amounts of data (because every phone call needs a voice transmission in either direction), but Internet-oriented traffic is more frequently one-way. For example, when browsing a website, the user will send commands, which are short, to the server, but the server will send whole files, that are generally larger than those commands, in response.
UMTS-TDD tends to be allocated frequency intended for mobile/wireless Internet services rather than used on existing cellular frequencies. This is, in part, because TDD duplexing is not normally allowed oncellular,PCS/PCN, and 3G frequencies. TDD technologies open up the usage of left-over unpaired spectrum.
Europe-wide, several bands are provided either specifically for UMTS-TDD or for similar technologies. These are 1900 MHz and 1920 MHz and between 2010 MHz and 2025 MHz. In several countries the 2500–2690 MHz band (also known as MMDS in the USA) have been used for UMTS-TDD deployments. Additionally, spectrum around the 3.5 GHz range has been allocated in some countries, notably Britain, in a technology-neutral environment. In the Czech Republic UTMS-TDD is also used in a frequency range around 872 MHz.[35]
UMTS-TDD has been deployed for public and/or private networks in at least nineteen countries around the world, with live systems in, amongst other countries, Australia, Czech Republic, France, Germany, Japan, New Zealand, Botswana, South Africa, the UK, and the USA.
Deployments in the US thus far have been limited. It has been selected for a public safety support network used by emergency responders in New York,[36]but outside of some experimental systems, notably one fromNextel, thus far the WiMAX standard appears to have gained greater traction as a general mobile Internet access system.
A variety of Internet-access systems exist which provide broadband speed access to the net. These include WiMAX andHIPERMAN. UMTS-TDD has the advantages of being able to use an operator's existing UMTS/GSM infrastructure, should it have one, and that it includes UMTS modes optimized for circuit switching should, for example, the operator want to offer telephone service. UMTS-TDD's performance is also more consistent. However, UMTS-TDD deployers often have regulatory problems with taking advantage of some of the services UMTS compatibility provides. For example, the UMTS-TDD spectrum in the UK cannot be used to provide telephone service, though the regulatorOFCOMis discussing the possibility of allowing it at some point in the future. Few operators considering UMTS-TDD have existing UMTS/GSM infrastructure.
Additionally, the WiMAX and HIPERMAN systems provide significantly larger bandwidths when the mobile station is near the tower.
Like most mobile Internet access systems, many users who might otherwise choose UMTS-TDD will find their needs covered by the ad hoc collection of unconnected Wi-Fi access points at many restaurants and transportation hubs, and/or by Internet access already provided by their mobile phone operator. By comparison, UMTS-TDD (and systems like WiMAX) offers mobile, and more consistent, access than the former, and generally faster access than the latter.
UMTS also specifies the Universal Terrestrial Radio Access Network (UTRAN), which is composed of multiple base stations, possibly using different terrestrial air interface standards and frequency bands.
UMTS and GSM/EDGE can share a Core Network (CN), making UTRAN an alternative radio access network toGERAN(GSM/EDGE RAN), and allowing (mostly) transparent switching between the RANs according to available coverage and service needs. Because of that, UMTS's and GSM/EDGE's radio access networks are sometimes collectively referred to as UTRAN/GERAN.
UMTS networks are often combined with GSM/EDGE, the latter of which is also a part of IMT-2000.
The UE (User Equipment) interface of theRAN(Radio Access Network) primarily consists ofRRC(Radio Resource Control),PDCP(Packet Data Convergence Protocol),RLC(Radio Link Control) and MAC (Media Access Control) protocols. RRC protocol handles connection establishment, measurements, radio bearer services, security and handover decisions. RLC protocol primarily divides into three Modes – Transparent Mode (TM), Unacknowledge Mode (UM), Acknowledge Mode (AM). The functionality of AM entity resembles TCP operation whereas UM operation resembles UDP operation. In TM mode, data will be sent to lower layers without adding any header toSDUof higher layers. MAC handles the scheduling of data on air interface depending on higher layer (RRC) configured parameters.
The set of properties related to data transmission is called Radio Bearer (RB). This set of properties decides the maximum allowed data in a TTI (Transmission Time Interval). RB includes RLC information and RB mapping. RB mapping decides the mapping between RB<->logical channel<->transport channel. Signaling messages are sent on Signaling Radio Bearers (SRBs) and data packets (either CS or PS) are sent on data RBs. RRC andNASmessages go on SRBs.
Security includes two procedures: integrity and ciphering. Integrity validates the resource of messages and also makes sure that no one (third/unknown party) on the radio interface has modified the messages. Ciphering ensures that no one listens to your data on the air interface. Both integrity and ciphering are applied for SRBs whereas only ciphering is applied for data RBs.
With Mobile Application Part, UMTS uses the same core network standard as GSM/EDGE. This allows a simple migration for existing GSM operators. However, the migration path to UMTS is still costly: while much of the core infrastructure is shared with GSM, the cost of obtaining new spectrum licenses and overlaying UMTS at existing towers is high.
The CN can be connected to variousbackbone networks, such as theInternetor anIntegrated Services Digital Network(ISDN) telephone network. UMTS (and GERAN) include the three lowest layers ofOSI model. The network layer (OSI 3) includes theRadio Resource Managementprotocol (RRM) that manages the bearer channels between the mobile terminals and the fixed network, including the handovers.
A UARFCN (abbreviationfor UTRA Absolute Radio Frequency Channel Number, where UTRA stands for UMTS Terrestrial Radio Access) is used to identify a frequency in theUMTS frequency bands.
Typically channel number is derived from the frequency in MHz through the formula Channel Number = Frequency * 5. However, this is only able to represent channels that are centered on a multiple of 200 kHz, which do not align with licensing in North America. 3GPP added several special values for the common North American channels.
Over 130 licenses had been awarded to operators worldwide, as of December 2004, specifying W-CDMA radio access technology that builds on GSM. In Europe, the license process occurred at the tail end of the technology bubble, and the auction mechanisms for allocation set up in some countries resulted in some extremely high prices being paid for the original 2100 MHz licenses, notably in the UK and Germany. InGermany, bidders paid a total €50.8 billion for six licenses, two of which were subsequently abandoned and written off by their purchasers (Mobilcom and theSonera/Telefónicaconsortium). It has been suggested that these huge license fees have the character of a very large tax paid on future income expected many years down the road. In any event, the high prices paid put some European telecom operators close to bankruptcy (most notablyKPN). Over the last few years some operators have written off some or all of the license costs. Between 2007 and 2009, all three Finnish carriers began to use 900 MHz UMTS in a shared arrangement with its surrounding 2G GSM base stations for rural area coverage, a trend that is expected to expand over Europe in the next 1–3 years.[needs update]
The 2100 MHz band (downlink around 2100 MHz and uplink around 1900 MHz) allocated for UMTS in Europe and most of Asia is already used in North America. The 1900 MHz range is used for 2G (PCS) services, and 2100 MHz range is used for satellite communications. Regulators have, however, freed up some of the 2100 MHz range for 3G services, together with a different range around 1700 MHz for the uplink.[needs update]
AT&T Wireless launched UMTS services in the United States by the end of 2004 strictly using the existing 1900 MHz spectrum allocated for 2G PCS services. Cingular acquired AT&T Wireless in 2004 and has since then launched UMTS in select US cities. Cingular renamed itself AT&T Mobility and rolled out[37]some cities with a UMTS network at 850 MHz to enhance its existing UMTS network at 1900 MHz and now offers subscribers a number of dual-band UMTS 850/1900 phones.
T-Mobile's rollout of UMTS in the US was originally focused on the 1700 MHz band. However, T-Mobile has been moving users from 1700 MHz to 1900 MHz (PCS) in order to reallocate the spectrum to 4GLTEservices.[38]
In Canada, UMTS coverage is being provided on the 850 MHz and 1900 MHz bands on the Rogers and Bell-Telus networks. Bell and Telus share the network. Recently, new providersWind Mobile,MobilicityandVideotronhave begun operations in the 1700 MHz band.
In 2008, Australian telco Telstra replaced its existing CDMA network with a national UMTS-based 3G network, branded asNextG, operating in the 850 MHz band. Telstra currently provides UMTS service on this network, and also on the 2100 MHz UMTS network, through a co-ownership of the owning and administrating company 3GIS. This company is also co-owned byHutchison 3G Australia, and this is the primary network used by their customers.Optusis currently rolling out a 3G network operating on the 2100 MHz band in cities and most large towns, and the 900 MHz band in regional areas.Vodafoneis also building a 3G network using the 900 MHz band.
In India,BSNLhas started its 3G services since October 2009, beginning with the larger cities and then expanding over to smaller cities. The 850 MHz and 900 MHz bands provide greater coverage compared to equivalent 1700/1900/2100 MHz networks, and are best suited to regional areas where greater distances separate base station and subscriber.
Carriers in South America are now also rolling out 850 MHz networks.
UMTS phones (and data cards) are highly portable – they have been designed to roam easily onto other UMTS networks (if the providers have roaming agreements in place). In addition, almost all UMTS phones are UMTS/GSM dual-mode devices, so if a UMTS phone travels outside of UMTS coverage during a call the call may be transparently handed off to available GSM coverage. Roaming charges are usually significantly higher than regular usage charges.
Most UMTS licensees consider ubiquitous, transparent globalroamingan important issue. To enable a high degree of interoperability, UMTS phones usually support several different frequencies in addition to their GSM fallback. Different countries support different UMTS frequency bands – Europe initially used 2100 MHz while the most carriers in the USA use 850 MHz and 1900 MHz. T-Mobile has launched a network in the US operating at 1700 MHz (uplink) /2100 MHz (downlink), and these bands also have been adopted elsewhere in the US and in Canada and Latin America. A UMTS phone and network must support a common frequency to work together. Because of the frequencies used, early models of UMTS phones designated for the United States will likely not be operable elsewhere and vice versa. There are now 11 different frequency combinations used around the world – including frequencies formerly used solely for 2G services.
UMTS phones can use aUniversal Subscriber Identity Module, USIM (based on GSM'sSIM card) and also work (including UMTS services) with GSM SIM cards. This is a global standard of identification, and enables a network to identify and authenticate the (U)SIM in the phone. Roaming agreements between networks allow for calls to a customer to be redirected to them while roaming and determine the services (and prices) available to the user. In addition to user subscriber information and authentication information, the (U)SIM provides storage space for phone book contact. Handsets can store their data on their own memory or on the (U)SIM card (which is usually more limited in its phone book contact information). A (U)SIM can be moved to another UMTS or GSM phone, and the phone will take on the user details of the (U)SIM, meaning it is the (U)SIM (not the phone) which determines the phone number of the phone and the billing for calls made from the phone.
Japan was the first country to adopt 3G technologies, and since they had not used GSM previously they had no need to build GSM compatibility into their handsets and their 3G handsets were smaller than those available elsewhere. In 2002, NTT DoCoMo's FOMA 3G network was the first commercial UMTS network – using a pre-release specification,[39]it was initially incompatible with the UMTS standard at the radio level but used standard USIM cards, meaning USIM card based roaming was possible (transferring the USIM card into a UMTS or GSM phone when travelling). Both NTT DoCoMo and SoftBank Mobile (which launched 3G in December 2002) now use standard UMTS.
All of the major 2G phone manufacturers (that are still in business) are now manufacturers of 3G phones. The early 3G handsets and modems were specific to the frequencies required in their country, which meant they could only roam to other countries on the same 3G frequency (though they can fall back to the older GSM standard). Canada and USA have a common share of frequencies, as do most European countries. The article UMTS frequency bands is an overview of UMTS network frequencies around the world.
Using acellular router, PCMCIA or USB card, customers are able to access 3G broadband services, regardless of their choice of computer (such as atablet PCor aPDA). Some softwareinstalls itselffrom the modem, so that in some cases absolutely no knowledge of technology is required to getonlinein moments. Using a phone that supports 3G and Bluetooth 2.0, multiple Bluetooth-capable laptops can be connected to the Internet. Some smartphones can also act as a mobileWLAN access point.
There are very few 3G phones or modems available supporting all 3G frequencies (UMTS850/900/1700/1900/2100 MHz). In 2010, Nokia released a range of phones withPentaband3G coverage, including theN8andE7. Many other phones are offering more than one band which still enables extensive roaming. For example, Apple'siPhone 4contains a quadband chipset operating on 850/900/1900/2100 MHz, allowing usage in the majority of countries where UMTS-FDD is deployed.
The main competitor to UMTS is CDMA2000 (IMT-MC), which is developed by the3GPP2. Unlike UMTS, CDMA2000 is an evolutionary upgrade to an existing 2G standard, cdmaOne, and is able to operate within the same frequency allocations. This and CDMA2000's narrower bandwidth requirements make it easier to deploy in existing spectra. In some, but not all, cases, existing GSM operators only have enough spectrum to implement either UMTS or GSM, not both. For example, in the US D, E, and F PCS spectrum blocks, the amount of spectrum available is 5 MHz in each direction. A standard UMTS system would saturate that spectrum. Where CDMA2000 is deployed, it usually co-exists with UMTS. In many markets however, the co-existence issue is of little relevance, as legislative hurdles exist to co-deploying two standards in the same licensed slice of spectrum.
Another competitor to UMTS isEDGE(IMT-SC), which is an evolutionary upgrade to the 2G GSM system, leveraging existing GSM spectrums. It is also much easier, quicker, and considerably cheaper for wireless carriers to "bolt-on" EDGE functionality by upgrading their existing GSM transmission hardware to support EDGE rather than having to install almost all brand-new equipment to deliver UMTS. However, being developed by 3GPP just as UMTS, EDGE is not a true competitor. Instead, it is used as a temporary solution preceding UMTS roll-out or as a complement for rural areas. This is facilitated by the fact that GSM/EDGE and UMTS specifications are jointly developed and rely on the same core network, allowing dual-mode operation includingvertical handovers.
China'sTD-SCDMAstandard is often seen as a competitor, too. TD-SCDMA has been added to UMTS' Release 4 as UTRA-TDD 1.28 Mcps Low Chip Rate (UTRA-TDD LCR). UnlikeTD-CDMA(UTRA-TDD 3.84 Mcps High Chip Rate, UTRA-TDD HCR) which complements W-CDMA (UTRA-FDD), it is suitable for both micro and macrocells. However, the lack of vendors' support is preventing it from being a real competitor.
While DECT is technically capable of competing with UMTS and other cellular networks in densely populated, urban areas, it has only been deployed for domestic cordless phones and private in-house networks.
All of these competitors have been accepted by ITU as part of the IMT-2000 family of 3G standards, along with UMTS-FDD.
On the Internet access side, competing systems include WiMAX andFlash-OFDM.
From a GSM/GPRS network, the following network elements can be reused:
From a GSM/GPRS communication radio network, the following elements cannot be reused:
They can remain in the network and be used in dual network operation where 2G and 3G networks co-exist while network migration and new 3G terminals become available for use in the network.
The UMTS network introduces new network elements that function as specified by 3GPP:
The functionality of MSC changes when going to UMTS. In a GSM system the MSC handles all the circuit switched operations like connecting A- and B-subscriber through the network. In UMTS the Media gateway (MGW) takes care of data transfer in circuit switched networks. MSC controls MGW operations.
Some countries, including the United States, have allocated spectrum differently from theITUrecommendations, so that the standard bands most commonly used for UMTS (UMTS-2100) have not been available.[citation needed]In those countries, alternative bands are used, preventing the interoperability of existing UMTS-2100 equipment, and requiring the design and manufacture of different equipment for the use in these markets. As is the case with GSM900 today[when?], standard UMTS 2100 MHz equipment will not work in those markets. However, it appears as though UMTS is not suffering as much from handset band compatibility issues as GSM did, as many UMTS handsets are multi-band in both UMTS and GSM modes. Penta-band (850, 900, 1700, 2100, and 1900 MHz bands), quad-band GSM (850, 900, 1800, and 1900 MHz bands) and tri-band UMTS (850, 1900, and 2100 MHz bands) handsets are becoming more commonplace.[40]
In its early days[when?], UMTS had problems in many countries: Overweight handsets with poor battery life were first to arrive on a market highly sensitive to weight and form factor.[citation needed]The Motorola A830, a debut handset on Hutchison's 3 network, weighed more than 200 grams and even featured a detachable camera to reduce handset weight. Another significant issue involved call reliability, related to problems with handover from UMTS to GSM. Customers found their connections being dropped as handovers were possible only in one direction (UMTS → GSM), with the handset only changing back to UMTS after hanging up. In most networks around the world this is no longer an issue.[citation needed]
Compared to GSM, UMTS networks initially required a higherbase stationdensity. For fully-fledged UMTS incorporatingvideo on demandfeatures, one base station needed to be set up every 1–1.5 km (0.62–0.93 mi). This was the case when only the 2100 MHz band was being used, however with the growing use of lower-frequency bands (such as 850 and 900 MHz) this is no longer so. This has led to increasing rollout of the lower-band networks by operators since 2006.[citation needed]
Even with current technologies and low-band UMTS, telephony and data over UMTS requires more power than on comparable GSM networks.Apple Inc.cited[41]UMTS power consumption as the reason that the first generationiPhoneonly supported EDGE. Their release of the iPhone 3G quotes talk time on UMTS as half that available when the handset is set to use GSM. Other manufacturers indicate different battery lifetime for UMTS mode compared to GSM mode as well. As battery and network technology improve, this issue is diminishing.
As early as 2008, it was known that carrier networks can be used to surreptitiously gather user location information.[42]In August 2014, theWashington Postreported on widespread marketing of surveillance systems usingSignalling System No. 7(SS7) protocols to locate callers anywhere in the world.[42]
In December 2014, news broke that SS7's very own functions can be repurposed for surveillance, because of its relaxed security, in order to listen to calls in real time or to record encrypted calls and texts for later decryption, or to defraud users and cellular carriers.[43]
Deutsche Telekomand Vodafone declared the same day that they had fixed gaps in their networks, but that the problem is global and can only be fixed with a telecommunication system-wide solution.[44]
The evolution of UMTS progresses according to planned releases. Each release is designed to introduce new features and improve upon existing ones.
|
https://en.wikipedia.org/wiki/W-CDMA
|
TheUniversal Mobile Telecommunications System(UMTS) is a3Gmobile cellular system for networks based on theGSMstandard.[1]UMTS useswideband code-division multiple access(W-CDMA) radio access technology to offer greater spectral efficiency and bandwidth tomobile network operatorscompared to previous2Gsystems likeGPRSandCSD.[2]UMTS on its provides a peak theoretical data rate of 2Mbit/s.[3]
Developed and maintained by the3GPP(3rd Generation Partnership Project), UMTS is a component of theInternational Telecommunication UnionIMT-2000standard set and compares with theCDMA2000standard set for networks based on the competingcdmaOnetechnology. The technology described in UMTS is sometimes also referred to asFreedom of Mobile Multimedia Access(FOMA)[4]or 3GSM.
UMTS specifies a complete network system, which includes theradio access network(UMTS Terrestrial Radio Access Network, or UTRAN), thecore network(Mobile Application Part, or MAP) and the authentication of users via SIM (subscriber identity module) cards. UnlikeEDGE(IMT Single-Carrier, based on GSM) and CDMA2000 (IMT Multi-Carrier), UMTS requires new base stations and new frequency allocations. UMTS has since been enhanced asHigh Speed Packet Access(HSPA).[5]
UMTS supports theoretical maximum datatransfer ratesof 42Mbit/swhenEvolved HSPA(HSPA+) is implemented in the network.[6]Users in deployed networks can expect a transfer rate of up to 384 kbit/s for Release '99 (R99) handsets (the original UMTS release), and 7.2 Mbit/s forHigh-Speed Downlink Packet Access(HSDPA) handsets in the downlink connection. These speeds are significantly faster than the 9.6 kbit/s of a single GSM error-corrected circuit switched data channel, multiple 9.6 kbit/s channels inHigh-Speed Circuit-Switched Data(HSCSD) and 14.4 kbit/s for CDMAOne channels.
Since 2006, UMTS networks in many countries have been or are in the process of being upgraded with High-Speed Downlink Packet Access (HSDPA), sometimes known as3.5G. Currently, HSDPA enablesdownlinktransfer speeds of up to 21 Mbit/s. Work is also progressing on improving the uplink transfer speed with theHigh-Speed Uplink Packet Access(HSUPA). The 3GPPLTEstandard succeeds UMTS and initially provided 4G speeds of 100 Mbit/s down and 50 Mbit/s up, with scalability up to 3 Gbps, using a next generation air interface technology based uponorthogonal frequency-division multiplexing.
The first national consumer UMTS networks launched in 2002 with a heavy emphasis on telco-provided mobile applications such as mobile TV andvideo calling. The high data speeds of UMTS are now most often utilised for Internet access: experience in Japan and elsewhere has shown that user demand for video calls is not high, and telco-provided audio/video content has declined in popularity in favour of high-speed access to the World Wide Web – either directly on a handset or connected to a computer viaWi-Fi,BluetoothorUSB.[citation needed]
UMTS combines three different terrestrialair interfaces,GSM'sMobile Application Part(MAP) core, and the GSM family ofspeech codecs.
The air interfaces are called UMTS Terrestrial Radio Access (UTRA).[7]All air interface options are part ofITU'sIMT-2000. In the currently most popular variant for cellular mobile telephones, W-CDMA (IMT Direct Spread) is used. It is also called "Uu interface", as it links User Equipment to the UMTS Terrestrial Radio Access Network.
Please note that the termsW-CDMA,TD-CDMAandTD-SCDMAare misleading. While they suggest covering just achannel access method(namely a variant ofCDMA), they are actually the common names for the whole air interface standards.[8]
W-CDMA (WCDMA; WidebandCode-Division Multiple Access), along with UMTS-FDD, UTRA-FDD, or IMT-2000 CDMA Direct Spread is an air interface standard found in3Gmobile telecommunicationsnetworks. It supports conventional cellular voice, text andMMSservices, but can also carry data at high speeds, allowing mobile operators to deliver higher bandwidth applications including streaming and broadband Internet access.[9]
W-CDMA uses theDS-CDMAchannel access method with a pair of 5 MHz wide channels. In contrast, the competingCDMA2000system uses one or more available 1.25 MHz channels for each direction of communication. W-CDMA systems are widely criticized for their large spectrum usage, which delayed deployment in countries that acted relatively slowly in allocating new frequencies specifically for 3G services (such as the United States).
The specificfrequency bandsoriginally defined by the UMTS standard are 1885–2025 MHz for the mobile-to-base (uplink) and 2110–2200 MHz for the base-to-mobile (downlink). In the US, 1710–1755 MHz and 2110–2155 MHz are used instead, as the 1900 MHz band was already used.[10]While UMTS2100 is the most widely deployed UMTS band, some countries' UMTS operators use the 850 MHz (900 MHz in Europe) and/or 1900 MHz bands (independently, meaning uplink and downlink are within the same band), notably in the US byAT&T Mobility, New Zealand byTelecom New Zealandon theXT Mobile Networkand in Australia byTelstraon theNext Gnetwork. Some carriers such asT-Mobileuse band numbers to identify the UMTS frequencies. For example, Band I (2100 MHz), Band IV (1700/2100 MHz), and Band V (850 MHz).
UMTS-FDD is an acronym for Universal Mobile Telecommunications System (UMTS) –frequency-division duplexing(FDD) and a3GPPstandardizedversion of UMTS networks that makes use of frequency-division duplexing forduplexingover an UMTS Terrestrial Radio Access (UTRA) air interface.[11]
W-CDMA is the basis of Japan'sNTT DoCoMo'sFOMAservice and the most-commonly used member of the Universal Mobile Telecommunications System (UMTS) family and sometimes used as a synonym for UMTS.[12]It uses the DS-CDMA channel access method and the FDD duplexing method to achieve higher speeds and support more users compared to most previously usedtime-division multiple access(TDMA) andtime-division duplex(TDD) schemes.
While not an evolutionary upgrade on the airside, it uses the samecore networkas the2GGSM networks deployed worldwide, allowingdual-mode mobileoperation along with GSM/EDGE; a feature it shares with other members of the UMTS family.
In the late 1990s, W-CDMA was developed by NTT DoCoMo as the air interface for their 3G networkFOMA. Later NTT DoCoMo submitted the specification to theInternational Telecommunication Union(ITU) as a candidate for the international 3G standard known as IMT-2000. The ITU eventually accepted W-CDMA as part of the IMT-2000 family of 3G standards, as an alternative to CDMA2000, EDGE, and the short rangeDECTsystem. Later, W-CDMA was selected as an air interface forUMTS.
As NTT DoCoMo did not wait for the finalisation of the 3G Release 99 specification, their network was initially incompatible with UMTS.[13]However, this has been resolved by NTT DoCoMo updating their network.
Code-Division Multiple Access communication networks have been developed by a number of companies over the years, but development of cell-phone networks based on CDMA (prior to W-CDMA) was dominated byQualcomm, the first company to succeed in developing a practical and cost-effective CDMA implementation for consumer cell phones and its earlyIS-95air interface standard has evolved into the current CDMA2000 (IS-856/IS-2000) standard. Qualcomm created an experimental wideband CDMA system called CDMA2000 3x which unified the W-CDMA (3GPP) and CDMA2000 (3GPP2) network technologies into a single design for a worldwide standard air interface. Compatibility with CDMA2000 would have beneficially enabled roaming on existing networks beyond Japan, since Qualcomm CDMA2000 networks are widely deployed, especially in the Americas, with coverage in 58 countries as of 2006[update]. However, divergent requirements resulted in the W-CDMA standard being retained and deployed globally. W-CDMA has then become the dominant technology with 457 commercial networks in 178 countries as of April 2012.[14]Several CDMA2000 operators have even converted their networks to W-CDMA for international roaming compatibility and smooth upgrade path toLTE.
Despite incompatibility with existing air-interface standards, late introduction and the high upgrade cost of deploying an all-new transmitter technology, W-CDMA has become the dominant standard.
W-CDMA transmits on a pair of 5 MHz-wide radio channels, while CDMA2000 transmits on one or several pairs of 1.25 MHz radio channels. Though W-CDMA does use adirect-sequenceCDMA transmission technique like CDMA2000, W-CDMA is not simply a wideband version of CDMA2000 and differs in many aspects from CDMA2000. From an engineering point of view, W-CDMA provides a different balance of trade-offs between cost, capacity, performance, and density[citation needed]; it also promises to achieve a benefit of reduced cost for video phone handsets. W-CDMA may also be better suited for deployment in the very dense cities of Europe and Asia. However, hurdles remain, andcross-licensingofpatentsbetween Qualcomm and W-CDMA vendors has not eliminated possible patent issues due to the features of W-CDMA which remain covered by Qualcomm patents.[15]
W-CDMA has been developed into a complete set of specifications, a detailed protocol that defines how a mobile phone communicates with the tower, how signals are modulated, how datagrams are structured, and system interfaces are specified allowing free competition on technology elements.
The world's first commercial W-CDMA service, FOMA, was launched by NTT DoCoMo inJapanin 2001.
Elsewhere, W-CDMA deployments are usually marketed under the UMTS brand.
W-CDMA has also been adapted for use in satellite communications on the U.S.Mobile User Objective Systemusing geosynchronous satellites in place of cell towers.
J-PhoneJapan (onceVodafoneand nowSoftBank Mobile) soon followed by launching their own W-CDMA based service, originally branded "Vodafone Global Standard" and claiming UMTS compatibility. The name of the service was changed to "Vodafone 3G" (now "SoftBank 3G") in December 2004.
Beginning in 2003,Hutchison Whampoagradually launched their upstart UMTS networks.
Most countries have, since the ITU approved of the 3G mobile service, either "auctioned" the radio frequencies to the company willing to pay the most, or conducted a "beauty contest" – asking the various companies to present what they intend to commit to if awarded the licences. This strategy has been criticised for aiming to drain the cash of operators to the brink of bankruptcy in order to honour their bids or proposals. Most of them have a time constraint for the rollout of the service – where a certain "coverage" must be achieved within a given date or the licence will be revoked.
Vodafone launched several UMTS networks in Europe in February 2004.MobileOneofSingaporecommercially launched its 3G (W-CDMA) services in February 2005.New Zealandin August 2005 andAustraliain October 2005.
AT&T Mobilityutilized a UMTS network, with HSPA+, from 2005 until its shutdown in February 2022.
Rogers inCanadaMarch 2007 has launched HSDPA in the Toronto Golden Horseshoe district on W-CDMA at 850/1900 MHz and plan the launch the service commercial in the top 25 cities October, 2007.
TeliaSoneraopened W-CDMA service inFinlandOctober 13, 2004, with speeds up to 384 kbit/s. Availability only in main cities. Pricing is approx. €2/MB.[citation needed]
SK TelecomandKTF, two largest mobile phone service providers inSouth Korea, have each started offering W-CDMA service in December 2003. Due to poor coverage and lack of choice in handhelds, the W-CDMA service has barely made a dent in the Korean market which was dominated by CDMA2000. By October 2006 both companies are covering more than 90 cities whileSK Telecomhas announced that it will provide nationwide coverage for its WCDMA network in order for it to offer SBSM (Single Band Single Mode) handsets by the first half of 2007.KT Freecelwill thus cut funding to its CDMA2000 network development to the minimum.
InNorway,Telenorintroduced W-CDMA in major cities by the end of 2004, while their competitor,NetCom, followed suit a few months later. Both operators have 98% national coverage on EDGE, but Telenor has parallel WLAN roaming networks on GSM, where the UMTS service is competing with this. For this reason Telenor is dropping support of their WLAN service in Austria (2006).
Maxis CommunicationsandCelcom, two mobile phone service providers inMalaysia, started offering W-CDMA services in 2005.
InSweden,Teliaintroduced W-CDMA in March 2004.
UMTS-TDD, an acronym for Universal Mobile Telecommunications System (UMTS) – time-division duplexing (TDD), is a 3GPP standardized version of UMTS networks that use UTRA-TDD.[11]UTRA-TDD is a UTRA that usestime-division duplexingfor duplexing.[11]While a full implementation of UMTS, it is mainly used to provide Internet access in circumstances similar to those whereWiMAXmight be used.[citation needed]UMTS-TDD is not directly compatible with UMTS-FDD: a device designed to use one standard cannot, unless specifically designed to, work on the other, because of the difference in air interface technologies and frequencies used.[citation needed]It is more formally as IMT-2000 CDMA-TDD or IMT 2000 Time-Division (IMT-TD).[16][17]
The two UMTS air interfaces (UTRAs) for UMTS-TDD are TD-CDMA and TD-SCDMA. Both air interfaces use a combination of two channel access methods,code-division multiple access(CDMA) and time-division multiple access (TDMA): the frequency band is divided into time slots (TDMA), which are further divided into channels using CDMA spreading codes. These air interfaces are classified as TDD, because time slots can be allocated to either uplink or downlink traffic.
TD-CDMA, an acronym for Time-Division-Code-Division Multiple Access, is a channel-access method based on usingspread-spectrummultiple-access (CDMA) across multiple time slots (TDMA). TD-CDMA is the channel access method for UTRA-TDD HCR, which is an acronym for UMTS Terrestrial Radio Access-Time Division Duplex High Chip Rate.[16]
UMTS-TDD's air interfaces that use the TD-CDMA channel access technique are standardized as UTRA-TDD HCR, which uses increments of 5MHzof spectrum, each slice divided into 10 ms frames containing fifteen time slots (1500 per second).[16]The time slots (TS) are allocated in fixed percentage for downlink and uplink. TD-CDMA is used to multiplex streams from or to multiple transceivers. Unlike W-CDMA, it does not need separate frequency bands for up- and downstream, allowing deployment in tightfrequency bands.[18]
TD-CDMA is a part of IMT-2000, defined as IMT-TD Time-Division (IMT CDMA TDD), and is one of the three UMTS air interfaces (UTRAs), as standardized by the 3GPP in UTRA-TDD HCR. UTRA-TDD HCR is closely related to W-CDMA, and provides the same types of channels where possible. UMTS's HSDPA/HSUPA enhancements are also implemented under TD-CDMA.[19]
In the United States, the technology has been used for public safety and government use in theNew York Cityand a few other areas.[needs update][20]In Japan, IPMobile planned to provide TD-CDMA service in year 2006, but it was delayed, changed to TD-SCDMA, and bankrupt before the service officially started.
Time-Division Synchronous Code-Division Multiple Access(TD-SCDMA) or UTRA TDD 1.28Mcpslow chip rate (UTRA-TDD LCR)[17][8]is an air interface[17]found in UMTS mobile telecommunications networks in China as an alternative to W-CDMA.
TD-SCDMA uses the TDMA channel access method combined with an adaptivesynchronous CDMAcomponent[17]on 1.6 MHz slices of spectrum, allowing deployment in even tighter frequency bands than TD-CDMA. It is standardized by the 3GPP and also referred to as "UTRA-TDD LCR". However, the main incentive for development of this Chinese-developed standard was avoiding or reducing the license fees that have to be paid to non-Chinese patent owners. Unlike the other air interfaces, TD-SCDMA was not part of UMTS from the beginning but has been added in Release 4 of the specification.
Like TD-CDMA, TD-SCDMA is known as IMT CDMA TDD within IMT-2000.
The term "TD-SCDMA" is misleading. While it suggests covering only a channel access method, it is actually the common name for the whole air interface specification.[8]
TD-SCDMA / UMTS-TDD (LCR) networks are incompatible with W-CDMA / UMTS-FDD and TD-CDMA / UMTS-TDD (HCR) networks.
TD-SCDMA was developed in the People's Republic of China by the Chinese Academy of Telecommunications Technology (CATT),Datang TelecomandSiemensin an attempt to avoid dependence on Western technology. This is likely primarily for practical reasons, since other 3G formats require the payment of patent fees to a large number of Western patent holders.
TD-SCDMA proponents also claim it is better suited for densely populated areas.[17]Further, it is supposed to cover all usage scenarios, whereas W-CDMA is optimised for symmetric traffic and macro cells, while TD-CDMA is best used in low mobility scenarios within micro or pico cells.[17]
TD-SCDMA is based on spread-spectrum technology which makes it unlikely that it will be able to completely escape the payment of license fees to western patent holders. The launch of a national TD-SCDMA network was initially projected by 2005[21]but only reached large scale commercial trials with 60,000 users across eight cities in 2008.[22]
On January 7, 2009, China granted a TD-SCDMA 3G licence toChina Mobile.[23]
On September 21, 2009, China Mobile officially announced that it had 1,327,000 TD-SCDMA subscribers as of the end of August, 2009.
TD-SCDMA is not commonly used outside of China.[24]
TD-SCDMA uses TDD, in contrast to the FDD scheme used byW-CDMA. By dynamically adjusting the number of timeslots used for downlink anduplink, the system can more easily accommodate asymmetric traffic with different data rate requirements on downlink and uplink than FDD schemes. Since it does not require paired spectrum for downlink and uplink, spectrum allocation flexibility is also increased. Using the same carrier frequency for uplink and downlink also means that the channel condition is the same on both directions, and thebase stationcan deduce the downlink channel information from uplink channel estimates, which is helpful to the application ofbeamformingtechniques.
TD-SCDMA also uses TDMA in addition to the CDMA used in WCDMA. This reduces the number of users in each timeslot, which reduces the implementation complexity ofmultiuser detectionand beamforming schemes, but the non-continuous transmission also reducescoverage(because of the higher peak power needed), mobility (because of lowerpower controlfrequency) and complicatesradio resource managementalgorithms.
The "S" in TD-SCDMA stands for "synchronous", which means that uplink signals are synchronized at the base station receiver, achieved by continuous timing adjustments. This reduces theinterferencebetween users of the same timeslot using different codes by improving theorthogonalitybetween the codes, therefore increasing system capacity, at the cost of some hardware complexity in achieving uplink synchronization.
On January 20, 2006,Ministry of Information Industryof the People's Republic of China formally announced that TD-SCDMA is the country's standard of 3G mobile telecommunication. On February 15, 2006, a timeline for deployment of the network in China was announced, stating pre-commercial trials would take place starting after completion of a number of test networks in select cities. These trials ran from March to October, 2006, but the results were apparently unsatisfactory. In early 2007, the Chinese government instructed the dominant cellular carrier, China Mobile, to build commercial trial networks in eight cities, and the two fixed-line carriers,China TelecomandChina Netcom, to build one each in two other cities. Construction of these trial networks was scheduled to finish during the fourth quarter of 2007, but delays meant that construction was not complete until early 2008.
The standard has been adopted by 3GPP since Rel-4, known as "UTRA TDD 1.28 Mcps Option".[17]
On March 28, 2008, China Mobile Group announced TD-SCDMA "commercial trials" for 60,000 test users in eight cities from April 1, 2008. Networks using other 3G standards (WCDMA and CDMA2000 EV/DO) had still not been launched in China, as these were delayed until TD-SCDMA was ready for commercial launch.
In January 2009, theMinistry of Industry and Information Technology(MIIT) in China took the unusual step of assigning licences for 3 different third-generation mobile phone standards to three carriers in a long-awaited step that is expected to prompt $41 billion in spending on new equipment. The Chinese-developed standard, TD-SCDMA, was assigned to China Mobile, the world's biggest phone carrier by subscribers. That appeared to be an effort to make sure the new system has the financial and technical backing to succeed. Licences for two existing 3G standards, W-CDMA andCDMA2000 1xEV-DO, were assigned toChina Unicomand China Telecom, respectively. Third-generation, or 3G, technology supports Web surfing, wireless video and other services and the start of service is expected to spur new revenue growth.
The technical split by MIIT has hampered the performance of China Mobile in the 3G market, with users and China Mobile engineers alike pointing to the lack of suitable handsets to use on the network.[25]Deployment of base stations has also been slow, resulting in lack of improvement of service for users.[26]The network connection itself has consistently been slower than that from the other two carriers, leading to a sharp decline in market share. By 2011 China Mobile has already moved its focus onto TD-LTE.[27][28]Gradual closures of TD-SCDMA stations started in 2016.[29][30]
The following is a list ofmobile telecommunicationsnetworks using third-generation TD-SCDMA / UMTS-TDD (LCR) technology.
In Europe,CEPTallocated the 2010–2020 MHz range for a variant of UMTS-TDD designed for unlicensed, self-provided use.[33]Some telecom groups and jurisdictions have proposed withdrawing this service in favour of licensed UMTS-TDD,[34]due to lack of demand, and lack of development of a UMTS TDD air interface technology suitable for deployment in this band.
Ordinary UMTS uses UTRA-FDD as an air interface and is known asUMTS-FDD. UMTS-FDD uses W-CDMA for multiple access andfrequency-division duplexfor duplexing, meaning that the up-link and down-link transmit on different frequencies. UMTS is usually transmitted on frequencies assigned for1G,2G, or 3G mobile telephone service in the countries of operation.
UMTS-TDD uses time-division duplexing, allowing the up-link and down-link to share the same spectrum. This allows the operator to more flexibly divide the usage of available spectrum according to traffic patterns. For ordinary phone service, you would expect the up-link and down-link to carry approximately equal amounts of data (because every phone call needs a voice transmission in either direction), but Internet-oriented traffic is more frequently one-way. For example, when browsing a website, the user will send commands, which are short, to the server, but the server will send whole files, that are generally larger than those commands, in response.
UMTS-TDD tends to be allocated frequency intended for mobile/wireless Internet services rather than used on existing cellular frequencies. This is, in part, because TDD duplexing is not normally allowed oncellular,PCS/PCN, and 3G frequencies. TDD technologies open up the usage of left-over unpaired spectrum.
Europe-wide, several bands are provided either specifically for UMTS-TDD or for similar technologies. These are 1900 MHz and 1920 MHz and between 2010 MHz and 2025 MHz. In several countries the 2500–2690 MHz band (also known as MMDS in the USA) have been used for UMTS-TDD deployments. Additionally, spectrum around the 3.5 GHz range has been allocated in some countries, notably Britain, in a technology-neutral environment. In the Czech Republic UTMS-TDD is also used in a frequency range around 872 MHz.[35]
UMTS-TDD has been deployed for public and/or private networks in at least nineteen countries around the world, with live systems in, amongst other countries, Australia, Czech Republic, France, Germany, Japan, New Zealand, Botswana, South Africa, the UK, and the USA.
Deployments in the US thus far have been limited. It has been selected for a public safety support network used by emergency responders in New York,[36]but outside of some experimental systems, notably one fromNextel, thus far the WiMAX standard appears to have gained greater traction as a general mobile Internet access system.
A variety of Internet-access systems exist which provide broadband speed access to the net. These include WiMAX andHIPERMAN. UMTS-TDD has the advantages of being able to use an operator's existing UMTS/GSM infrastructure, should it have one, and that it includes UMTS modes optimized for circuit switching should, for example, the operator want to offer telephone service. UMTS-TDD's performance is also more consistent. However, UMTS-TDD deployers often have regulatory problems with taking advantage of some of the services UMTS compatibility provides. For example, the UMTS-TDD spectrum in the UK cannot be used to provide telephone service, though the regulatorOFCOMis discussing the possibility of allowing it at some point in the future. Few operators considering UMTS-TDD have existing UMTS/GSM infrastructure.
Additionally, the WiMAX and HIPERMAN systems provide significantly larger bandwidths when the mobile station is near the tower.
Like most mobile Internet access systems, many users who might otherwise choose UMTS-TDD will find their needs covered by the ad hoc collection of unconnected Wi-Fi access points at many restaurants and transportation hubs, and/or by Internet access already provided by their mobile phone operator. By comparison, UMTS-TDD (and systems like WiMAX) offers mobile, and more consistent, access than the former, and generally faster access than the latter.
UMTS also specifies the Universal Terrestrial Radio Access Network (UTRAN), which is composed of multiple base stations, possibly using different terrestrial air interface standards and frequency bands.
UMTS and GSM/EDGE can share a Core Network (CN), making UTRAN an alternative radio access network toGERAN(GSM/EDGE RAN), and allowing (mostly) transparent switching between the RANs according to available coverage and service needs. Because of that, UMTS's and GSM/EDGE's radio access networks are sometimes collectively referred to as UTRAN/GERAN.
UMTS networks are often combined with GSM/EDGE, the latter of which is also a part of IMT-2000.
The UE (User Equipment) interface of theRAN(Radio Access Network) primarily consists ofRRC(Radio Resource Control),PDCP(Packet Data Convergence Protocol),RLC(Radio Link Control) and MAC (Media Access Control) protocols. RRC protocol handles connection establishment, measurements, radio bearer services, security and handover decisions. RLC protocol primarily divides into three Modes – Transparent Mode (TM), Unacknowledge Mode (UM), Acknowledge Mode (AM). The functionality of AM entity resembles TCP operation whereas UM operation resembles UDP operation. In TM mode, data will be sent to lower layers without adding any header toSDUof higher layers. MAC handles the scheduling of data on air interface depending on higher layer (RRC) configured parameters.
The set of properties related to data transmission is called Radio Bearer (RB). This set of properties decides the maximum allowed data in a TTI (Transmission Time Interval). RB includes RLC information and RB mapping. RB mapping decides the mapping between RB<->logical channel<->transport channel. Signaling messages are sent on Signaling Radio Bearers (SRBs) and data packets (either CS or PS) are sent on data RBs. RRC andNASmessages go on SRBs.
Security includes two procedures: integrity and ciphering. Integrity validates the resource of messages and also makes sure that no one (third/unknown party) on the radio interface has modified the messages. Ciphering ensures that no one listens to your data on the air interface. Both integrity and ciphering are applied for SRBs whereas only ciphering is applied for data RBs.
With Mobile Application Part, UMTS uses the same core network standard as GSM/EDGE. This allows a simple migration for existing GSM operators. However, the migration path to UMTS is still costly: while much of the core infrastructure is shared with GSM, the cost of obtaining new spectrum licenses and overlaying UMTS at existing towers is high.
The CN can be connected to variousbackbone networks, such as theInternetor anIntegrated Services Digital Network(ISDN) telephone network. UMTS (and GERAN) include the three lowest layers ofOSI model. The network layer (OSI 3) includes theRadio Resource Managementprotocol (RRM) that manages the bearer channels between the mobile terminals and the fixed network, including the handovers.
A UARFCN (abbreviationfor UTRA Absolute Radio Frequency Channel Number, where UTRA stands for UMTS Terrestrial Radio Access) is used to identify a frequency in theUMTS frequency bands.
Typically channel number is derived from the frequency in MHz through the formula Channel Number = Frequency * 5. However, this is only able to represent channels that are centered on a multiple of 200 kHz, which do not align with licensing in North America. 3GPP added several special values for the common North American channels.
Over 130 licenses had been awarded to operators worldwide, as of December 2004, specifying W-CDMA radio access technology that builds on GSM. In Europe, the license process occurred at the tail end of the technology bubble, and the auction mechanisms for allocation set up in some countries resulted in some extremely high prices being paid for the original 2100 MHz licenses, notably in the UK and Germany. InGermany, bidders paid a total €50.8 billion for six licenses, two of which were subsequently abandoned and written off by their purchasers (Mobilcom and theSonera/Telefónicaconsortium). It has been suggested that these huge license fees have the character of a very large tax paid on future income expected many years down the road. In any event, the high prices paid put some European telecom operators close to bankruptcy (most notablyKPN). Over the last few years some operators have written off some or all of the license costs. Between 2007 and 2009, all three Finnish carriers began to use 900 MHz UMTS in a shared arrangement with its surrounding 2G GSM base stations for rural area coverage, a trend that is expected to expand over Europe in the next 1–3 years.[needs update]
The 2100 MHz band (downlink around 2100 MHz and uplink around 1900 MHz) allocated for UMTS in Europe and most of Asia is already used in North America. The 1900 MHz range is used for 2G (PCS) services, and 2100 MHz range is used for satellite communications. Regulators have, however, freed up some of the 2100 MHz range for 3G services, together with a different range around 1700 MHz for the uplink.[needs update]
AT&T Wireless launched UMTS services in the United States by the end of 2004 strictly using the existing 1900 MHz spectrum allocated for 2G PCS services. Cingular acquired AT&T Wireless in 2004 and has since then launched UMTS in select US cities. Cingular renamed itself AT&T Mobility and rolled out[37]some cities with a UMTS network at 850 MHz to enhance its existing UMTS network at 1900 MHz and now offers subscribers a number of dual-band UMTS 850/1900 phones.
T-Mobile's rollout of UMTS in the US was originally focused on the 1700 MHz band. However, T-Mobile has been moving users from 1700 MHz to 1900 MHz (PCS) in order to reallocate the spectrum to 4GLTEservices.[38]
In Canada, UMTS coverage is being provided on the 850 MHz and 1900 MHz bands on the Rogers and Bell-Telus networks. Bell and Telus share the network. Recently, new providersWind Mobile,MobilicityandVideotronhave begun operations in the 1700 MHz band.
In 2008, Australian telco Telstra replaced its existing CDMA network with a national UMTS-based 3G network, branded asNextG, operating in the 850 MHz band. Telstra currently provides UMTS service on this network, and also on the 2100 MHz UMTS network, through a co-ownership of the owning and administrating company 3GIS. This company is also co-owned byHutchison 3G Australia, and this is the primary network used by their customers.Optusis currently rolling out a 3G network operating on the 2100 MHz band in cities and most large towns, and the 900 MHz band in regional areas.Vodafoneis also building a 3G network using the 900 MHz band.
In India,BSNLhas started its 3G services since October 2009, beginning with the larger cities and then expanding over to smaller cities. The 850 MHz and 900 MHz bands provide greater coverage compared to equivalent 1700/1900/2100 MHz networks, and are best suited to regional areas where greater distances separate base station and subscriber.
Carriers in South America are now also rolling out 850 MHz networks.
UMTS phones (and data cards) are highly portable – they have been designed to roam easily onto other UMTS networks (if the providers have roaming agreements in place). In addition, almost all UMTS phones are UMTS/GSM dual-mode devices, so if a UMTS phone travels outside of UMTS coverage during a call the call may be transparently handed off to available GSM coverage. Roaming charges are usually significantly higher than regular usage charges.
Most UMTS licensees consider ubiquitous, transparent globalroamingan important issue. To enable a high degree of interoperability, UMTS phones usually support several different frequencies in addition to their GSM fallback. Different countries support different UMTS frequency bands – Europe initially used 2100 MHz while the most carriers in the USA use 850 MHz and 1900 MHz. T-Mobile has launched a network in the US operating at 1700 MHz (uplink) /2100 MHz (downlink), and these bands also have been adopted elsewhere in the US and in Canada and Latin America. A UMTS phone and network must support a common frequency to work together. Because of the frequencies used, early models of UMTS phones designated for the United States will likely not be operable elsewhere and vice versa. There are now 11 different frequency combinations used around the world – including frequencies formerly used solely for 2G services.
UMTS phones can use aUniversal Subscriber Identity Module, USIM (based on GSM'sSIM card) and also work (including UMTS services) with GSM SIM cards. This is a global standard of identification, and enables a network to identify and authenticate the (U)SIM in the phone. Roaming agreements between networks allow for calls to a customer to be redirected to them while roaming and determine the services (and prices) available to the user. In addition to user subscriber information and authentication information, the (U)SIM provides storage space for phone book contact. Handsets can store their data on their own memory or on the (U)SIM card (which is usually more limited in its phone book contact information). A (U)SIM can be moved to another UMTS or GSM phone, and the phone will take on the user details of the (U)SIM, meaning it is the (U)SIM (not the phone) which determines the phone number of the phone and the billing for calls made from the phone.
Japan was the first country to adopt 3G technologies, and since they had not used GSM previously they had no need to build GSM compatibility into their handsets and their 3G handsets were smaller than those available elsewhere. In 2002, NTT DoCoMo's FOMA 3G network was the first commercial UMTS network – using a pre-release specification,[39]it was initially incompatible with the UMTS standard at the radio level but used standard USIM cards, meaning USIM card based roaming was possible (transferring the USIM card into a UMTS or GSM phone when travelling). Both NTT DoCoMo and SoftBank Mobile (which launched 3G in December 2002) now use standard UMTS.
All of the major 2G phone manufacturers (that are still in business) are now manufacturers of 3G phones. The early 3G handsets and modems were specific to the frequencies required in their country, which meant they could only roam to other countries on the same 3G frequency (though they can fall back to the older GSM standard). Canada and USA have a common share of frequencies, as do most European countries. The article UMTS frequency bands is an overview of UMTS network frequencies around the world.
Using acellular router, PCMCIA or USB card, customers are able to access 3G broadband services, regardless of their choice of computer (such as atablet PCor aPDA). Some softwareinstalls itselffrom the modem, so that in some cases absolutely no knowledge of technology is required to getonlinein moments. Using a phone that supports 3G and Bluetooth 2.0, multiple Bluetooth-capable laptops can be connected to the Internet. Some smartphones can also act as a mobileWLAN access point.
There are very few 3G phones or modems available supporting all 3G frequencies (UMTS850/900/1700/1900/2100 MHz). In 2010, Nokia released a range of phones withPentaband3G coverage, including theN8andE7. Many other phones are offering more than one band which still enables extensive roaming. For example, Apple'siPhone 4contains a quadband chipset operating on 850/900/1900/2100 MHz, allowing usage in the majority of countries where UMTS-FDD is deployed.
The main competitor to UMTS is CDMA2000 (IMT-MC), which is developed by the3GPP2. Unlike UMTS, CDMA2000 is an evolutionary upgrade to an existing 2G standard, cdmaOne, and is able to operate within the same frequency allocations. This and CDMA2000's narrower bandwidth requirements make it easier to deploy in existing spectra. In some, but not all, cases, existing GSM operators only have enough spectrum to implement either UMTS or GSM, not both. For example, in the US D, E, and F PCS spectrum blocks, the amount of spectrum available is 5 MHz in each direction. A standard UMTS system would saturate that spectrum. Where CDMA2000 is deployed, it usually co-exists with UMTS. In many markets however, the co-existence issue is of little relevance, as legislative hurdles exist to co-deploying two standards in the same licensed slice of spectrum.
Another competitor to UMTS isEDGE(IMT-SC), which is an evolutionary upgrade to the 2G GSM system, leveraging existing GSM spectrums. It is also much easier, quicker, and considerably cheaper for wireless carriers to "bolt-on" EDGE functionality by upgrading their existing GSM transmission hardware to support EDGE rather than having to install almost all brand-new equipment to deliver UMTS. However, being developed by 3GPP just as UMTS, EDGE is not a true competitor. Instead, it is used as a temporary solution preceding UMTS roll-out or as a complement for rural areas. This is facilitated by the fact that GSM/EDGE and UMTS specifications are jointly developed and rely on the same core network, allowing dual-mode operation includingvertical handovers.
China'sTD-SCDMAstandard is often seen as a competitor, too. TD-SCDMA has been added to UMTS' Release 4 as UTRA-TDD 1.28 Mcps Low Chip Rate (UTRA-TDD LCR). UnlikeTD-CDMA(UTRA-TDD 3.84 Mcps High Chip Rate, UTRA-TDD HCR) which complements W-CDMA (UTRA-FDD), it is suitable for both micro and macrocells. However, the lack of vendors' support is preventing it from being a real competitor.
While DECT is technically capable of competing with UMTS and other cellular networks in densely populated, urban areas, it has only been deployed for domestic cordless phones and private in-house networks.
All of these competitors have been accepted by ITU as part of the IMT-2000 family of 3G standards, along with UMTS-FDD.
On the Internet access side, competing systems include WiMAX andFlash-OFDM.
From a GSM/GPRS network, the following network elements can be reused:
From a GSM/GPRS communication radio network, the following elements cannot be reused:
They can remain in the network and be used in dual network operation where 2G and 3G networks co-exist while network migration and new 3G terminals become available for use in the network.
The UMTS network introduces new network elements that function as specified by 3GPP:
The functionality of MSC changes when going to UMTS. In a GSM system the MSC handles all the circuit switched operations like connecting A- and B-subscriber through the network. In UMTS the Media gateway (MGW) takes care of data transfer in circuit switched networks. MSC controls MGW operations.
Some countries, including the United States, have allocated spectrum differently from theITUrecommendations, so that the standard bands most commonly used for UMTS (UMTS-2100) have not been available.[citation needed]In those countries, alternative bands are used, preventing the interoperability of existing UMTS-2100 equipment, and requiring the design and manufacture of different equipment for the use in these markets. As is the case with GSM900 today[when?], standard UMTS 2100 MHz equipment will not work in those markets. However, it appears as though UMTS is not suffering as much from handset band compatibility issues as GSM did, as many UMTS handsets are multi-band in both UMTS and GSM modes. Penta-band (850, 900, 1700, 2100, and 1900 MHz bands), quad-band GSM (850, 900, 1800, and 1900 MHz bands) and tri-band UMTS (850, 1900, and 2100 MHz bands) handsets are becoming more commonplace.[40]
In its early days[when?], UMTS had problems in many countries: Overweight handsets with poor battery life were first to arrive on a market highly sensitive to weight and form factor.[citation needed]The Motorola A830, a debut handset on Hutchison's 3 network, weighed more than 200 grams and even featured a detachable camera to reduce handset weight. Another significant issue involved call reliability, related to problems with handover from UMTS to GSM. Customers found their connections being dropped as handovers were possible only in one direction (UMTS → GSM), with the handset only changing back to UMTS after hanging up. In most networks around the world this is no longer an issue.[citation needed]
Compared to GSM, UMTS networks initially required a higherbase stationdensity. For fully-fledged UMTS incorporatingvideo on demandfeatures, one base station needed to be set up every 1–1.5 km (0.62–0.93 mi). This was the case when only the 2100 MHz band was being used, however with the growing use of lower-frequency bands (such as 850 and 900 MHz) this is no longer so. This has led to increasing rollout of the lower-band networks by operators since 2006.[citation needed]
Even with current technologies and low-band UMTS, telephony and data over UMTS requires more power than on comparable GSM networks.Apple Inc.cited[41]UMTS power consumption as the reason that the first generationiPhoneonly supported EDGE. Their release of the iPhone 3G quotes talk time on UMTS as half that available when the handset is set to use GSM. Other manufacturers indicate different battery lifetime for UMTS mode compared to GSM mode as well. As battery and network technology improve, this issue is diminishing.
As early as 2008, it was known that carrier networks can be used to surreptitiously gather user location information.[42]In August 2014, theWashington Postreported on widespread marketing of surveillance systems usingSignalling System No. 7(SS7) protocols to locate callers anywhere in the world.[42]
In December 2014, news broke that SS7's very own functions can be repurposed for surveillance, because of its relaxed security, in order to listen to calls in real time or to record encrypted calls and texts for later decryption, or to defraud users and cellular carriers.[43]
Deutsche Telekomand Vodafone declared the same day that they had fixed gaps in their networks, but that the problem is global and can only be fixed with a telecommunication system-wide solution.[44]
The evolution of UMTS progresses according to planned releases. Each release is designed to introduce new features and improve upon existing ones.
|
https://en.wikipedia.org/wiki/TD-SCDMA
|
A5/2is astream cipherused to provide voice privacy in theGSMcellular telephoneprotocol. It was designed in 1992-1993 (finished March 1993) as a replacement for the relatively stronger (but still weak)A5/1, to allow the GSM standard to be exported to countries "with restrictions on the import of products with cryptographic security features".[1]
The cipher is based on a combination of fourlinear-feedback shift registerswith irregular clocking and anon-linearcombiner.
In 1999,Ian GoldbergandDavid A. Wagnercryptanalyzed A5/2 in the same month it was reverse engineered, and showed that it was extremely weak – so much so that low end equipment can probably break it in real time.[2]
In 2003, Elad Barkan,Eli Bihamand Nathan Keller presented a ciphertext-only attack based on theerror correcting codesused in GSM communication. They also demonstrated a vulnerability in the GSM protocols that allows aman-in-the-middle attackto work whenever the mobile phone supports A5/2, regardless of whether it was actually being used.[3]
Since July 1, 2006, theGSMA(GSM Association) mandated that GSM Mobile Phones will not support the A5/2 Cipher any longer, due to its weakness, and the fact thatA5/1is deemed mandatory by the3GPPassociation. In July 2007, the3GPPhas approved a change request to prohibit the implementation of A5/2 in any new mobile phones, stating: "It is mandatory for A5/1 and non encrypted mode to be implemented in mobile stations. It is prohibited to implement A5/2 in mobile stations."[4]If the network does not support A5/1 then an unencrypted connection can be used.
|
https://en.wikipedia.org/wiki/A5/2
|
KASUMIis ablock cipherused inUMTS,GSM, andGPRSmobile communicationssystems.
In UMTS, KASUMI is used in theconfidentiality(f8) andintegrityalgorithms (f9) with names UEA1 and UIA1, respectively.[1]In GSM, KASUMI is used in theA5/3key stream generator and in GPRS in theGEA3key stream generator.
KASUMI was designed for3GPPto be used in UMTS security system by theSecurity Algorithms Group of Experts(SAGE), a part of the European standards bodyETSI.[2]Because of schedule pressures in 3GPP standardization, instead of developing a new cipher, SAGE agreed with
3GPP technical specification group (TSG) for system aspects of 3G security (SA3) to base the development
on an existing algorithm that had already undergone some evaluation.[2]They chose the cipher algorithmMISTY1developed[3]and patented[4]byMitsubishi Electric Corporation.
The original algorithm was slightly modified for easier hardware implementation and to
meet other requirements set for 3G mobile communications security.
KASUMI is named after the original algorithmMISTY1—霞み(hiraganaかすみ, romajikasumi) is theJapaneseword for "mist".
In January 2010,Orr Dunkelman, Nathan Keller andAdi Shamirreleased a paper showing that they could break Kasumi with arelated-key attackand very modest computational resources; this attack is ineffective againstMISTY1.[5]
KASUMI algorithm is specified in a 3GPP technical specification.[6]KASUMI is a block cipher with 128-bit key and 64-bit input and output.
The core of KASUMI is an eight-roundFeistel network. The round functions
in the main Feistel network are irreversible Feistel-like network
transformations. In each round the round function uses a round key
which consists of eight 16-bit sub keys
derived from the original 128-bit key using a fixed key schedule.
The 128-bit keyKis divided into eight 16-bit sub keysKi:
K=K1‖K2‖K3‖K4‖K5‖K6‖K7‖K8{\displaystyle K=K_{1}\|K_{2}\|K_{3}\|K_{4}\|K_{5}\|K_{6}\|K_{7}\|K_{8}\,}
Additionally a modified keyK', similarly divided into 16-bit
sub keysK'i, is used. The modified key is derived from
the original key by XORing with 0x123456789ABCDEFFEDCBA9876543210 (chosen as a"nothing up my sleeve" number).
Round keys are either derived from the sub keys by bitwise rotation to left
by a given amount and from the modified sub keys (unchanged).
The round keys are as follows:
KLi,1=ROL(Ki,1)KLi,2=Ki+2′KOi,1=ROL(Ki+1,5)KOi,2=ROL(Ki+5,8)KOi,3=ROL(Ki+6,13)KIi,1=Ki+4′KIi,2=Ki+3′KIi,3=Ki+7′{\displaystyle {\begin{array}{lcl}KL_{i,1}&=&{\rm {ROL}}(K_{i},1)\\KL_{i,2}&=&K'_{i+2}\\KO_{i,1}&=&{\rm {ROL}}(K_{i+1},5)\\KO_{i,2}&=&{\rm {ROL}}(K_{i+5},8)\\KO_{i,3}&=&{\rm {ROL}}(K_{i+6},13)\\KI_{i,1}&=&K'_{i+4}\\KI_{i,2}&=&K'_{i+3}\\KI_{i,3}&=&K'_{i+7}\end{array}}}
Sub key index additions are cyclic so that ifi+jis greater than 8
one has to subtract 8 from the result to get the actual sub key index.
KASUMI algorithm processes the 64-bit word in two 32-bit halves, left (Li{\displaystyle L_{i}})
and right (Ri{\displaystyle R_{i}}).
The input word is concatenation of the left and right halves of the first round:
input=R0‖L0{\displaystyle {\rm {input}}=R_{0}\|L_{0}\,}.
In each round the right half is XOR'ed with the output of the round function
after which the halves are swapped:
Li=Fi(KLi,KOi,KIi,Li−1)⊕Ri−1Ri=Li−1{\displaystyle {\begin{array}{rcl}L_{i}&=&F_{i}(KL_{i},KO_{i},KI_{i},L_{i-1})\oplus R_{i-1}\\R_{i}&=&L_{i-1}\end{array}}}
whereKLi,KOi,KIiare round keys
for theithround.
The round functions for even and odd rounds are slightly different. In each case
the round function is a composition of two functionsFLiandFOi.
For an odd round
Fi(Ki,Li−1)=FO(KOi,KIi,FL(KLi,Li−1)){\displaystyle F_{i}(K_{i},L_{i-1})=FO(KO_{i},KI_{i},FL(KL_{i},L_{i-1}))\,}
and for an even round
Fi(Ki,Li−1)=FL(KLi,FO(KOi,KIi,Li−1)){\displaystyle F_{i}(K_{i},L_{i-1})=FL(KL_{i},FO(KO_{i},KI_{i},L_{i-1}))\,}.
The output is the concatenation of the outputs of the last round.
output=R8‖L8{\displaystyle {\rm {output}}=R_{8}\|L_{8}\,}.
BothFLandFOfunctions divide the 32-bit input data to two 16-bit halves.
TheFLfunction is an irreversible bit manipulation while theFOfunction is
an irreversible three round Feistel-like network.
The 32-bit inputxofFL(KLi,x){\displaystyle FL(KL_{i},x)}is divided to two 16-bit halvesx=l‖r{\displaystyle x=l\|r}.
First the left half of the inputl{\displaystyle l}is ANDed bitwise with round keyKLi,1{\displaystyle KL_{i,1}}and rotated
left by one bit. The result of that is XOR'ed to the right half of the inputr{\displaystyle r}to get the right
half of the outputr′{\displaystyle r'}.
r′=ROL(l∧KLi,1,1)⊕r{\displaystyle r'={\rm {ROL}}(l\wedge KL_{i,1},1)\oplus r}
Then the right half of the outputr′{\displaystyle r'}is ORed bitwise with the round keyKLi,2{\displaystyle KL_{i,2}}and rotated
left by one bit. The result of that is XOR'ed to the left half of the inputl{\displaystyle l}to get the left
half of the outputl′{\displaystyle l'}.
l′=ROL(r′∨KLi,2,1)⊕l{\displaystyle l'={\rm {ROL}}(r'\vee KL_{i,2},1)\oplus l}
Output of the function is concatenation of the left and right halvesx′=l′‖r′{\displaystyle x'=l'\|r'}.
The 32-bit inputxofFO(KOi,KIi,x){\displaystyle FO(KO_{i},KI_{i},x)}is divided into two 16-bit halvesx=l0‖r0{\displaystyle x=l_{0}\|r_{0}}, and passed through three rounds of a Feistel network.
In each of the three rounds (indexed byjthat takes values 1, 2, and 3) the left half is modified
to get the new right half and the right half is made the left half of the next round.
rj=FI(KIi,j,lj−1⊕KOi,j)⊕rj−1lj=rj−1{\displaystyle {\begin{array}{lcl}r_{j}&=&FI(KI_{i,j},l_{j-1}\oplus KO_{i,j})\oplus r_{j-1}\\l_{j}&=&r_{j-1}\end{array}}}
The output of the function isx′=l3‖r3{\displaystyle x'=l_{3}\|r_{3}}.
The functionFIis an irregular Feistel-like network.
The 16-bit inputx{\displaystyle x}of the functionFI(Ki,x){\displaystyle FI(Ki,x)}is divided to two halvesx=l0‖r0{\displaystyle x=l_{0}\|r_{0}}of whichl0{\displaystyle l_{0}}is 9 bits wide andr0{\displaystyle r_{0}}is 7 bits wide.
Bits in the left halfl0{\displaystyle l_{0}}are first shuffled by 9-bitsubstitution box(S-box)S9and the result is XOR'ed with
the zero-extended right halfr0{\displaystyle r_{0}}to get the new 9-bit right halfr1{\displaystyle r_{1}}.
r1=S9(l0)⊕(00‖r0){\displaystyle r_{1}=S9(l_{0})\oplus (00\|r_{0})\,}
Bits of the right halfr0{\displaystyle r_{0}}are shuffled by 7-bit S-boxS7and the result is XOR'ed with
the seven least significant bits (LS7) of the new right halfr1{\displaystyle r_{1}}to get the new 7-bit left halfl1{\displaystyle l_{1}}.
l1=S7(r0)⊕LS7(r1){\displaystyle l_{1}=S7(r_{0})\oplus LS7(r_{1})\,}
The intermediate wordx1=l1‖r1{\displaystyle x_{1}=l_{1}\|r_{1}}is XORed with the round key KI to getx2=l2‖r2{\displaystyle x_{2}=l_{2}\|r_{2}}of whichl2{\displaystyle l_{2}}is 7 bits wide andr2{\displaystyle r_{2}}is 9 bits wide.
x2=KI⊕x1{\displaystyle x_{2}=KI\oplus x_{1}}
Bits in the right halfr2{\displaystyle r_{2}}are then shuffled by 9-bit S-boxS9and the result is XOR'ed with
the zero-extended left halfl2{\displaystyle l_{2}}to get the new 9-bit right half of the outputr3{\displaystyle r_{3}}.
r3=S9(r2)⊕(00‖l2){\displaystyle r_{3}=S9(r_{2})\oplus (00\|l_{2})\,}
Finally the bits of the left halfl2{\displaystyle l_{2}}are shuffled by 7-bit S-boxS7and the result is XOR'ed with
the seven least significant bits (LS7) of the right half of the outputr3{\displaystyle r_{3}}to get the 7-bit left
halfl3{\displaystyle l_{3}}of the output.
l3=S7(l2)⊕LS7(r3){\displaystyle l_{3}=S7(l_{2})\oplus LS7(r_{3})\,}
The output is the concatenation of the final left and right halvesx′=l3‖r3{\displaystyle x'=l_{3}\|r_{3}}.
Thesubstitution boxes(S-boxes) S7 and S9 are defined by both bit-wise AND-XOR expressions and look-up tables in the specification.
The bit-wise expressions are intended to hardware implementation but nowadays it is customary to use
the look-up tables even in the HW design.
S7 is defined by the following array:
S9 is defined by the following array:
In 2001, animpossible differential attackon six rounds of KASUMI was presented by Kühn (2001).[7]
In 2003 Elad Barkan,Eli Bihamand Nathan Keller demonstratedman-in-the-middle attacksagainst theGSMprotocol which avoided the A5/3 cipher and thus breaking the protocol. This approach does not attack the A5/3 cipher, however.[8]The full version of their paper was published later in 2006.[9]
In 2005, Israeli researchersEli Biham,Orr Dunkelmanand Nathan Keller published arelated-keyrectangle (boomerang) attackon KASUMI that can break all 8 rounds faster than exhaustive search.[10]The attack requires 254.6chosen plaintexts, each of which has been encrypted under one of four related keys, and has a time complexity equivalent to 276.1KASUMI encryptions. While this is obviously not a practical attack, it invalidates some proofs about the security of the 3GPP protocols that had relied on the presumed strength of KASUMI.
In 2010, Dunkelman, Keller and Shamir published a new attack that allows an adversary to recover a full A5/3 key byrelated-key attack.[5]The time and space complexities of the attack are low enough that the authors carried out the attack in two hours on anIntel Core 2 Duodesktop computer even using the unoptimized reference KASUMI implementation. The authors note that this attack may not be applicable to the way A5/3 is used in 3G systems; their main purpose was to discredit 3GPP's assurances that their changes to MISTY wouldn't significantly impact the security of the algorithm.
|
https://en.wikipedia.org/wiki/KASUMI_(block_cipher)
|
Incryptography, theCellular Message Encryption Algorithm(CMEA) is ablock cipherwhich was used for securingmobile phonesin theUnited States. CMEA is one of fourcryptographicprimitives specified in aTelecommunications Industry Association(TIA) standard, and is designed toencryptthe control channel, rather than the voice data. In 1997, a group of cryptographers published attacks on theciphershowing it had several weaknesses which give it a trivial effective strength of a 24-bit to 32-bit cipher.[1]Some accusations were made that theNSAhad pressured the original designers into crippling CMEA, but the NSA has denied any role in the design or selection of the algorithm. TheECMEAandSCEMAciphers are derived from CMEA.
CMEA is described inU.S. patent 5,159,634. It isbyte-oriented, with variableblock size, typically 2 to 6 bytes. Thekey sizeis only 64 bits. Both of these are unusually small for a modern cipher. The algorithm consists of only 3 passes over the data: a non-linear left-to-right diffusion operation, an unkeyed linear mixing, and another non-linear diffusion that is in fact the inverse of the first. The non-linear operations use a keyedlookup tablecalled theT-box, which uses an unkeyed lookup table called theCaveTable. The algorithm isself-inverse; re-encrypting the ciphertext with the same key is equivalent to decrypting it.
CMEA is severely insecure. There is achosen-plaintext attack, effective for all block sizes, using 338 chosen plaintexts. For 3-byte blocks (typically used to encrypt each dialled digit), there is aknown-plaintext attackusing 40 to 80 known plaintexts. For 2-byte blocks, 4 known plaintexts suffice.
The "improved" CMEA, CMEA-I, is not much better: chosen-plaintext attack of it requires less than 850 plaintexts in its adaptive version.[2]
|
https://en.wikipedia.org/wiki/Cellular_Message_Encryption_Algorithm
|
Inmathematics, more specifically inring theory, aEuclidean domain(also called aEuclidean ring) is anintegral domainthat can be endowed with aEuclidean functionwhich allows a suitable generalization ofEuclidean divisionofintegers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in theringof integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute thegreatest common divisorof any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination
of them (Bézout's identity). In particular, the existence of efficient algorithms for Euclidean division of integers and ofpolynomialsin one variable over afieldis of basic importance incomputer algebra.
It is important to compare theclassof Euclidean domains with the larger class ofprincipal ideal domains(PIDs). An arbitrary PID has much the same "structural properties" of a Euclidean domain (or, indeed, even of the ring of integers), but lacks an analogue of theEuclidean algorithmandextended Euclidean algorithmto compute greatest common divisors. So, given an integral domainR, it is often very useful to know thatRhas a Euclidean function: in particular, this implies thatRis a PID. However, if there is no "obvious" Euclidean function, then determining whetherRis a PID is generally a much easier problem than determining whether it is a Euclidean domain.
Everyidealin a Euclidean domain isprincipal, which implies a suitable generalization of thefundamental theorem of arithmetic: every Euclidean domain is also aunique factorization domain. Euclidean domains appear in the following chain ofclass inclusions:
LetRbe an integral domain. AEuclidean functiononRis afunctionffromR\ {0}to the non-negative integers satisfying the following fundamental division-with-remainder property:
AEuclidean domainis an integral domain which can be endowed with at least one Euclidean function. A particular Euclidean functionfisnotpart of the definition of a Euclidean domain, as, in general, a Euclidean domain may admit many different Euclidean functions.
In this context,qandrare called respectively aquotientand aremainderof thedivision(orEuclidean division) ofabyb. In contrast with the case ofintegersandpolynomials, the quotient is generally not uniquely defined, but when a quotient has been chosen, the remainder is uniquely defined.
Most algebra texts require a Euclidean function to have the following additional property:
However, one can show that (EF1) alone suffices to define a Euclidean domain; if an integral domainRis endowed with a functiongsatisfying (EF1), thenRcan also be endowed with a function satisfying both (EF1) and (EF2) simultaneously. Indeed, forainR\ {0}, one can definef(a)as follows:[1]
In words, one may definef(a)to be the minimum value attained bygon the set of all non-zero elements of the principal ideal generated bya.
A Euclidean functionfismultiplicativeiff(ab) =f(a)f(b)andf(a)is never zero. It follows thatf(1) = 1. More generally,f(a) = 1if and only ifais aunit.
Many authors use other terms in place of "Euclidean function", such as "degree function", "valuation function", "gauge function" or "norm function".[2]Some authors also require thedomainof the Euclidean function to be the entire ringR;[2]however, this does not essentially affect the definition, since (EF1) does not involve the value off(0). The definition is sometimes generalized by allowing the Euclidean function to take its values in anywell-ordered set; this weakening does not affect the most important implications of the Euclidean property.
The property (EF1) can be restated as follows: for any principal idealIofRwith nonzero generatorb, all nonzero classes of thequotient ringR/Ihave a representativerwithf(r) <f(b). Since the possible values offare well-ordered, this property can be established by showing thatf(r) <f(b)for anyr∉Iwith minimal value off(r)in its class. Note that, for a Euclidean function that is so established, there need not exist an effective method to determineqandrin (EF1).
Examples of Euclidean domains include:
Examples of domains that arenotEuclidean domains include:
LetRbe a domain andfa Euclidean function onR. Then:
However, in manyfinite extensionsofQwithtrivialclass group, the ring of integers is Euclidean (not necessarily with respect to the absolute value of the field norm; see below).
Assuming theextended Riemann hypothesis, ifKis a finiteextensionofQand the ring of integers ofKis a PID with an infinite number of units, then the ring of integers is Euclidean.[12]In particular this applies to the case oftotally realquadratic number fieldswith trivial class group.
In addition (and without assuming ERH), if the fieldKis aGalois extensionofQ, has trivial class group andunit rankstrictly greater than three, then the ring of integers is Euclidean.[13]An immediatecorollaryof this is that if thenumber fieldis Galois overQ, its class group is trivial and the extension hasdegreegreater than 8 then the ring of integers is necessarily Euclidean.
Algebraic number fieldsKcome with a canonical norm function on them: the absolute value of thefield normNthat takes analgebraic elementαto the product of all theconjugatesofα. This norm maps thering of integersof a number fieldK, sayOK, to the nonnegativerational integers, so it is a candidate to be a Euclidean norm on this ring. If this norm satisfies the axioms of a Euclidean function then the number fieldKis callednorm-Euclideanor simplyEuclidean.[14][15]Strictly speaking it is the ring of integers that is Euclidean since fields are trivially Euclidean domains, but the terminology is standard.
If a field is not norm-Euclidean then that does not mean the ring of integers is not Euclidean, just that the field norm does not satisfy the axioms of a Euclidean function. In fact, the rings of integers of number fields may be divided in several classes:
The norm-Euclideanquadratic fieldshave been fully classified; they areQ(d){\displaystyle \mathbf {Q} ({\sqrt {d}}\,)}whered{\displaystyle d}takes the values
Every Euclidean imaginary quadratic field is norm-Euclidean and is one of the five first fields in the preceding list.
|
https://en.wikipedia.org/wiki/Euclidean_domain
|
Inmathematics, theChinese remainder theoremstates that if one knows theremainders of the Euclidean divisionof anintegernby several integers, then one can determine uniquely the remainder of the division ofnby the product of these integers, under the condition that thedivisorsarepairwise coprime(no two divisors share a common factor other than 1).[1]
The theorem is sometimes calledSunzi's theorem. Both names of the theorem refer to its earliest known statement that appeared inSunzi Suanjing, a Chinese manuscript written during the 3rd to 5th century CE. This first statement was restricted to the following example:
If one knows that the remainder ofndivided by 3 is 2, the remainder ofndivided by 5 is 3, and the remainder ofndivided by 7 is 2, then with no other information, one can determine the remainder ofndivided by 105 (the product of 3, 5, and 7) without knowing the value ofn. In this example, the remainder is 23. Moreover, this remainder is the only possible positive value ofnthat is less than 105.
The Chinese remainder theorem is widely used for computing with large integers, as it allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers.
The Chinese remainder theorem (expressed in terms ofcongruences) is true over everyprincipal ideal domain. It has been generalized to anyring, with a formulation involvingtwo-sided ideals.
The earliest known statement of the problem appears in the 5th-century bookSunzi Suanjingby the Chinese mathematician Sunzi:[2]
There are certain things whose number is unknown. If we count them by threes, we have two left over; by fives, we have three left over; and by sevens, two are left over. How many things are there?[3]
Sunzi's work would not be considered atheoremby modern standards; it only gives one particular problem, without showing how to solve it, much less anyproofabout the general case or a generalalgorithmfor solving it.[4]An algorithm for solving this problem was described byAryabhata(6th century).[5]Special cases of the Chinese remainder theorem were also known toBrahmagupta(7th century) and appear inFibonacci'sLiber Abaci(1202).[6]The result was later generalized with a complete solution calledDa-yan-shu(大衍術) inQin Jiushao's 1247Mathematical Treatise in Nine Sections[7]which was translated into English in early 19th century by British missionaryAlexander Wylie.[8]
The notion of congruences was first introduced and used byCarl Friedrich Gaussin hisDisquisitiones Arithmeticaeof 1801.[10]Gauss illustrates the Chinese remainder theorem on a problem involving calendars, namely, "to find the years that have a certain period number with respect to the solar and lunar cycle and the Roman indiction."[11]Gauss introduces a procedure for solving the problem that had already been used byLeonhard Eulerbut was in fact an ancient method that had appeared several times.[12]
Letn1, ...,nkbe integers greater than 1, which are often calledmoduliordivisors. Let us denote byNthe product of theni.
The Chinese remainder theorem asserts that if theniarepairwise coprime, and ifa1, ...,akare integers such that 0 ≤ai<nifor everyi, then there is one and only one integerx, such that 0 ≤x<Nand the remainder of theEuclidean divisionofxbyniisaifor everyi.
This may be restated as follows in terms ofcongruences:
If theni{\displaystyle n_{i}}are pairwise coprime, and ifa1, ...,akare any integers, then the system
has a solution, and any two solutions, sayx1andx2, are congruent moduloN, that is,x1≡x2(modN).[13]
Inabstract algebra, the theorem is often restated as: if theniare pairwise coprime, the map
defines aring isomorphism[14]
between theringofintegers moduloNand thedirect productof the rings of integers modulo theni. This means that for doing a sequence of arithmetic operations inZ/NZ,{\displaystyle \mathbb {Z} /N\mathbb {Z} ,}one may do the same computation independently in eachZ/niZ{\displaystyle \mathbb {Z} /n_{i}\mathbb {Z} }and then get the result by applying the isomorphism (from the right to the left). This may be much faster than the direct computation ifNand the number of operations are large. This is widely used, under the namemulti-modular computation, forlinear algebraover the integers or therational numbers.
The theorem can also be restated in the language ofcombinatoricsas the fact that the infinitearithmetic progressionsof integers form aHelly family.[15]
The existence and the uniqueness of the solution may be proven independently. However, the first proof of existence, given below, uses this uniqueness.
Suppose thatxandyare both solutions to all the congruences. Asxandygive the same remainder, when divided byni, their differencex−yis a multiple of eachni. As theniare pairwise coprime, their productNalso dividesx−y, and thusxandyare congruent moduloN. Ifxandyare supposed to be non-negative and less thanN(as in the first statement of the theorem), then their difference may be a multiple ofNonly ifx=y.
The map
mapscongruence classesmoduloNto sequences of congruence classes moduloni. The proof of uniqueness shows that this map isinjective. As thedomainand thecodomainof this map have the same number of elements, the map is alsosurjective, which proves the existence of the solution.
This proof is very simple but does not provide any direct way for computing a solution. Moreover, it cannot be generalized to other situations where the following proof can.
Existence may be established by an explicit construction ofx.[16]This construction may be split into two steps, first solving the problem in the case of two moduli, and then extending this solution to the general case byinductionon the number of moduli.
We want to solve the system:
wheren1{\displaystyle n_{1}}andn2{\displaystyle n_{2}}arecoprime.
Bézout's identityasserts the existence of two integersm1{\displaystyle m_{1}}andm2{\displaystyle m_{2}}such that
The integersm1{\displaystyle m_{1}}andm2{\displaystyle m_{2}}may be computed by theextended Euclidean algorithm.
A solution is given by
Indeed,
implying thatx≡a1(modn1).{\displaystyle x\equiv a_{1}{\pmod {n_{1}}}.}The second congruence is proved similarly, by exchanging the subscripts 1 and 2.
Consider a sequence of congruence equations:
where theni{\displaystyle n_{i}}are pairwise coprime. The two first equations have a solutiona1,2{\displaystyle a_{1,2}}provided by the method of the previous section. The set of the solutions of these two first equations is the set of all solutions of the equation
As the otherni{\displaystyle n_{i}}are coprime withn1n2,{\displaystyle n_{1}n_{2},}this reduces solving the initial problem ofkequations to a similar problem withk−1{\displaystyle k-1}equations. Iterating the process, one gets eventually the solutions of the initial problem.
For constructing a solution, it is not necessary to make an induction on the number of moduli. However, such a direct construction involves more computation with large numbers, which makes it less efficient and less used. Nevertheless,Lagrange interpolationis a special case of this construction, applied topolynomialsinstead of integers.
LetNi=N/ni{\displaystyle N_{i}=N/n_{i}}be the product of all moduli but one. As theni{\displaystyle n_{i}}are pairwise coprime,Ni{\displaystyle N_{i}}andni{\displaystyle n_{i}}are coprime. ThusBézout's identityapplies, and there exist integersMi{\displaystyle M_{i}}andmi{\displaystyle m_{i}}such that
A solution of the system of congruences is
In fact, asNj{\displaystyle N_{j}}is a multiple ofni{\displaystyle n_{i}}fori≠j,{\displaystyle i\neq j,}we have
for everyi.{\displaystyle i.}
Consider a system of congruences:
where theni{\displaystyle n_{i}}arepairwise coprime, and letN=n1n2⋯nk.{\displaystyle N=n_{1}n_{2}\cdots n_{k}.}In this section several methods are described for computing the unique solution forx{\displaystyle x}, such that0≤x<N,{\displaystyle 0\leq x<N,}and these methods are applied on the example
Several methods of computation are presented. The two first ones are useful for small examples, but become very inefficient when the productn1⋯nk{\displaystyle n_{1}\cdots n_{k}}is large. The third one uses the existence proof given in§ Existence (constructive proof). It is the most convenient when the productn1⋯nk{\displaystyle n_{1}\cdots n_{k}}is large, or for computer computation.
It is easy to check whether a value ofxis a solution: it suffices to compute the remainder of theEuclidean divisionofxby eachni. Thus, to find the solution, it suffices to check successively the integers from0toNuntil finding the solution.
Although very simple, this method is very inefficient. For the simple example considered here,40integers (including0) have to be checked for finding the solution, which is39. This is anexponential timealgorithm, as the size of the input is, up to a constant factor, the number of digits ofN, and the average number of operations is of the order ofN.
Therefore, this method is rarely used, neither for hand-written computation nor on computers.
The search of the solution may be made dramatically faster by sieving. For this method, we suppose, without loss of generality, that0≤ai<ni{\displaystyle 0\leq a_{i}<n_{i}}(if it were not the case, it would suffice to replace eachai{\displaystyle a_{i}}by the remainder of its division byni{\displaystyle n_{i}}). This implies that the solution belongs to thearithmetic progression
By testing the values of these numbers modulon2,{\displaystyle n_{2},}one eventually finds a solutionx2{\displaystyle x_{2}}of the two first congruences. Then the solution belongs to the arithmetic progression
Testing the values of these numbers modulon3,{\displaystyle n_{3},}and continuing until every modulus has been tested eventually yields the solution.
This method is faster if the moduli have been ordered by decreasing value, that is ifn1>n2>⋯>nk.{\displaystyle n_{1}>n_{2}>\cdots >n_{k}.}For the example, this gives the following computation. We consider first the numbers that are congruent to 4 modulo 5 (the largest modulus), which are 4,9 = 4 + 5,14 = 9 + 5, ... For each of them, compute the remainder by 4 (the second largest modulus) until getting a number congruent to 3 modulo 4. Then one can proceed by adding20 = 5 × 4at each step, and computing only the remainders by 3. This gives
This method works well for hand-written computation with a product of moduli that is not too big. However, it is much slower than other methods, for very large products of moduli. Although dramatically faster than the systematic search, this method also has anexponential timecomplexity and is therefore not used on computers.
Theconstructive existence proofshows that, in thecase of two moduli, the solution may be obtained by the computation of theBézout coefficientsof the moduli, followed by a few multiplications, additions andreductions modulon1n2{\displaystyle n_{1}n_{2}}(for getting a result in theinterval(0,n1n2−1){\displaystyle (0,n_{1}n_{2}-1)}). As the Bézout's coefficients may be computed with theextended Euclidean algorithm, the whole computation, at most, has aquadratic timecomplexityofO((s1+s2)2),{\displaystyle O((s_{1}+s_{2})^{2}),}wheresi{\displaystyle s_{i}}denotes the number of digits ofni.{\displaystyle n_{i}.}
For more than two moduli, the method for two moduli allows the replacement of any two congruences by a single congruence modulo the product of the moduli. Iterating this process provides eventually the solution with a complexity, which is quadratic in the number of digits of the product of all moduli. This quadratic time complexity does not depend on the order in which the moduli are regrouped. One may regroup the two first moduli, then regroup the resulting modulus with the next one, and so on. This strategy is the easiest to implement, but it also requires more computation involving large numbers.
Another strategy consists in partitioning the moduli in pairs whose product have comparable sizes (as much as possible), applying, in parallel, the method of two moduli to each pair, and iterating with a number of moduli approximatively divided by two. This method allows an easy parallelization of the algorithm. Also, if fast algorithms (that is, algorithms working inquasilinear time) are used for the basic operations, this method provides an algorithm for the whole computation that works in quasilinear time.
On the current example (which has only three moduli), both strategies are identical and work as follows.
Bézout's identityfor 3 and 4 is
Putting this in the formula given for proving the existence gives
for a solution of the two first congruences, the other solutions being obtained by adding to −9 any multiple of3 × 4 = 12. One may continue with any of these solutions, but the solution3 = −9 +12is smaller (inabsolute value) and thus leads probably to an easier computation
Bézout identity for 5 and 3 × 4 = 12 is
Applying the same formula again, we get a solution of the problem:
The other solutions are obtained by adding any multiple of3 × 4 × 5 = 60, and the smallest positive solution is−21 + 60 = 39.
The system of congruences solved by the Chinese remainder theorem may be rewritten as asystem of linear Diophantine equations:
where the unknown integers arex{\displaystyle x}and thexi.{\displaystyle x_{i}.}Therefore, every general method for solving such systems may be used for finding the solution of Chinese remainder theorem, such as the reduction of thematrixof the system toSmith normal formorHermite normal form. However, as usual when using a general algorithm for a more specific problem, this approach is less efficient than the method of the preceding section, based on a direct use ofBézout's identity.
In§ Statement, the Chinese remainder theorem has been stated in three different ways: in terms of remainders, of congruences, and of aring isomorphism. The statement in terms of remainders does not apply, in general, toprincipal ideal domains, as remainders are not defined in suchrings. However, the two other versions make sense over a principal ideal domainR: it suffices to replace "integer" by "element of the domain" andZ{\displaystyle \mathbb {Z} }byR. These two versions of the theorem are true in this context, because the proofs (except for the first existence proof), are based onEuclid's lemmaandBézout's identity, which are true over every principal domain.
However, in general, the theorem is only an existence theorem and does not provide any way for computing the solution, unless one has an algorithm for computing the coefficients of Bézout's identity.
The statement in terms of remainders given in§ Theorem statementcannot be generalized to any principal ideal domain, but its generalization toEuclidean domainsis straightforward. Theunivariate polynomialsover afieldis the typical example of a Euclidean domain which is not the integers. Therefore, we state the theorem for the case of the ringR=K[X]{\displaystyle R=K[X]}for a fieldK.{\displaystyle K.}For getting the theorem for a general Euclidean domain, it suffices to replace thedegreeby theEuclidean functionof the Euclidean domain.
The Chinese remainder theorem for polynomials is thus: LetPi(X){\displaystyle P_{i}(X)}(the moduli) be, fori=1,…,k{\displaystyle i=1,\dots ,k}, pairwisecoprime polynomialsinR=K[X]{\displaystyle R=K[X]}. Letdi=degPi{\displaystyle d_{i}=\deg P_{i}}be the degree ofPi(X){\displaystyle P_{i}(X)}, andD{\displaystyle D}be the sum of thedi.{\displaystyle d_{i}.}IfAi(X),…,Ak(X){\displaystyle A_{i}(X),\ldots ,A_{k}(X)}are polynomials such thatAi(X)=0{\displaystyle A_{i}(X)=0}ordegAi<di{\displaystyle \deg A_{i}<d_{i}}for everyi, then, there is one and only one polynomialP(X){\displaystyle P(X)}, such thatdegP<D{\displaystyle \deg P<D}and the remainder of theEuclidean divisionofP(X){\displaystyle P(X)}byPi(X){\displaystyle P_{i}(X)}isAi(X){\displaystyle A_{i}(X)}for everyi.
The construction of the solution may be done as in§ Existence (constructive proof)or§ Existence (direct proof). However, the latter construction may be simplified by using, as follows,partial fraction decompositioninstead of theextended Euclidean algorithm.
Thus, we want to find a polynomialP(X){\displaystyle P(X)}, which satisfies the congruences
fori=1,…,k.{\displaystyle i=1,\ldots ,k.}
Consider the polynomials
The partial fraction decomposition of1/Q(X){\displaystyle 1/Q(X)}giveskpolynomialsSi(X){\displaystyle S_{i}(X)}with degreesdegSi(X)<di,{\displaystyle \deg S_{i}(X)<d_{i},}such that
and thus
Then a solution of the simultaneous congruence system is given by the polynomial
In fact, we have
for1≤i≤k.{\displaystyle 1\leq i\leq k.}
This solution may have a degree larger thanD=∑i=1kdi.{\displaystyle D=\sum _{i=1}^{k}d_{i}.}The unique solution of degree less thanD{\displaystyle D}may be deduced by considering the remainderBi(X){\displaystyle B_{i}(X)}of the Euclidean division ofAi(X)Si(X){\displaystyle A_{i}(X)S_{i}(X)}byPi(X).{\displaystyle P_{i}(X).}This solution is
A special case of Chinese remainder theorem for polynomials isLagrange interpolation. For this, considerkmonic polynomialsof degree one:
They are pairwise coprime if thexi{\displaystyle x_{i}}are all different. The remainder of the division byPi(X){\displaystyle P_{i}(X)}of a polynomialP(X){\displaystyle P(X)}isP(xi){\displaystyle P(x_{i})}, by thepolynomial remainder theorem.
Now, letA1,…,Ak{\displaystyle A_{1},\ldots ,A_{k}}be constants (polynomials of degree 0) inK.{\displaystyle K.}Both Lagrange interpolation and Chinese remainder theorem assert the existence of a unique polynomialP(X),{\displaystyle P(X),}of degree less thank{\displaystyle k}such that
for everyi.{\displaystyle i.}
Lagrange interpolation formula is exactly the result, in this case, of the above construction of the solution. More precisely, let
Thepartial fraction decompositionof1Q(X){\displaystyle {\frac {1}{Q(X)}}}is
In fact, reducing the right-hand side to a common denominator one gets
and the numerator is equal to one, as being a polynomial of degree less thank,{\displaystyle k,}which takes the value one fork{\displaystyle k}different values ofX.{\displaystyle X.}
Using the above general formula, we get the Lagrange interpolation formula:
Hermite interpolationis an application of the Chinese remainder theorem for univariate polynomials, which may involve moduli of arbitrary degrees (Lagrange interpolation involves only moduli of degree one).
The problem consists of finding a polynomial of the least possible degree, such that the polynomial and its firstderivativestake given values at some fixed points.
More precisely, letx1,…,xk{\displaystyle x_{1},\ldots ,x_{k}}bek{\displaystyle k}elements of the groundfieldK,{\displaystyle K,}and, fori=1,…,k,{\displaystyle i=1,\ldots ,k,}letai,0,ai,1,…,ai,ri−1{\displaystyle a_{i,0},a_{i,1},\ldots ,a_{i,r_{i}-1}}be the values of the firstri{\displaystyle r_{i}}derivatives of the sought polynomial atxi{\displaystyle x_{i}}(including the 0th derivative, which is the value of the polynomial itself). The problem is to find a polynomialP(X){\displaystyle P(X)}such that itsjth derivative takes the valueai,j{\displaystyle a_{i,j}}atxi,{\displaystyle x_{i},}fori=1,…,k{\displaystyle i=1,\ldots ,k}andj=0,…,rj.{\displaystyle j=0,\ldots ,r_{j}.}
Consider the polynomial
This is theTaylor polynomialof orderri−1{\displaystyle r_{i}-1}atxi{\displaystyle x_{i}}, of the unknown polynomialP(X).{\displaystyle P(X).}Therefore, we must have
Conversely, any polynomialP(X){\displaystyle P(X)}that satisfies thesek{\displaystyle k}congruences, in particular verifies, for anyi=1,…,k{\displaystyle i=1,\ldots ,k}
thereforePi(X){\displaystyle P_{i}(X)}is its Taylor polynomial of orderri−1{\displaystyle r_{i}-1}atxi{\displaystyle x_{i}}, that is,P(X){\displaystyle P(X)}solves the initial Hermite interpolation problem.
The Chinese remainder theorem asserts that there exists exactly one polynomial of degree less than the sum of theri,{\displaystyle r_{i},}which satisfies thesek{\displaystyle k}congruences.
There are several ways for computing the solutionP(X).{\displaystyle P(X).}One may use the method described at the beginning of§ Over univariate polynomial rings and Euclidean domains. One may also use the constructions given in§ Existence (constructive proof)or§ Existence (direct proof).
The Chinese remainder theorem can be generalized to non-coprime moduli. Letm,n,a,b{\displaystyle m,n,a,b}be any integers, letg=gcd(m,n){\displaystyle g=\gcd(m,n)};M=lcm(m,n){\displaystyle M=\operatorname {lcm} (m,n)}, and consider the system of congruences:
Ifa≡b(modg){\displaystyle a\equiv b{\pmod {g}}}, then this system has a unique solution moduloM=mn/g{\displaystyle M=mn/g}. Otherwise, it has no solutions.
If one usesBézout's identityto writeg=um+vn{\displaystyle g=um+vn}, then the solution is given by
This defines an integer, asgdivides bothmandn. Otherwise, the proof is very similar to that for coprime moduli.[17]
The Chinese remainder theorem can be generalized to anyring, by usingcoprime ideals(also calledcomaximal ideals). TwoidealsIandJare coprime if there are elementsi∈I{\displaystyle i\in I}andj∈J{\displaystyle j\in J}such thati+j=1.{\displaystyle i+j=1.}This relation plays the role ofBézout's identityin the proofs related to this generalization, which otherwise are very similar. The generalization may be stated as follows.[18][19]
LetI1, ...,Ikbe two-sided ideals of a ringR{\displaystyle R}and letIbe theirintersection. If the ideals are pairwise coprime, we have theisomorphism:
between thequotient ringR/I{\displaystyle R/I}and thedirect productof theR/Ii,{\displaystyle R/I_{i},}where "xmodI{\displaystyle x{\bmod {I}}}" denotes theimageof the elementx{\displaystyle x}in the quotient ring defined by the idealI.{\displaystyle I.}Moreover, ifR{\displaystyle R}iscommutative, then the ideal intersection of pairwise coprime ideals is equal to theirproduct; that is
ifIiandIjare coprime for alli≠j.
LetI1,I2,…,Ik{\displaystyle I_{1},I_{2},\dots ,I_{k}}be pairwise coprime two-sided ideals with⋂i=1kIi=0,{\displaystyle \bigcap _{i=1}^{k}I_{i}=0,}and
be the isomorphism defined above. Letfi=(0,…,1,…,0){\displaystyle f_{i}=(0,\ldots ,1,\ldots ,0)}be the element of(R/I1)×⋯×(R/Ik){\displaystyle (R/I_{1})\times \cdots \times (R/I_{k})}whose components are all0except theith which is1, andei=φ−1(fi).{\displaystyle e_{i}=\varphi ^{-1}(f_{i}).}
Theei{\displaystyle e_{i}}arecentral idempotentsthat are pairwiseorthogonal; this means, in particular, thatei2=ei{\displaystyle e_{i}^{2}=e_{i}}andeiej=ejei=0{\displaystyle e_{i}e_{j}=e_{j}e_{i}=0}for everyiandj. Moreover, one hase1+⋯+en=1,{\textstyle e_{1}+\cdots +e_{n}=1,}andIi=R(1−ei).{\displaystyle I_{i}=R(1-e_{i}).}
In summary, this generalized Chinese remainder theorem is the equivalence between giving pairwise coprime two-sided ideals with a zero intersection, and giving central and pairwise orthogonal idempotents that sum to1.[20]
The Chinese remainder theorem has been used to construct aGödel numbering for sequences, which is involved in the proof ofGödel's incompleteness theorems.
Theprime-factor FFT algorithm(also called Good-Thomas algorithm) uses the Chinese remainder theorem for reducing the computation of afast Fourier transformof sizen1n2{\displaystyle n_{1}n_{2}}to the computation of two fast Fourier transforms of smaller sizesn1{\displaystyle n_{1}}andn2{\displaystyle n_{2}}(providing thatn1{\displaystyle n_{1}}andn2{\displaystyle n_{2}}are coprime).
Mostimplementations of RSA use the Chinese remainder theoremduring signing ofHTTPScertificates and during decryption.
The Chinese remainder theorem can also be used insecret sharing, which consists of distributing a set of shares among a group of people who, all together (but no one alone), can recover a certain secret from the given set of shares. Each of the shares is represented in a congruence, and the solution of the system of congruences using the Chinese remainder theorem is the secret to be recovered.Secret sharing using the Chinese remainder theoremuses, along with the Chinese remainder theorem, special sequences of integers that guarantee the impossibility of recovering the secret from a set of shares with less than a certaincardinality.
Therange ambiguity resolutiontechniques used withmedium pulse repetition frequencyradar can be seen as a special case of the Chinese remainder theorem.
Given asurjectionZ/n→Z/m{\displaystyle \mathbb {Z} /n\to \mathbb {Z} /m}offiniteabelian groups, we can use the Chinese remainder theorem to give a complete description of any such map. First of all, the theorem gives isomorphisms
where{pm1,…,pmj}⊆{pn1,…,pni}{\displaystyle \{p_{m_{1}},\ldots ,p_{m_{j}}\}\subseteq \{p_{n_{1}},\ldots ,p_{n_{i}}\}}. In addition, for any induced map
from the original surjection, we haveak≥bl{\displaystyle a_{k}\geq b_{l}}andpnk=pml,{\displaystyle p_{n_{k}}=p_{m_{l}},}since for a pair ofprimesp,q{\displaystyle p,q}, the only non-zero surjections
can be defined ifp=q{\displaystyle p=q}anda≥b{\displaystyle a\geq b}.
These observations are pivotal for constructing the ring ofprofinite integers, which is given as aninverse limitof all such maps.
Dedekind's theorem on the linear independence of characters.LetMbe amonoidandkanintegral domain, viewed as a monoid by considering the multiplication onk. Then any finite family(fi)i∈Iof distinctmonoid homomorphismsfi:M→kislinearly independent. In other words, every family(αi)i∈Iof elementsαi∈ksatisfying
must be equal to the family(0)i∈I.
Proof.First assume thatkis afield, otherwise, replace the integral domainkby itsquotient field, and nothing will change. We can linearly extend the monoid homomorphismsfi:M→ktok-algebra homomorphismsFi:k[M] →k, wherek[M]is themonoid ringofMoverk. Then, by linearity, the condition
yields
Next, fori,j∈I;i≠jthe twok-linear mapsFi:k[M] →kandFj:k[M] →kare not proportional to each other. Otherwisefiandfjwould also be proportional, and thus equal since as monoid homomorphisms they satisfy:fi(1) = 1 =fj(1), which contradicts the assumption that they are distinct.
Therefore, thekernelsKerFiandKerFjare distinct. Sincek[M]/KerFi≅Fi(k[M]) =kis a field,KerFiis amaximal idealofk[M]for everyiinI. Because they are distinct and maximal the idealsKerFiandKerFjare coprime wheneveri≠j. The Chinese Remainder Theorem (for general rings) yields an isomorphism:
where
Consequently, the map
is surjective. Under the isomorphismsk[M]/KerFi→Fi(k[M]) =k,the mapΦcorresponds to:
Now,
yields
for every vector(ui)i∈Iin theimageof the mapψ. Sinceψis surjective, this means that
for every vector
Consequently,(αi)i∈I= (0)i∈I. QED.
|
https://en.wikipedia.org/wiki/Linear_congruence_theorem
|
Kuṭṭakais analgorithmfor findingintegersolutions oflinearDiophantine equations. A linear Diophantine equation is anequationof the formax+by=cwherexandyareunknown quantitiesanda,b, andcare known quantities with integer values. The algorithm was originally invented by the Indian astronomer-mathematicianĀryabhaṭa(476–550 CE) and is described very briefly in hisĀryabhaṭīya. Āryabhaṭa did not give the algorithm the nameKuṭṭaka, and his description of the method was mostly obscure and incomprehensible. It wasBhāskara I(c. 600 – c. 680) who gave a detailed description of the algorithm with several examples from astronomy in hisĀryabhatiyabhāṣya, who gave the algorithm the nameKuṭṭaka. InSanskrit, the word Kuṭṭaka meanspulverization(reducing to powder), and it indicates the nature of the algorithm. The algorithm in essence is a process where the coefficients in a given linear Diophantine equation are broken up into smaller numbers to get a linear Diophantine equation with smaller coefficients. In general, it is easy to find integer solutions of linear Diophantine equations with small coefficients. From a solution to the reduced equation, a solution to the original equation can be determined. Many Indian mathematicians after Aryabhaṭa have discussed the Kuṭṭaka method with variations and refinements. The Kuṭṭaka method was considered to be so important that the entire subject of algebra used to be calledKuṭṭaka-ganitaor simplyKuṭṭaka. Sometimes the subject of solving linear Diophantine equations is also calledKuṭṭaka.
In literature, there are several other names for the Kuṭṭaka algorithm likeKuṭṭa,KuṭṭakāraandKuṭṭikāra. There is also a treatise devoted exclusively to a discussion of Kuṭṭaka. Such specialized treatises are very rare in the mathematical literature of ancient India.[1]The treatise written in Sanskrit is titledKuṭṭākāra Śirōmaṇiand is authored by one Devaraja.[2]
The Kuṭṭaka algorithm has much similarity with and can be considered as a precursor of the modern dayextended Euclidean algorithm. The latter algorithm is a procedure for finding integersxandysatisfying the conditionax+by=gcd(a,b).[3]
The problem that can supposedly be solved by the Kuṭṭaka method was not formulated by Aryabhaṭa as a problem of solving the linear Diophantine equation. Aryabhaṭa considered the following problems all of which are equivalent to the problem of solving the linear Diophantine equation:
Aryabhata and other Indian writers had noted the following property of linear Diophantine equations: "The linear Diophantine equationax+by=chas a solution if and only if gcd(a,b) is adivisorofc." So the first stage in thepulverizationprocess is to cancel out the common factor gcd(a,b) froma,bandc, and obtain an equation with smaller coefficients in which the coefficients ofxandyarerelatively prime.
For example, Bhāskara I observes: "The dividend and the divisor shall become prime to each other, on being divided by the residue of their mutual division. The operation of the pulveriser should be considered in relation to them."[1]
Aryabhata gave the algorithm for solving the linear Diophantine equation in verses 32–33 of Ganitapada of Aryabhatiya.[1]Taking Bhāskara I's explanation of these verses also into consideration, Bibhutibbhushan Datta has given the following translation of these verses:
Some comments are in order.
Without loss of generality, letax−by=c{\displaystyle ax-by=c}be our Diophantine equation wherea,bare positive integers andcis an integer. Divide both sides of the equation bygcd(a,b){\displaystyle \gcd(a,b)}. Ifcis not divisible bygcd(a,b){\displaystyle \gcd(a,b)}then there are no integer solutions to this equation. After the division, we get the equationa′x−b′y=c′{\displaystyle a'x-b'y=c'}. The solution to this equation is the solution toax−by=c{\displaystyle ax-by=c}. Without loss of generality, let us consider a > b.
UsingEuclidean division, follow these recursive steps:
Now, define quantitiesxn+2,xn+1,xn,... by backward induction as follows:
Ifnis odd, takexn+2= 0 andxn+1= 1.
Ifnis even, takexn+2=1 andxn+1=rn−1−1.
Now, calculate allxm(n≥m≥1) byxm=amxm+1+xm+2. Theny=c′x1andx=c′x2.
Consider the following problem:
The required number is 334.
The number 334 is thesmallestinteger which leaves remainders 15 and 19 when divided by 29 and 45 respectively.
The following example taken fromLaghubhāskarīyaofBhāskara I[4]illustrates how the Kuttaka algorithm was used in the astronomical calculations in India.[5]
The sum, the difference and the product increased by unity, of the residues of the revolutions of Saturn and Mars – each is a perfect square. Taking the equations furnished by the above and applying the methods of such quadratics obtain the (simplest) solution by the substitution of 2, 3, etc. successively (in the general solution). Then calculate theaharganaand the revolutions performed by Saturn and Mars in that time together with the number of solar years elapsed.
In the Indian astronomical tradition, aYugais a period consisting of 1,577,917,500 civil days. Saturn makes 146,564 revolutions and Mars makes 229,6824 revolutions in a Yuga. So Saturn makes 146,564/1,577,917,500 = 36,641/394,479,375 revolutions in a day. By saying that the residue of the revolution of Saturn isx, what is meant is that the fractional number of revolutions isx/394,479,375. Similarly, Mars makes 229,6824/1,577,917,500 = 190,412/131,493,125 revolutions in a day. By saying that the residue of the revolution of Mars isy, what is meant is that the fractional number of revolutions isy/131,493,125.
Letxandydenote the residues of the revolutions of Saturn and Mars respectively satisfying the conditions stated in the problem. They must be such that each ofx+y.x−yandxy+ 1is a perfect square.
Setting
one obtains
and so
Forxy+ 1 also to be a perfect square we must have
Thus the following general solution is obtained:
The valueq= 2 yields the special solutionx= 40,y= 24.
Aharganais the number of days elapsed since the beginning of the Yuga.
Letube the value of the ahargana corresponding the residue 24 for Saturn. Duringudays, saturn would have completed (36,641/394,479,375)×unumber of revolutions. Since there is a residue of 24, this number would include the fractional number 24/394,479,375 of revolutions also. Hence during the aharganau, the number of revolutions completed would be
which would be an integer. Denoting this integer byv, the problem reduces to solving the following linear Diophantine equation:
Kuttaka may be applied to solve this equation. The smallest solution is
Letube the value of the ahargana corresponding the residue 40 for Mars. Duringudays, Mars would have completed (190,412/131,493,125) ×unumber of revolutions. Since there is a residue of 40, this number would include the fractional number 40/131,493,125 of revolutions also. Hence during the aharganau, the number of revolutions completed would be
which would be an integer. Denoting this integer byv, the problem reduces to solving the following linear Diophantine equation:
Kuttaka may be applied to solve this equation. The smallest solution is
|
https://en.wikipedia.org/wiki/Ku%E1%B9%AD%E1%B9%ADaka
|
Incomputer security, anattribute certificate, orauthorization certificate(AC) is adigital documentcontaining attributes associated to the holder by the issuer.[1]When the associated attributes are mainly used for the purpose ofauthorization, AC is calledauthorization certificate. AC is standardized inX.509. RFC 5755 further specifies the usage for authorization purpose in the Internet.
The authorization certificate works in conjunction with apublic key certificate(PKC). While the PKC is issued by acertificate authority(CA) and is used as a proof of identity of its holder like apassport, the authorization certificate is issued by an attribute authority (AA) and is used to characterize or entitle its holder like avisa. Because identity information seldom changes and has a long validity time while attribute information frequently changes or has a short validity time, separate certificates with different security rigours, validity times and issuers are necessary.[2]
An AC resembles a PKC but contains nopublic keybecause an AC verifier is under the control of the AC issuer, and therefore, trusts the issuer directly by having the public key of the issuer preinstalled. This means that once the AC issuer'sprivate keyis compromised, the issuer has to generate a newkey pairand replaces the old public key in all verifiers under its control with the new one.
The verification of an AC requires the presence of the PKC that is referred as the AC holder in the AC.
As with a PKC, an AC can be chained to delegate attributions. For example, an authorization certificate issued for Alice authorizes her to use a particular service. Alice can delegate this privilege to her assistant Bob by issuing an AC for Bob's PKC. When Bob wants to use the service, he presents his PKC and a chain of ACs starting from his own AC issued by Alice and then Alice's AC issued by the issuer that the service trusts. In this way, the service can verify that Alice has delegated her privilege to Bob and that Alice has been authorized to use the service by the issuer that controls the service. RFC 3281, however, does not recommend the use of AC chains because of the complexity in administering and processing the chain and there is little use of AC in the Internet.
To use a service or a resource that the issuer of an AC controls, a user presents both the PKC and the AC to a part of the service or resource that functions as an AC verifier. The verifier will first check the identity of the user using the PKC, for example, by asking the user to decrypt a message encrypted by the user's public key in the PKC. If the authentication is successful, the verifier will use the preinstalled public key of the AC issuer to check the validity of the presented AC. If the AC is valid, the verifier will check whether or not the PKC specified in the AC matches the presented PKC. If it matches, the verifier will check the validity period of the AC. If the AC is still valid, the verifier can perform additional checks before offering the user a particular level of service or resource usage in accordance to the attributes contained in the AC.
For example, a software developer that already has aPKCwants to deploy its software in a computing device employingDRMlikeiPadwhere software can only be run in the device after the software has been approved by the device manufacturer. The software developer signs the software with theprivate keyof the PKC and sends the signed software to the device manufacturer for approval. After authenticating the developer using the PKC and reviewing the software, the manufacturer may decide to issue an AC granting the software the basic capability to install itself and be executed as well as an additional capability to use the Wi-Fi device following theprinciple of least privilege. In this example, the AC does not refer to the PKC of the developer as the holder but to the software, for example, by storing the developer's signature of the software in the holder field of the AC. When the software is put into the computing device, the device will verify the integrity of the software using the developer's PKC before checking the validity of the AC and granting the software access to the device functionalities.
A user may also need to obtain several ACs from different issuers to use a particular service. For example, a company gives one of its employees a company-wide AC that specifies engineering department as the work area. To access engineering data, however, the employee also needs a security clearance AC from the head of the engineering department. In this example, the resource of engineering data needs to be preinstalled with the public keys of both the company-wide and the engineering department AC issuers.
Using attribute certificate, the service or resourcehostdoes not need to maintain anaccess control listthat can potentially be large or to always be connected to a network to access a central server like when usingKerberos. It is similar to the idea ofcapabilitiesin which the permission (or permissions) to use a service or resource is not stored in the service or resource itself but in the users using atamper resistancemechanism.
|
https://en.wikipedia.org/wiki/Authorization_certificate
|
RFC4210(CMPv2, 2005)RFC9480(CMPv3, 2023)
RFC2510(CMPv1, 1999)
TheCertificate Management Protocol(CMP) is an Internet protocol standardized by theIETFused for obtaining X.509digital certificatesin apublic key infrastructure(PKI).
CMP is a very feature-rich and flexible protocol, supporting many types of cryptography.
CMP messages are self-contained, which, as opposed toEST, makes the protocol independent of the transport mechanism and provides end-to-end security.
CMP messages are encoded inASN.1, using theDERmethod.
CMP is described inRFC4210. Enrollment request messages employ the Certificate Request Message Format (CRMF), described inRFC4211.
The only other protocol so far using CRMF isCertificate Management over CMS(CMC), described inRFC5273.
An obsolete version of CMP is described inRFC2510, the respective CRMF version inRFC2511.
In November 2023,CMP Updates,CMP Algorithms, andCoAP transfer for CMP, have been published as well as theLightweight CMP Profilefocusing on industrial use.
In apublic key infrastructure(PKI), so-calledend entities(EEs) act as CMP client, requesting one or more certificates for themselves from acertificate authority(CA), which issues the legal certificates and acts as a CMP server. None or any number ofregistration authorities(RA), can be used to mediate between the EEs and CAs, having both a downstream CMP server interface and an upstream CMP client interface. Using a "cross-certification request" a CA can get a certificate signed by another CA.
CMP messages are usually transferred using HTTP, but any reliable means of transportation can be used.
TheContent-Typeused isapplication/pkixcmp; older versions of the draft usedapplication/pkixcmp-poll,application/x-pkixcmporapplication/x-pkixcmp-poll.
|
https://en.wikipedia.org/wiki/Certificate_Management_Protocol
|
RFC 5272
RFC 2797
TheCertificate Management over CMS(CMC) is anInternet Standardpublished by theIETF, defining transport mechanisms for theCryptographic Message Syntax(CMS). It is defined inRFC5272, its transport mechanisms inRFC5273.
Similarly to theCertificate Management Protocol(CMP), it can be used for obtaining X.509digital certificatesin apublic key infrastructure(PKI).
CMS is one of two protocols utilizing the Certificate Request Message Format (CRMF), described inRFC4211, with the other protocol being CMP.
TheEnrollment over Secure Transport(EST) protocol, described inRFC7030, can be seen as a profile of CMC for use in provisioning certificates to end entities. As such, EST can play a similar role toSCEP.
|
https://en.wikipedia.org/wiki/Certificate_Management_over_CMS
|
Simple Certificate Enrollment Protocol(SCEP) is described by the informationalRFC8894. Older versions of thisprotocolbecame a de facto industrial standard for pragmatic provisioning of digital certificates mostly for network equipment.
The protocol has been designed to make the request and issuing ofdigital certificatesas simple as possible for any standard network user. These processes have usually required intensive input fromnetwork administrators, and so have not been suited to large-scale deployments.
The Simple Certificate Enrollment Protocol still is the most popular and widely available certificate enrollment protocol, being used by numerous manufacturers of network equipment and software who are developing simplified means of handling certificates for large-scale implementation to everyday users.[citation needed]It is used, for example, by theCisco Internetworking Operating System(IOS), though Cisco promotes theEnrollment over Secure Transport(EST), with additional features, andiPhones(iOS) to enroll in enterprisepublic key infrastructure(PKI).[1]Most PKI software (specifically RA implementations) supports it, including the Network Device Enrollment Service (NDES) ofActive DirectoryCertificate Service andIntune.[2]
SCEP was designed by Verisign for Cisco[3]as a lean alternative toCertificate Management over CMS(CMC) and the very powerful but also rather bulkyCertificate Management Protocol(CMP). It had support from Microsoft early with its continuous inclusion in Windows starting withWindows 2000.[4]In around 2010,Ciscosuspended work on SCEP and developedESTinstead. In 2015,Peter Gutmannrevived theInternet Draftdue to SCEP widespread use in industry and in other standards.[5]He updated the draft with more modern algorithms and corrected numerous issues in the original specification. In September 2020, the draft was published as informationalRFC8894, more than twenty years after the beginning of the standardization effort.[6]The new version also supports enrollment of non-RSA certificates (e.g., forECCpublic keys).
|
https://en.wikipedia.org/wiki/Simple_Certificate_Enrollment_Protocol
|
TheEnrollment over Secure Transport, orESTis acryptographicprotocolthat describes anX.509certificate management protocol targetingpublic key infrastructure(PKI) clients that need to acquire client certificates and associatedcertificate authority(CA) certificates. EST is described inRFC7030. EST has been put forward as a replacement forSCEP, being easier to implement on devices already having an HTTPS stack. EST uses HTTPS as transport and leveragesTLSfor many of its security attributes. EST has described standardized URLs and uses the well-known Uniform Resource Identifiers (URIs) definition codified inRFC5785.
EST has a following set of operations:
The basic functions of EST were designed to be easy to use and although not aREST API, it can be used in a REST-like manner using simple tools such asOpenSSLandcURL. A simple command to make initial enrollment with a pre-generated PKCS#10Certificate Signing Request(stored as device.b64), using one of the authentication mechanisms (username:password) specified in EST is:
curl-v--cacertManagementCA.cacert.pem--userusername:password--data@device.b64-odevice-p7.b64-H"Content-Type: application/pkcs10"-H"Content-Transfer-Encoding: base64"https://hostname.tld/.well-known/est/simpleenroll
The issued certificate, returned as a Base64-encodedPKCS#7message, is stored as device-p7.b64.
This cryptography-related article is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/Enrollment_over_Secure_Transport
|
TheAutomatic Certificate Management Environment(ACME) protocol is acommunications protocolfor automating interactions betweencertificate authoritiesand their users' servers, allowing the automated deployment ofpublic key infrastructureat very low cost.[1][2]It was designed by theInternet Security Research Group(ISRG) for theirLet's Encryptservice.[1]
The protocol, based on passingJSON-formatted messages overHTTPS,[2][3]has been published as an Internet Standard inRFC8555[4]by its own charteredIETFworking group.[5]
The ISRG providesfree and open-sourcereference implementations for ACME:certbotis aPython-based implementation of server certificate management software using the ACME protocol,[6][7][8]andboulderis acertificate authorityimplementation, written inGo.[9]
Since 2015 a large variety of client options have appeared for all operating systems.[10]
API v1 specification was published on April 12, 2016. It supports issuing certificates for fully-qualified domain names, such asexample.comorcluster.example.com, but not wildcards like*.example.com. Let's Encrypt turned off API v1 support on 1 June 2021.[11]
API v2 was released March 13, 2018 after being pushed back several times. ACME v2 is not backwards compatible with v1. Version 2 supports wildcard domains, such as*.example.com, allowing for many subdomains to have trustedTLS, e.g.https://cluster01.example.com,https://cluster02.example.com,https://example.com, on private networks under a single domain using a single shared "wildcard" certificate.[12]A major new requirement in v2 is that requests for wildcard certificates require the modification of a Domain Name ServiceTXT record, verifying control over the domain.
Changes to ACME v2 protocol since v1 include:[13]
|
https://en.wikipedia.org/wiki/Automatic_Certificate_Management_Environment
|
Resource Public Key Infrastructure(RPKI), also known asResource Certification, is a specializedpublic key infrastructure(PKI) framework to support improved security for theInternet'sBGProutinginfrastructure.
RPKI provides a way to connect Internet number resource information (such asAutonomous Systemnumbers andIP addresses) to atrust anchor. The certificate structure mirrors the way in whichInternet numberresources are distributed. That is, resources are initially distributed by theIANAto theregional Internet registries(RIRs), who in turn distribute them tolocal Internet registries(LIRs), who then distribute the resources to their customers. RPKI can be used by the legitimate holders of the resources to control the operation of Internetrouting protocolsto preventroute hijackingand other attacks. In particular, RPKI is used to secure theBorder Gateway Protocol(BGP) through BGP Route Origin Validation (ROV), as well asNeighbor Discovery Protocol(ND) forIPv6through theSecure Neighbor Discoveryprotocol (SEND).
The RPKI architecture is documented in RFC 6480. The RPKI specification is documented in a spread out series of RFCs: RFC 6481, RFC 6482, RFC 6483, RFC 6484, RFC 6485, RFC 6486, RFC 6487, RFC 6488, RFC 6489, RFC 6490, RFC 6491, RFC 6492, and RFC 6493. SEND is documented in RFC 6494 and RFC 6495. These RFCs are a product of theIETF's SIDR ("Secure Inter-Domain Routing") working group,[1]and are based on a threat analysis which was documented in RFC 4593. These standards cover BGP origin validation, while path validation is provided byBGPsec, which has been standardized separately in RFC 8205. Several implementations for prefix origin validation already exist.[2]
RPKI usesX.509PKI certificates (RFC 5280) with extensions for IP addresses and AS identifiers (RFC 3779). It allows the members ofregional Internet registries, known aslocal Internet registries(LIRs), to obtain a resource certificate listing theInternet numberresources they hold. This offers them validatable proof of holdership, though the certificate does not contain identity information. Using the resource certificate, LIRs can create cryptographic attestations about the route announcements they authorise to be made with the prefixes and ASNs they hold. These attestations are described below.
ARoute Origin Authorization(ROA)[3]states whichautonomous system(AS) is authorised to originate certainIP prefixes. In addition, it can determine the maximum length of the prefix that the AS is authorised to advertise.
The maximum prefix length is an optional field. When not defined, the AS is only authorised to advertise exactly the prefix specified. Any more specific announcement of the prefix will be considered invalid. This is a way to enforce aggregation and prevent hijacking through the announcement of a more specific prefix.
When present, this specifies the length of the most specific IP prefix that the AS is authorised to advertise. For example, if the IP address prefix is10.0.0.0/16and the maximum length is 22, the AS is authorised to advertise any prefix under10.0.0.0/16, as long as it is no more specific than/22. So, in this example, the AS would be authorised to advertise10.0.0.0/16,10.0.128.0/20or10.0.252.0/22, but not10.0.255.0/24.
AnAutonomous System Provider Authorization(ASPA) states which networks are permitted to appear as direct upstream adjacencies of an autonomous system in BGP AS_PATHs.[4]
When a ROA is created for a certain combination of origin AS and prefix, this will have an effect on the RPKI validity[5]of one or more route announcements. They can be:
Note that invalid BGP updates may also be due to incorrectly configured ROAs.[6]
There are open source tools[7]available to run the certificate authority and manage the resource certificate and child objects such as ROAs. In addition, the RIRs have a hosted RPKI platform available in their member portals. This allows LIRs to choose to rely on a hosted system, or run their own software.
The system does not use a single repository publication point to publish RPKI objects. Instead, the RPKI repository system consists of multiple distributed and delegated repository publication points. Each repository publication point is associated with one or more RPKI certificates' publication points. In practice this means that when running a certificate authority, an LIR can either publish all cryptographic material themselves, or they can rely on a third party for publication. When an LIR chooses to use the hosted system provided by the RIR, in principle publication is done in the RIR repository.
Relying party software will fetch, cache, and validate repository data usingrsyncor the RPKI Repository Delta Protocol (RFC 8182).[8]It is important for a relying party to regularly synchronize with all the publication points to maintain a complete and timely view of repository data. Incomplete or stale data can lead to erroneous routing decisions.[9][10]
After validation of ROAs, the attestations can be compared to BGP routing and aid network operators in their decision-making process. This can be done manually, but the validated prefix origin data can also be sent to a supported router using the RPKI to Router Protocol (RFC 6810),[11]Cisco Systemsoffers native support on many platforms[12]for fetching the RPKI data set and using it in the router configuration.[13]Juniper offers support on all platforms[14]that run version 12.2 or newer.Quaggaobtains this functionality through BGP Secure Routing Extensions (BGP-SRx)[15]or a RPKI implementation[16]fully RFC-compliant based on RTRlib. The RTRlib[17]provides an open source C implementation of the RTR protocol and prefix origin verification. The library is useful for developers of routing software but also for network operators.[18]Developers can integrate the RTRlib into the BGP daemon to extend their implementation towards RPKI. Network operators may use the RTRlib to develop monitoring tools (e.g., to check the proper operation of caches or to evaluate their performance).
RFC 6494 updates the certificate validation method of theSecure Neighbor Discoveryprotocol (SEND) security mechanisms forNeighbor Discovery Protocol(ND) to use RPKI for use in IPv6. It defines a SEND certificate profile utilizing a modified RFC 6487 RPKI certificate profile which must include a single RFC 3779 IP address delegation extension.
|
https://en.wikipedia.org/wiki/Resource_Public_Key_Infrastructure
|
In mathematics, asemigroupis analgebraic structureconsisting of asettogether with anassociativeinternalbinary operationon it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmeticmultiplication):x⋅y, or simplyxy, denotes the result of applying the semigroup operation to theordered pair(x,y). Associativity is formally expressed as that(x⋅y) ⋅z=x⋅ (y⋅z)for allx,yandzin the semigroup.
Semigroups may be considered a special case ofmagmas, where the operation is associative, or as a generalization ofgroups, without requiring the existence of an identity element or inverses.[a]As in the case of groups or magmas, the semigroup operation need not becommutative, sox⋅yis not necessarily equal toy⋅x; a well-known example of an operation that is associative but non-commutative ismatrix multiplication. If the semigroup operation is commutative, then the semigroup is called acommutative semigroupor (less often than in theanalogous case of groups) it may be called anabelian semigroup.
Amonoidis an algebraic structure intermediate between semigroups and groups, and is a semigroup having anidentity element, thus obeying all but one of the axioms of a group: existence of inverses is not required of a monoid. A natural example isstringswithconcatenationas the binary operation, and the empty string as the identity element. Restricting to non-emptystringsgives an example of a semigroup that is not a monoid. Positiveintegerswith addition form a commutative semigroup that is not a monoid, whereas the non-negativeintegersdo form a monoid. A semigroup without an identity element can be easily turned into a monoid by just adding an identity element. Consequently, monoids are studied in the theory of semigroups rather than in group theory. Semigroups should not be confused withquasigroups, which are generalization of groups in a different direction; the operation in a quasigroup need not be associative but quasigroupspreserve from groupsthe notion ofdivision. Division in semigroups (or in monoids) is not possible in general.
The formal study of semigroups began in the early 20th century. Early results includea Cayley theorem for semigroupsrealizing any semigroup as atransformation semigroup, in which arbitrary functions replace the role of bijections in group theory. A deep result in the classification of finite semigroups isKrohn–Rhodes theory, analogous to theJordan–Hölder decompositionfor finite groups. Some other techniques for studying semigroups, likeGreen's relations, do not resemble anything in group theory.
The theory of finite semigroups has been of particular importance intheoretical computer sciencesince the 1950s because of the natural link between finite semigroups andfinite automatavia thesyntactic monoid. Inprobability theory, semigroups are associated withMarkov processes.[1]In other areas ofapplied mathematics, semigroups are fundamental models forlinear time-invariant systems. Inpartial differential equations, a semigroup is associated to any equation whose spatial evolution is independent of time.
There are numerousspecial classes of semigroups, semigroups with additional properties, which appear in particular applications. Some of these classes are even closer to groups by exhibiting some additional but not all properties of a group. Of these we mention:regular semigroups,orthodox semigroups,semigroups with involution,inverse semigroupsandcancellative semigroups. There are also interesting classes of semigroups that do not contain any groups except thetrivial group; examples of the latter kind arebandsand their commutative subclass –semilattices, which are alsoordered algebraic structures.
A semigroup is asetStogether with abinary operation⋅ (that is, afunction⋅ :S×S→S) that satisfies theassociative property:
More succinctly, a semigroup is an associativemagma.
Aleft identityof a semigroupS(or more generally,magma) is an elementesuch that for allxinS,e⋅x=x. Similarly, aright identityis an elementfsuch that for allxinS,x⋅f=x. Left and right identities are both calledone-sided identities. A semigroup may have one or more left identities but no right identity, and vice versa.
Atwo-sided identity(or justidentity) is an element that is both a left and right identity. Semigroups with a two-sided identity are calledmonoids. A semigroup may have at most one two-sided identity. If a semigroup has a two-sided identity, then the two-sided identity is the only one-sided identity in the semigroup. If a semigroup has both a left identity and a right identity, then it has a two-sided identity (which is therefore the unique one-sided identity).
A semigroupSwithout identity may beembeddedin a monoid formed by adjoining an elemente∉StoSand defininge⋅s=s⋅e=sfor alls∈S∪ {e}.[2][3]The notationS1denotes a monoid obtained fromSby adjoining an identityif necessary(S1=Sfor a monoid).[3]
Similarly, every magma has at most oneabsorbing element, which in semigroup theory is called azero. Analogous to the above construction, for every semigroupS, one can defineS0, a semigroup with 0 that embedsS.
The semigroup operation induces an operation on the collection of its subsets: given subsetsAandBof a semigroupS, their productA·B, written commonly asAB, is the set{ab|a∈Aandb∈B}.(This notion is defined identically asit is for groups.) In terms of this operation, a subsetAis called
IfAis both a left ideal and a right ideal then it is called anideal(or atwo-sided ideal).
IfSis a semigroup, then the intersection of any collection of subsemigroups ofSis also a subsemigroup ofS.
So the subsemigroups ofSform acomplete lattice.
An example of a semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of acommutativesemigroup, when it exists, is a group.
Green's relations, a set of fiveequivalence relationsthat characterise the elements in terms of theprincipal idealsthey generate, are important tools for analysing the ideals of a semigroup and related notions of structure.
The subset with the property that every element commutes with any other element of the semigroup is called thecenterof the semigroup.[4]The center of a semigroup is actually a subsemigroup.[5]
Asemigrouphomomorphismis a function that preserves semigroup structure. A functionf:S→Tbetween two semigroups is a homomorphism if the equation
holds for all elementsa,binS, i.e. the result is the same when performing the semigroup operation after or before applying the mapf.
A semigroup homomorphism between monoids preserves identity if it is amonoid homomorphism. But there are semigroup homomorphisms that are not monoid homomorphisms, e.g. the canonical embedding of a semigroupSwithout identity intoS1. Conditions characterizing monoid homomorphisms are discussed further. Letf:S0→S1be a semigroup homomorphism. The image offis also a semigroup. IfS0is a monoid with an identity elemente0, thenf(e0) is the identity element in the image off. IfS1is also a monoid with an identity elemente1ande1belongs to the image off, thenf(e0) =e1, i.e.fis a monoid homomorphism. Particularly, iffissurjective, then it is a monoid homomorphism.
Two semigroupsSandTare said to beisomorphicif there exists abijectivesemigroup homomorphismf:S→T. Isomorphic semigroups have the same structure.
Asemigroup congruence~ is anequivalence relationthat is compatible with the semigroup operation. That is, a subset~ ⊆S×Sthat is an equivalence relation andx~yandu~vimpliesxu~yvfor everyx,y,u,vinS. Like any equivalence relation, a semigroup congruence ~ inducescongruence classes
and the semigroup operation induces a binary operation ∘ on the congruence classes:
Because ~ is a congruence, the set of all congruence classes of ~ forms a semigroup with ∘, called thequotient semigrouporfactor semigroup, and denotedS/ ~. The mappingx↦ [x]~is a semigroup homomorphism, called thequotient map,canonicalsurjectionorprojection; ifSis a monoid then quotient semigroup is a monoid with identity [1]~. Conversely, thekernelof any semigroup homomorphism is a semigroup congruence. These results are nothing more than a particularization of thefirst isomorphism theorem in universal algebra. Congruence classes and factor monoids are the objects of study instring rewriting systems.
Anuclear congruenceonSis one that is the kernel of an endomorphism ofS.[6]
A semigroupSsatisfies themaximal condition on congruencesif any family of congruences onS, ordered by inclusion, has a maximal element. ByZorn's lemma, this is equivalent to saying that theascending chain conditionholds: there is no infinite strictly ascending chain of congruences onS.[7]
Every idealIof a semigroup induces a factor semigroup, theRees factor semigroup, via the congruence ρ defined byxρyif eitherx=y, or bothxandyare inI.
The following notions[8]introduce the idea that a semigroup is contained in another one.
A semigroupTis a quotient of a semigroupSif there is a surjective semigroup morphism fromStoT. For example,(Z/2Z, +)is a quotient of(Z/4Z, +), using the morphism consisting of taking the remainder modulo 2 of an integer.
A semigroupTdivides a semigroupS, denotedT≼SifTis a quotient of a subsemigroupS. In particular, subsemigroups ofSdividesT, while it is not necessarily the case that there are a quotient ofS.
Both of those relations are transitive.
For any subsetAofSthere is a smallest subsemigroupTofSthat containsA, and we say thatAgeneratesT. A single elementxofSgenerates the subsemigroup{xn|n∈Z+}. If this is finite, thenxis said to be offinite order, otherwise it is ofinfinite order.
A semigroup is said to beperiodicif all of its elements are of finite order.
A semigroup generated by a single element is said to bemonogenic(orcyclic). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positiveintegerswith the operation of addition.
If it is finite and nonempty, then it must contain at least oneidempotent.
It follows that every nonempty periodic semigroup has at least one idempotent.
A subsemigroup that is also a group is called asubgroup. There is a close relationship between the subgroups of a semigroup and its idempotents. Each subgroup contains exactly one idempotent, namely the identity element of the subgroup. For each idempotenteof the semigroup there is a unique maximal subgroup containinge. Each maximal subgroup arises in this way, so there is a one-to-one correspondence between idempotents and maximal subgroups. Here the termmaximal subgroupdiffers from its standard use in group theory.
More can often be said when the order is finite. For example, every nonempty finite semigroup is periodic, and has a minimalidealand at least one idempotent. The number of finite semigroups of a given size (greater than 1) is (obviously) larger than the number of groups of the same size. For example, of the sixteen possible "multiplication tables" for a set of two elements{a,b}, eight form semigroups[b]whereas only four of these are monoids and only two form groups. For more on the structure of finite semigroups, seeKrohn–Rhodes theory.
There is a structure theorem for commutative semigroups in terms ofsemilattices.[10]A semilattice (or more precisely a meet-semilattice)(L, ≤)is apartially ordered setwhere every pair of elementsa,b∈Lhas agreatest lower bound, denoteda∧b. The operation ∧ makesLinto a semigroup that satisfies the additionalidempotencelawa∧a=a.
Given a homomorphismf:S→Lfrom an arbitrary semigroup to a semilattice, each inverse imageSa=f−1{a}is a (possibly empty) semigroup. Moreover,SbecomesgradedbyL, in the sense thatSaSb⊆Sa∧b.
Iffis onto, the semilatticeLis isomorphic to thequotientofSby the equivalence relation ~ such thatx~yif and only iff(x) =f(y). This equivalence relation is a semigroup congruence, as defined above.
Whenever we take the quotient of a commutative semigroup by a congruence, we get another commutative semigroup. The structure theorem says that for any commutative semigroupS, there is a finest congruence ~ such that the quotient ofSby this equivalence relation is a semilattice. Denoting this semilattice byL, we get a homomorphismffromSontoL. As mentioned,Sbecomes graded by this semilattice.
Furthermore, the componentsSaare allArchimedean semigroups. An Archimedean semigroup is one where given any pair of elementsx,y, there exists an elementzandn> 0such thatxn=yz.
The Archimedean property follows immediately from the ordering in the semilatticeL, since with this ordering we havef(x) ≤f(y)if and only ifxn=yzfor somezandn> 0.
Thegroup of fractionsorgroup completionof a semigroupSis thegroupG=G(S)generated by the elements ofSas generators and all equationsxy=zthat hold true inSasrelations.[11]There is an obvious semigroup homomorphismj:S→G(S)that sends each element ofSto the corresponding generator. This has auniversal propertyfor morphisms fromSto a group:[12]given any groupHand any semigroup homomorphismk:S→H, there exists a uniquegroup homomorphismf:G→Hwithk=fj. We may think ofGas the "most general" group that contains a homomorphic image ofS.
An important question is to characterize those semigroups for which this map is an embedding. This need not always be the case: for example, takeSto be the semigroup of subsets of some setXwithset-theoretic intersectionas the binary operation (this is an example of a semilattice). SinceA.A=Aholds for all elements ofS, this must be true for all generators ofG(S) as well, which is therefore thetrivial group. It is clearly necessary for embeddability thatShave thecancellation property. WhenSis commutative this condition is also sufficient[13]and theGrothendieck groupof the semigroup provides a construction of the group of fractions. The problem for non-commutative semigroups can be traced to the first substantial paper on semigroups.[14][15]Anatoly Maltsevgave necessary and sufficient conditions for embeddability in 1937.[16]
Semigroup theory can be used to study some problems in the field ofpartial differential equations. Roughly speaking, the semigroup approach is to regard a time-dependent partial differential equation as anordinary differential equationon a function space. For example, consider the following initial/boundary value problem for theheat equationon the spatialinterval(0, 1) ⊂Rand timest≥ 0:
LetX=L2((0, 1)R)be theLpspaceof square-integrable real-valued functions with domain the interval(0, 1)and letAbe the second-derivative operator withdomain
whereH2{\displaystyle H^{2}}is aSobolev space. Then the above initial/boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the spaceX:
On an heuristic level, the solution to this problem "ought" to beu(t)=exp(tA)u0.{\displaystyle u(t)=\exp(tA)u_{0}.}However, for a rigorous treatment, a meaning must be given to theexponentialoftA. As a function oft, exp(tA) is a semigroup of operators fromXto itself, taking the initial stateu0at timet= 0to the stateu(t) = exp(tA)u0at timet. The operatorAis said to be theinfinitesimal generatorof the semigroup.
The study of semigroups trailed behind that of other algebraic structures with more complex axioms such asgroupsorrings. A number of sources[17][18]attribute the first use of the term (in French) to J.-A. de Séguier inÉlements de la Théorie des Groupes Abstraits(Elements of the Theory of Abstract Groups) in 1904. The term is used in English in 1908 in Harold Hinton'sTheory of Groups of Finite Order.
Anton Sushkevichobtained the first non-trivial results about semigroups. His 1928 paper "Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit" ("On finite groups without the rule of unique invertibility") determined the structure of finitesimple semigroupsand showed that the minimal ideal (orGreen's relationsJ-class) of a finite semigroup is simple.[18]From that point on, the foundations of semigroup theory were further laid byDavid Rees,James Alexander Green,Evgenii Sergeevich Lyapin[fr],Alfred H. CliffordandGordon Preston. The latter two published a two-volume monograph on semigroup theory in 1961 and 1967 respectively. In 1970, a new periodical calledSemigroup Forum(currently published bySpringer Verlag) became one of the few mathematical journals devoted entirely to semigroup theory.
Therepresentation theoryof semigroups was developed in 1963 byBoris Scheinusingbinary relationson a setAandcomposition of relationsfor the semigroup product.[19]At an algebraic conference in 1972 Schein surveyed the literature on BA, the semigroup of relations onA.[20]In 1997 Schein andRalph McKenzieproved that every semigroup is isomorphic to a transitive semigroup of binary relations.[21]
In recent years researchers in the field have become more specialized with dedicated monographs appearing on important classes of semigroups, likeinverse semigroups, as well as monographs focusing on applications inalgebraic automata theory, particularly for finite automata, and also infunctional analysis.
If the associativity axiom of a semigroup is dropped, the result is amagma, which is nothing more than a setMequipped with abinary operationthat is closedM×M→M.
Generalizing in a different direction, ann-ary semigroup(alson-semigroup,polyadic semigroupormultiary semigroup) is a generalization of a semigroup to a setGwith an-ary operationinstead of a binary operation.[22]The associative law is generalized as follows: ternary associativity is(abc)de=a(bcd)e=ab(cde), i.e. the stringabcdewith any three adjacent elements bracketed.n-ary associativity is a string of lengthn+ (n− 1)with anynadjacent elements bracketed. A 2-ary semigroup is just a semigroup. Further axioms lead to ann-ary group.
A third generalization is thesemigroupoid, in which the requirement that the binary relation be total is lifted. As categories generalize monoids in the same way, a semigroupoid behaves much like a category but lacks identities.
Infinitary generalizations of commutative semigroups have sometimes been considered by various authors.[c]
|
https://en.wikipedia.org/wiki/Semigroup
|
Inabstract algebra, amonoidis a set equipped with anassociativebinary operationand anidentity element. For example, the nonnegativeintegerswith addition form a monoid, the identity element being0.
Monoids aresemigroupswith identity. Suchalgebraic structuresoccur in several branches of mathematics.
The functions from a set into itself form a monoid with respect to function composition. More generally, incategory theory, the morphisms of anobjectto itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object.
Incomputer scienceandcomputer programming, the set ofstringsbuilt from a given set ofcharactersis afree monoid.Transition monoidsandsyntactic monoidsare used in describingfinite-state machines.Trace monoidsandhistory monoidsprovide a foundation forprocess calculiandconcurrent computing.
Intheoretical computer science, the study of monoids is fundamental forautomata theory(Krohn–Rhodes theory), andformal language theory(star height problem).
Seesemigroupfor the history of the subject, and some other general properties of monoids.
AsetSequipped with abinary operationS×S→S, which we will denote•, is amonoidif it satisfies the following two axioms:
In other words, a monoid is asemigroupwith anidentity element. It can also be thought of as amagmawith associativity and identity. The identity element of a monoid is unique.[a]For this reason the identity is regarded as aconstant, i. e.0-ary (or nullary) operation. The monoid therefore is characterized by specification of thetriple(S, • ,e).
Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted byjuxtaposition; for example, the monoid axioms may be written(ab)c=a(bc)andea=ae=a. This notation does not imply that it is numbers being multiplied.
A monoid in which each element has aninverseis agroup.
Asubmonoidof a monoid(M, •)is asubsetNofMthat is closed under the monoid operation and contains the identity elementeofM.[1][b]Symbolically,Nis a submonoid ofMife∈N⊆M, andx•y∈Nwheneverx,y∈N. In this case,Nis a monoid under the binary operation inherited fromM.
On the other hand, ifNis a subset of a monoid that isclosedunder the monoid operation, and is a monoid for this inherited operation, thenNis not always a submonoid, since the identity elements may differ. For example, thesingleton set{0}is closed under multiplication, and is not a submonoid of the (multiplicative) monoid of thenonnegative integers.
A subsetSofMis said togenerateMif the smallest submonoid ofMcontainingSisM. If there is a finite set that generatesM, thenMis said to be afinitely generated monoid.
A monoid whose operation iscommutativeis called acommutative monoid(or, less commonly, anabelian monoid). Commutative monoids are often written additively. Any commutative monoid is endowed with itsalgebraicpreordering≤, defined byx≤yif there existszsuch thatx+z=y.[2]Anorder-unitof a commutative monoidMis an elementuofMsuch that for any elementxofM, there existsvin the set generated byusuch thatx≤v. This is often used in caseMis thepositive coneof apartially orderedabelian groupG, in which case we say thatuis an order-unit ofG.
A monoid for which the operation is commutative for some, but not all elements is atrace monoid; trace monoids commonly occur in the theory ofconcurrent computation.
[012⋯n−2n−1123⋯n−1k]{\displaystyle {\begin{bmatrix}0&1&2&\cdots &n-2&n-1\\1&2&3&\cdots &n-1&k\end{bmatrix}}}or, equivalentlyf(i):={i+1,if0≤i<n−1k,ifi=n−1.{\displaystyle f(i):={\begin{cases}i+1,&{\text{if }}0\leq i<n-1\\k,&{\text{if }}i=n-1.\end{cases}}}
Multiplication of elements in⟨f⟩is then given by function composition.
Whenk= 0then the functionfis a permutation of{0, 1, 2, ...,n−1}, and gives the uniquecyclic groupof ordern.
The monoid axioms imply that the identity elementeis unique: Ifeandfare identity elements of a monoid, thene=ef=f.
For each nonnegative integern, one can define the productpn=∏i=1nai{\displaystyle p_{n}=\textstyle \prod _{i=1}^{n}a_{i}}of any sequence(a1, ...,an)ofnelements of a monoid recursively: letp0=eand letpm=pm−1•amfor1 ≤m≤n.
As a special case, one can define nonnegative integer powers of an elementxof a monoid:x0= 1andxn=xn−1•xforn≥ 1. Thenxm+n=xm•xnfor allm,n≥ 0.
An elementxis calledinvertibleif there exists an elementysuch thatx•y=eandy•x=e. The elementyis called the inverse ofx. Inverses, if they exist, are unique: ifyandzare inverses ofx, then by associativityy=ey= (zx)y=z(xy) =ze=z.[6]
Ifxis invertible, say with inversey, then one can define negative powers ofxby settingx−n=ynfor eachn≥ 1; this makes the equationxm+n=xm•xnhold for allm,n∈Z.
The set of all invertible elements in a monoid, together with the operation •, forms agroup.
Not every monoid sits inside a group. For instance, it is perfectly possible to have a monoid in which two elementsaandbexist such thata•b=aholds even thoughbis not the identity element. Such a monoid cannot be embedded in a group, because in the group multiplying both sides with the inverse ofawould get thatb=e, which is not true.
A monoid(M, •)has thecancellation property(or is cancellative) if for alla,bandcinM, the equalitya•b=a•cimpliesb=c, and the equalityb•a=c•aimpliesb=c.
A commutative monoid with the cancellation property can always be embedded in a group via theGrothendieck group construction. That is how the additive group of the integers (a group with operation+) is constructed from the additive monoid of natural numbers (a commutative monoid with operation+and cancellation property). However, a non-commutative cancellative monoid need not be embeddable in a group.
If a monoid has the cancellation property and isfinite, then it is in fact a group.[c]
The right- and left-cancellative elements of a monoid each in turn form a submonoid (i.e. are closed under the operation and obviously include the identity). This means that the cancellative elements of any commutative monoid can be extended to a group.
The cancellative property in a monoid is not necessary to perform the Grothendieck construction – commutativity is sufficient. However, if a commutative monoid does not have the cancellation property, the homomorphism of the monoid into its Grothendieck group is not injective. More precisely, ifa•b=a•c, thenbandchave the same image in the Grothendieck group, even ifb≠c. In particular, if the monoid has anabsorbing element, then its Grothendieck group is thetrivial group.
Aninverse monoidis a monoid where for everyainM, there exists a uniquea−1inMsuch thata=a•a−1•aanda−1=a−1•a•a−1. If an inverse monoid is cancellative, then it is a group.
In the opposite direction, azerosumfree monoidis an additively written monoid in whicha+b= 0implies thata= 0andb= 0:[7]equivalently, that no element other than zero has an additive inverse.
LetMbe a monoid, with the binary operation denoted by•and the identity element denoted bye. Then a (left)M-act(or left act overM) is a setXtogether with an operation⋅ :M×X→Xwhich is compatible with the monoid structure as follows:
This is the analogue in monoid theory of a (left)group action. RightM-acts are defined in a similar way. A monoid with an act is also known as anoperator monoid. Important examples includetransition systemsofsemiautomata. Atransformation semigroupcan be made into an operator monoid by adjoining the identity transformation.
Ahomomorphismbetween two monoids(M, ∗)and(N, •)is a functionf:M→Nsuch that
whereeMandeNare the identities onMandNrespectively. Monoid homomorphisms are sometimes simply calledmonoid morphisms.
Not everysemigroup homomorphismbetween monoids is a monoid homomorphism, since it may not map the identity to the identity of the target monoid, even though the identity is the identity of the image of the homomorphism.[d]For example, consider[Z]n, the set ofresidue classesmodulonequipped with multiplication. In particular,[1]nis the identity element. Functionf: [Z]3→ [Z]6given by[k]3↦ [3k]6is a semigroup homomorphism, since[3k⋅ 3l]6= [9kl]6= [3kl]6. However,f([1]3) = [3]6≠ [1]6, so a monoid homomorphism is a semigroup homomorphism between monoids that maps the identity of the first monoid to the identity of the second monoid and the latter condition cannot be omitted.
In contrast, a semigroup homomorphism between groups is always agroup homomorphism, as it necessarily preserves the identity (because, in the target group of the homomorphism, the identity element is the only elementxsuch thatx⋅x=x).
Abijectivemonoid homomorphism is called a monoidisomorphism. Two monoids are said to be isomorphic if there is a monoid isomorphism between them.
Monoids may be given apresentation, much in the same way that groups can be specified by means of agroup presentation. One does this by specifying a set of generatorsΣ, and a set of relations on thefree monoidΣ∗. One does this by extending (finite)binary relationsonΣ∗to monoid congruences, and then constructing the quotient monoid, as above.
Given a binary relationR⊂ Σ∗× Σ∗, one defines its symmetric closure asR∪R−1. This can be extended to a symmetric relationE⊂ Σ∗× Σ∗by definingx~Eyif and only ifx=sutandy=svtfor some stringsu,v,s,t∈ Σ∗with(u,v) ∈R∪R−1. Finally, one takes the reflexive and transitive closure ofE, which is then a monoid congruence.
In the typical situation, the relationRis simply given as a set of equations, so thatR= {u1=v1, ...,un=vn}. Thus, for example,
is the equational presentation for thebicyclic monoid, and
is theplactic monoidof degree2(it has infinite order). Elements of this plactic monoid may be written asaibj(ba)k{\displaystyle a^{i}b^{j}(ba)^{k}}for integersi,j,k, as the relations show thatbacommutes with bothaandb.
Monoids can be viewed as a special class ofcategories. Indeed, the axioms required of a monoid operation are exactly those required ofmorphismcomposition when restricted to the set of all morphisms whose source and target is a given object.[8]That is,
More precisely, given a monoid(M, •), one can construct a small category with only one object and whose morphisms are the elements ofM. The composition of morphisms is given by the monoid operation•.
Likewise, monoid homomorphisms are justfunctorsbetween single object categories.[8]So this construction gives anequivalencebetween thecategory of (small) monoidsMonand a full subcategory of the category of (small) categoriesCat. Similarly, thecategory of groupsis equivalent to another full subcategory ofCat.
In this sense, category theory can be thought of as an extension of the concept of a monoid. Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid.
Monoids, just like other algebraic structures, also form their own category,Mon, whose objects are monoids and whose morphisms are monoid homomorphisms.[8]
There is also a notion ofmonoid objectwhich is an abstract definition of what is a monoid in a category. A monoid object inSetis just a monoid.
In computer science, manyabstract data typescan be endowed with a monoid structure. In a common pattern, asequenceof elements of a monoid is "folded" or "accumulated" to produce a final value. For instance, many iterative algorithms need to update some kind of "running total" at each iteration; this pattern may be elegantly expressed by a monoid operation. Alternatively, the associativity of monoid operations ensures that the operation can beparallelizedby employing aprefix sumor similar algorithm, in order to utilize multiple cores or processors efficiently.
Given a sequence of values of typeMwith identity elementεand associative operation•, thefoldoperation is defined as follows:
In addition, anydata structurecan be 'folded' in a similar way, given a serialization of its elements. For instance, the result of "folding" abinary treemight differ depending on pre-order vs. post-ordertree traversal.
An application of monoids in computer science is the so-calledMapReduceprogramming model (seeEncoding Map-Reduce As A Monoid With Left Folding). MapReduce, in computing, consists of two or three operations. Given a dataset, "Map" consists of mapping arbitrary data to elements of a specific monoid. "Reduce" consists of folding those elements, so that in the end we produce just one element.
For example, if we have amultiset, in a program it is represented as a map from elements to their numbers. Elements are called keys in this case. The number of distinct keys may be too big, and in this case, themultisetis being sharded. To finalize reduction properly, the "Shuffling" stage regroups the data among the nodes. If we do not need this step, the whole Map/Reduce consists of mapping and reducing; both operations are parallelizable, the former due to its element-wise nature, the latter due to associativity of the monoid.
Acomplete monoidis a commutative monoid equipped with aninfinitarysum operationΣI{\displaystyle \Sigma _{I}}for anyindex setIsuch that[9][10][11][12]
and
Anordered commutative monoidis a commutative monoidMtogether with apartial ordering≤such thata≥ 0for everya∈M, anda≤bimpliesa+c≤b+cfor alla,b,c∈M.
Acontinuous monoidis an ordered commutative monoid(M, ≤)in which everydirected subsethas aleast upper bound, and these least upper bounds are compatible with the monoid operation:
for everya∈Mand directed subsetSofM.
If(M, ≤)is a continuous monoid, then for any index setIand collection of elements(ai)i∈I, one can define
andMtogether with this infinitary sum operation is a complete monoid.[12]
|
https://en.wikipedia.org/wiki/Monoid
|
Inmathematics,racksandquandlesare sets withbinary operationssatisfying axioms analogous to theReidemeister movesused to manipulateknotdiagrams.
While mainly used to obtain invariants of knots, they can be viewed asalgebraicconstructions in their own right. In particular, the definition of a quandle axiomatizes the properties ofconjugationin agroup.
In 1942,Mituhisa Takasaki[ja]introduced an algebraic structure which he called akei(圭),[1][2]which would later come to be known as an involutive quandle.[3]His motivation was to find a nonassociative algebraic structure to capture the notion of areflectionin the context offinite geometry.[2][3]The idea was rediscovered and generalized in an unpublished 1959 correspondence betweenJohn ConwayandGavin Wraith, who at the time were undergraduate students at theUniversity of Cambridge. It is here that the modern definitions of quandles and of racks first appear. Wraith had become interested in these structures (which he initially dubbedsequentials) while at school.[4]Conway renamed themwracks, partly as a pun on his colleague's name, and partly because they arise as the remnants (or 'wrack and ruin') of agroupwhen one discards the multiplicative structure and considers only theconjugationstructure. The spelling 'rack' has now become prevalent.
These constructs surfaced again in the 1980s: in a 1982 paper byDavid Joyce[5](where the termquandle, an arbitrary nonsense word, was coined),[6]in a 1982 paper bySergei Matveev(under the namedistributivegroupoids)[7]and in a 1986 conference paper byEgbert Brieskorn(where they were calledautomorphicsets).[8]A detailed overview of racks and their applications in knot theory may be found in the paper byColin RourkeandRoger Fenn.[9]
Arackmay be defined as a setR{\displaystyle \mathrm {R} }with a binary operation◃{\displaystyle \triangleleft }such that for everya,b,c∈R{\displaystyle a,b,c\in \mathrm {R} }theself-distributive lawholds:
and for everya,b∈R,{\displaystyle a,b\in \mathrm {R} ,}there exists a uniquec∈R{\displaystyle c\in \mathrm {R} }such that
This definition, while terse and commonly used, is suboptimal for certain purposes because it contains an existential quantifier which is not really necessary. To avoid this, we may write the uniquec∈R{\displaystyle c\in \mathrm {R} }such thata◃c=b{\displaystyle a\triangleleft c=b}asb▹a.{\displaystyle b\triangleright a.}We then have
and thus
and
Using this idea, a rack may be equivalently defined as a setR{\displaystyle \mathrm {R} }with two binary operations◃{\displaystyle \triangleleft }and▹{\displaystyle \triangleright }such that for alla,b,c∈R:{\displaystyle a,b,c\in \mathrm {R} {\text{:}}}
It is convenient to say that the elementa∈R{\displaystyle a\in \mathrm {R} }is acting from the left in the expressiona◃b,{\displaystyle a\triangleleft b,}and acting from the right in the expressionb▹a.{\displaystyle b\triangleright a.}The third and fourth rack axioms then say that these left and right actions are inverses of each other. Using this, we can eliminate either one of these actions from the definition of rack. If we eliminate the right action and keep the left one, we obtain the terse definition given initially.
Many different conventions are used in the literature on racks and quandles. For example, many authors prefer to work with just therightaction. Furthermore, the use of the symbols◃{\displaystyle \triangleleft }and▹{\displaystyle \triangleright }is by no means universal: many authors use exponential notation
and
while many others write
Yet another equivalent definition of a rack is that it is a set where each element acts on the left and right asautomorphismsof the rack, with the left action being the inverse of the right one. In this definition, the fact that each element acts as automorphisms encodes the left and right self-distributivity laws, and also these laws:
which are consequences of the definition(s) given earlier.
Aquandleis defined as anidempotentrack,Q,{\displaystyle \mathrm {Q} ,}such that for alla∈Q{\displaystyle a\in \mathrm {Q} }
or equivalently
Every group gives a quandle where the operations come from conjugation:
In fact, every equational law satisfied byconjugationin a group follows from the quandle axioms. So, one can think of a quandle as what is left of a group when we forget multiplication, the identity, and inverses, and only remember the operation of conjugation.
Everytame knotinthree-dimensionalEuclidean spacehas a 'fundamental quandle'. To define this, one can note that thefundamental groupof the knot complement, orknot group, has a presentation (theWirtinger presentation) in which the relations only involve conjugation. So, this presentation can also be used as a presentation of a quandle. The fundamental quandle is a very powerful invariant of knots. In particular, if two knots haveisomorphicfundamental quandles then there is ahomeomorphismof three-dimensional Euclidean space, which may beorientation reversing, taking one knot to the other.
Less powerful but more easily computable invariants of knots may be obtained by counting the homomorphisms from the knot quandle to a fixed quandleQ.{\displaystyle \mathrm {Q} .}Since the Wirtinger presentation has one generator for each strand in aknot diagram, these invariants can be computed by counting ways of labelling each strand by an element ofQ,{\displaystyle \mathrm {Q} ,}subject to certain constraints. More sophisticated invariants of this sort can be constructed with the help of quandlecohomology.
TheAlexander quandlesare also important, since they can be used to compute theAlexander polynomialof a knot. LetA{\displaystyle \mathrm {A} }be a module over the ringZ[t,t−1]{\displaystyle \mathbb {Z} [t,t^{-1}]}ofLaurent polynomialsin one variable. Then theAlexander quandleisA{\displaystyle \mathrm {A} }made into a quandle with the left action given by
Racks are a useful generalization of quandles in topology, since while quandles can represent knots on a round linear object (such as rope or a thread), racks can represent ribbons, which may be twisted as well as knotted.
A quandleQ{\displaystyle \mathrm {Q} }is said to beinvolutoryif for alla,b∈Q,{\displaystyle a,b\in \mathrm {Q} ,}
or equivalently,
Anysymmetric spacegives an involutory quandle, wherea◃b{\displaystyle a\triangleleft b}is the result of 'reflectingb{\displaystyle b}througha{\displaystyle a}'.
|
https://en.wikipedia.org/wiki/Racks_and_quandles
|
Inmathematics, especially inabstract algebra, aquasigroupis analgebraic structurethat resembles agroupin the sense that "division" is always possible. Quasigroups differ from groups mainly in that theassociativeandidentity elementproperties are optional. In fact, a nonempty associative quasigroup is a group.[1][2]
A quasigroup that has an identity element is called aloop.
There are at least two structurally equivalent formal definitions of quasigroup:
Thehomomorphicimageof a quasigroup that is defined with a single binary operation, however, need not be a quasigroup, in contrast to a quasigroup as having three primitive operations.[3]We begin with the first definition.
Aquasigroup(Q, ∗)is a non-emptysetQwith a binary operation∗(that is, amagma, indicating that a quasigroup has to satisfy the closure property), obeying theLatin square property. This states that, for eachaandbinQ, there exist unique elementsxandyinQsuch that botha∗x=b{\displaystyle a\ast x=b}y∗a=b{\displaystyle y\ast a=b}hold. (In other words: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, orCayley table. This property ensures that the Cayley table of a finite quasigroup, and, in particular, a finite group, is aLatin square.) The requirement thatxandybe unique can be replaced by the requirement that the magma becancellative.[4][a]
The unique solutions to these equations are writtenx=a\bandy=b/a. The operations '\' and '/' are called, respectively,left divisionandright division. With regard to the Cayley table, the first equation (left division) means that thebentry in thearow is in thexcolumn while the second equation (right division) means that thebentry in theacolumn is in theyrow.
Theempty setequipped with theempty binary operationsatisfies this definition of a quasigroup. Some authors accept the empty quasigroup but others explicitly exclude it.[5][6]
Given somealgebraic structure, anidentityis an equation in which all variables are tacitlyuniversally quantified, and in which alloperationsare among the primitive operations proper to the structure. Algebraic structures that satisfy axioms that are given solely by identities are called avariety. Many standard results inuniversal algebrahold only for varieties. Quasigroups form a variety if left and right division are taken as primitive.
Aright-quasigroup(Q, ∗, /)is a type(2, 2)algebra that satisfy both identities:y=(y/x)∗x{\displaystyle y=(y/x)\ast x}y=(y∗x)/x{\displaystyle y=(y\ast x)/x}
Aleft-quasigroup(Q, ∗, \)is a type(2, 2)algebra that satisfy both identities:y=x∗(x∖y){\displaystyle y=x\ast (x\backslash y)}y=x∖(x∗y){\displaystyle y=x\backslash (x\ast y)}
Aquasigroup(Q, ∗, \, /)is a type(2, 2, 2)algebra (i.e., equipped with three binary operations) that satisfy the identities:[b]y=(y/x)∗x{\displaystyle y=(y/x)\ast x}y=(y∗x)/x{\displaystyle y=(y\ast x)/x}y=x∗(x∖y){\displaystyle y=x\ast (x\backslash y)}y=x∖(x∗y){\displaystyle y=x\backslash (x\ast y)}
In other words: Multiplication and division in either order, one after the other, on the same side by the same element, have no net effect.
Hence if(Q, ∗)is a quasigroup according to the definition of the previous section, then(Q, ∗, \, /)is the same quasigroup in the sense of universal algebra. And vice versa: if(Q, ∗, \, /)is a quasigroup according to the sense of universal algebra, then(Q, ∗)is a quasigroup according to the first definition.
Aloopis a quasigroup with anidentity element; that is, an element,e, such that
It follows that the identity element,e, is unique, and that every element ofQhas uniqueleftandright inverses(which need not be the same).
A quasigroup with anidempotent elementis called apique("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given anabelian group,(A, +), taking its subtraction operation as quasigroup multiplication yields a pique(A, −)with the group identity (zero) turned into a "pointed idempotent". (That is, there is aprincipal isotopy(x,y,z) ↦ (x, −y,z).)
A loop that is associative is a group. A group can have a strictly nonassociative pique isotope, but it cannot have a strictly nonassociative loop isotope.
There are weaker associativity properties that have been given special names.
For instance, aBol loopis a loop that satisfies either:
or else
A loop that is both a left and right Bol loop is aMoufang loop. This is equivalent to any one of the following single Moufang identities holding for allx,y,z:
According to Jonathan D. H. Smith, "loops" were named after theChicago Loop, as their originators were studying quasigroups in Chicago at the time.[9]
Smith (2007)names the following important properties and subclasses:
A quasigroup issemisymmetricif any of the following equivalent identities hold for allx,y:[c]
Although this class may seem special, every quasigroupQinduces a semisymmetric quasigroupQΔ on the direct product cubeQ3via the following operation:
where "//" and "\\" are theconjugate division operationsgiven byy//x=x/yandy\\x=x\y.
A quasigroup may exhibit semisymmetrictriality.[10]
A narrower class is atotally symmetric quasigroup(sometimes abbreviatedTS-quasigroup) in which allconjugatescoincide as one operation:x∗y=x/y=x\y. Another way to define (the same notion of) totally symmetric quasigroup is as a semisymmetric quasigroup that is commutative, i.e.x∗y=y∗x.
Idempotent total symmetric quasigroups are precisely (i.e. in a bijection with)Steiner triples, so such a quasigroup is also called aSteiner quasigroup, and sometimes the latter is even abbreviated assquag. The termslooprefers to an analogue for loops, namely, totally symmetric loops that satisfyx∗x= 1instead ofx∗x=x. Without idempotency, total symmetric quasigroups correspond to the geometric notion ofextended Steiner triple, also called Generalized Elliptic Cubic Curve (GECC).
A quasigroup(Q, ∗)is calledweakly totally anti-symmetricif for allc,x,y∈Q, the following implication holds.[11]
A quasigroup(Q, ∗)is calledtotally anti-symmetricif, in addition, for allx,y∈Q, the following implication holds:[11]
This property is required, for example, in theDamm algorithm.
Quasigroups have thecancellation property: ifab=ac, thenb=c. This follows from the uniqueness of left division ofaboracbya. Similarly, ifba=ca, thenb=c.
The Latin square property of quasigroups implies that, given any two of the three variables inxy=z, the third variable is uniquely determined.
The definition of a quasigroup can be treated as conditions on the left and rightmultiplication operatorsLx,Rx:Q→Q, defined by
The definition says that both mappings arebijectionsfromQto itself. A magmaQis a quasigroup precisely when all these operators, for everyxinQ, are bijective. The inverse mappings are left and right division, that is,
In this notation the identities among the quasigroup's multiplication and division operations (stated in the section onuniversal algebra) are
whereiddenotes the identity mapping onQ.
The multiplication table of a finite quasigroup is aLatin square: ann×ntable filled withndifferent symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.
Conversely, every Latin square can be taken as the multiplication table of a quasigroup in many ways: the border row (containing the column headers) and the border column (containing the row headers) can each be any permutation of the elements. SeeSmall Latin squares and quasigroups.
For acountably infinitequasigroupQ, it is possible to imagine an infinite array in which every row and every column corresponds to some elementqofQ, and where the elementa∗bis in the row corresponding toaand the column responding tob. In this situation too, the Latin square property says that each row and each column of the infinite array will contain every possible value precisely once.
For anuncountably infinitequasigroup, such as the group of non-zeroreal numbersunder multiplication, the Latin square property still holds, although the name is somewhat unsatisfactory, as it is not possible to produce the array of combinations to which the above idea of an infinite array extends since the real numbers cannot all be written in asequence. (This is somewhat misleading however, as the reals can be written in a sequence of lengthc{\displaystyle {\mathfrak {c}}}, assuming thewell-ordering theorem.)
The binary operation of a quasigroup isinvertiblein the sense that bothLxandRx, theleft and right multiplication operators, are bijective, and henceinvertible.
Every loop element has a unique left and right inverse given by
A loop is said to have (two-sided)inversesifxλ=xρfor allx. In this case the inverse element is usually denoted byx−1.
There are some stronger notions of inverses in loops that are often useful:
A loop has theinverse propertyif it has both the left and right inverse properties. Inverse property loops also have the antiautomorphic and weak inverse properties. In fact, any loop that satisfies any two of the above four identities has the inverse property and therefore satisfies all four.
Any loop that satisfies the left, right, or antiautomorphic inverse properties automatically has two-sided inverses.
A quasigroup or loophomomorphismis amapf:Q→Pbetween two quasigroups such thatf(xy) =f(x)f(y). Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).
LetQandPbe quasigroups. Aquasigroup homotopyfromQtoPis a triple(α,β,γ)of maps fromQtoPsuch that
for allx,yinQ. A quasigroup homomorphism is just a homotopy for which the three maps are equal.
Anisotopyis a homotopy for which each of the three maps(α,β,γ)is abijection. Two quasigroups areisotopicif there is an isotopy between them. In terms of Latin squares, an isotopy(α,β,γ)is given by a permutation of rowsα, a permutation of columnsβ, and a permutation on the underlying element setγ.
Anautotopyis an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup forms a group with theautomorphism groupas a subgroup.
Every quasigroup is isotopic to a loop. If a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group. However, a quasigroup that is isotopic to a group need not be a group. For example, the quasigroup onRwith multiplication given by(x,y) ↦ (x+y)/2is isotopic to the additive group(R, +), but is not itself a group as it has no identity element. Everymedialquasigroup is isotopic to anabelian groupby theBruck–Toyoda theorem.
Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation ∗ (i.e.,x∗y=z) we can form five new operations:xoy:=y∗x(theoppositeoperation),/and\, and their opposites. That makes a total of six quasigroup operations, which are called theconjugatesorparastrophesof ∗. Any two of these operations are said to be "conjugate" or "parastrophic" to each other (and to themselves).
If the setQhas two quasigroup operations, ∗ and ·, and one of them is isotopic to a conjugate of the other, the operations are said to beisostrophicto each other. There are also many other names for this relation of "isostrophe", e.g.,paratopy.
Ann-ary quasigroupis a set with ann-ary operation,(Q,f)withf:Qn→Q, such that the equationf(x1, ...,xn) =yhas a unique solution for any one variable if all the othernvariables are specified arbitrarily.Polyadicormultiarymeansn-ary for some nonnegative integern.
A 0-ary, ornullary, quasigroup is just a constant element ofQ. A 1-ary, orunary, quasigroup is a bijection ofQto itself. Abinary, or 2-ary, quasigroup is an ordinary quasigroup.
An example of a multiary quasigroup is an iterated group operation,y=x1·x2· ··· ·xn; it is not necessary to use parentheses to specify the order of operations because the group is associative. One can also form a multiary quasigroup by carrying out any sequence of the same or different group or quasigroup operations, if the order of operations is specified.
There exist multiary quasigroups that cannot be represented in any of these ways. Ann-ary quasigroup isirreducibleif its operation cannot be factored into the composition of two operations in the following way:
where1 ≤i<j≤nand(i,j) ≠ (1,n). Finite irreduciblen-ary quasigroups exist for alln> 2; seeAkivis & Goldberg (2001)for details.
Ann-ary quasigroup with ann-ary version ofassociativityis called ann-ary group.
The number of isomorphism classes of small quasigroups (sequenceA057991in theOEIS) and loops (sequenceA057771in theOEIS) is given here:[14]
|
https://en.wikipedia.org/wiki/Quasigroup
|
The theory ofassociation schemesarose instatistics, in the theory ofexperimental designfor theanalysis of variance.[1][2][3]Inmathematics, association schemes belong to bothalgebraandcombinatorics. Inalgebraic combinatorics, association schemes provide a unified approach to many topics, for examplecombinatorial designsandthe theory of error-correcting codes.[4][5]In algebra, the theory of association schemes generalizes thecharacter theoryoflinear representations of groups.[6][7][8]
Ann-class association scheme consists of asetXtogether with apartitionSofX×Xinton+ 1binary relations,R0,R1, ...,Rnwhich satisfy:
An association scheme iscommutativeifpijk=pjik{\displaystyle p_{ij}^{k}=p_{ji}^{k}}for alli{\displaystyle i},j{\displaystyle j}andk{\displaystyle k}. Most authors assume this property.
Note, however, that while the notion of an association scheme generalizes the notion of a group, the notion of a commutative association scheme only generalizes the notion of acommutative group.
Asymmetricassociation scheme is one in which eachRi{\displaystyle R_{i}}is asymmetric relation. That is:
Every symmetric association scheme is commutative.
Two pointsxandyare calledith associates if(x,y)∈Ri{\displaystyle (x,y)\in R_{i}}. The definition states that ifxandyareith associates then so areyandx. Every pair of points areith associates for exactly onei{\displaystyle i}. Each point is its own zeroth associate while distinct points are never zeroth associates. Ifxandyarekth associates then the number of pointsz{\displaystyle z}which are bothith associates ofx{\displaystyle x}andjth associates ofy{\displaystyle y}is a constantpijk{\displaystyle p_{ij}^{k}}.
A symmetric association scheme can be visualized as acomplete graphwith labeled edges. The graph hasv{\displaystyle v}vertices, one for each point ofX{\displaystyle X}, and the edge joining verticesx{\displaystyle x}andy{\displaystyle y}is labeledi{\displaystyle i}ifx{\displaystyle x}andy{\displaystyle y}arei{\displaystyle i}th associates. Each edge has a unique label, and the number of triangles with a fixed base labeledk{\displaystyle k}having the other edges labeledi{\displaystyle i}andj{\displaystyle j}is a constantpijk{\displaystyle p_{ij}^{k}}, depending oni,j,k{\displaystyle i,j,k}but not on the choice of the base. In particular, each vertex is incident with exactlypii0=vi{\displaystyle p_{ii}^{0}=v_{i}}edges labeledi{\displaystyle i};vi{\displaystyle v_{i}}is thevalencyof therelationRi{\displaystyle R_{i}}. There are also loops labeled0{\displaystyle 0}at each vertexx{\displaystyle x}, corresponding toR0{\displaystyle R_{0}}.
Therelationsare described by theiradjacency matrices.Ai{\displaystyle A_{i}}is the adjacency matrix ofRi{\displaystyle R_{i}}fori=0,…,n{\displaystyle i=0,\ldots ,n}and is av×vmatrixwith rows and columns labeled by the points ofX{\displaystyle X}.
The definition of a symmetric association scheme is equivalent to saying that theAi{\displaystyle A_{i}}arev×v(0,1)-matriceswhich satisfy
The (x,y)-th entry of the left side of (IV) is the number of paths of length two betweenxandywith labelsiandjin the graph. Note that the rows and columns ofAi{\displaystyle A_{i}}containvi{\displaystyle v_{i}}1{\displaystyle 1}'s:
The termassociation schemeis due to (Bose & Shimamoto 1952) but the concept is already inherent in (Bose & Nair 1939).[9]These authors were studying what statisticians have calledpartially balanced incomplete block designs(PBIBDs). The subject became an object of algebraic interest with the publication of (Bose & Mesner 1959) and the introduction of theBose–Mesner algebra. The most important contribution to the theory was the thesis ofPh. Delsarte(Delsarte 1973) who recognized and fully used the connections with coding theory and design theory.[10]
A generalization calledcoherent configurationshas been studied by D. G. Higman.
Theadjacency matricesAi{\displaystyle A_{i}}of thegraphs(X,Ri){\displaystyle \left(X,R_{i}\right)}generate acommutativeandassociative algebraA{\displaystyle {\mathcal {A}}}(over therealorcomplex numbers) both for thematrix productand thepointwise product. This associative, commutative algebra is called theBose–Mesner algebraof the association scheme.
Since the matrices inA{\displaystyle {\mathcal {A}}}aresymmetricandcommutewith each other, they can bediagonalizedsimultaneously. Therefore,A{\displaystyle {\mathcal {A}}}issemi-simpleand has a unique basis of primitiveidempotentsJ0,…,Jn{\displaystyle J_{0},\ldots ,J_{n}}.
There is another algebra of(n+1)×(n+1){\displaystyle (n+1)\times (n+1)}matrices which isisomorphictoA{\displaystyle {\mathcal {A}}}, and is often easier to work with.
TheHamming schemeand theJohnson schemeare of major significance in classicalcoding theory.
Incoding theory, association scheme theory is mainly concerned with thedistanceof acode. Thelinear programmingmethod produces upper bounds for the size of acodewith given minimumdistance, and lower bounds for the size of adesignwith a given strength. The most specific results are obtained in the case where the underlying association scheme satisfies certainpolynomialproperties; this leads one into the realm oforthogonal polynomials. In particular, some universal bounds are derived forcodesanddesignsin polynomial-type association schemes.
In classicalcoding theory, dealing withcodesin aHamming scheme, the MacWilliams transform involves a family of orthogonal polynomials known as theKrawtchouk polynomials. These polynomials give theeigenvaluesof the distance relation matrices of theHamming scheme.
|
https://en.wikipedia.org/wiki/Association_scheme
|
Inmathematics, specificallygroup theory,Cauchy's theoremstates that ifGis afinite groupandpis aprime numberdividing theorderofG(the number of elements inG), thenGcontains an element of orderp. That is, there isxinGsuch thatpis the smallest positiveintegerwithxp=e, whereeis theidentity elementofG. It is named afterAugustin-Louis Cauchy, who discovered it in 1845.[1][2]
The theorem is a partial converse toLagrange's theorem, which states that the order of anysubgroupof a finite groupGdivides the order ofG. In general, not every divisor of|G|{\displaystyle |G|}arises as the order of a subgroup ofG{\displaystyle G}.[3]Cauchy's theorem states that for anyprimedivisorpof the order ofG, there is a subgroup ofGwhose order isp—thecyclic groupgeneratedby the element in Cauchy's theorem.
Cauchy's theorem is generalized bySylow's first theorem, which implies that ifpnis the maximal power ofpdividing the order ofG, thenGhas a subgroup of orderpn(and using the fact that ap-group issolvable, one can show thatGhas subgroups of orderprfor anyrless than or equal ton).
Many texts prove the theorem with the use ofstrong inductionand theclass equation, though considerably less machinery is required to prove the theorem in theabeliancase. One can also invokegroup actionsfor the proof.[4]
Cauchy's theorem—LetGbe afinite groupandpbe aprime. Ifpdivides theorderofG, thenGhas an element of orderp.
We first prove the special case that whereGisabelian, and then the general case; both proofs are by induction onn= |G|, and have as starting casen=pwhich is trivial because any non-identity element now has orderp. Suppose first thatGis abelian. Take any non-identity elementa, and letHbe thecyclic groupit generates. Ifpdivides |H|, thena|H|/pis an element of orderp. Ifpdoes not divide |H|, then it divides the order [G:H] of thequotient groupG/H, which therefore contains an element of orderpby the inductive hypothesis. That element is a classxHfor somexinG, and ifmis the order ofxinG, thenxm=einGgives (xH)m=eHinG/H, sopdividesm; as beforexm/pis now an element of orderpinG, completing the proof for the abelian case.
In the general case, letZbe thecenterofG, which is an abelian subgroup. Ifpdivides |Z|, thenZcontains an element of orderpby the case of abelian groups, and this element works forGas well. So we may assume thatpdoes not divide the order ofZ. Sincepdoes divide |G|, andGis the disjoint union ofZand of theconjugacy classesof non-central elements, there exists a conjugacy class of a non-central elementawhose size is not divisible byp. But theclass equationshows that size is [G:CG(a)], sopdivides the order of thecentralizerCG(a) ofainG, which is a proper subgroup becauseais not central. This subgroup contains an element of orderpby the inductive hypothesis, and we are done.
This proof uses the fact that for anyactionof a (cyclic) group of prime orderp, the only possible orbit sizes are 1 andp, which is immediate from theorbit stabilizer theorem.
The set that our cyclic group shall act on is the set
ofp-tuples of elements ofGwhose product (in order) gives the identity. Such ap-tuple is uniquely determined by all its components except the last one, as the last element must be the inverse of the product of those preceding elements. One also sees that thosep− 1elements can be chosen freely, soXhas |G|p−1elements, which is divisible byp.
Now from the fact that in a group ifab=ethenba=e, it follows that anycyclic permutationof the components of an element ofXagain gives an element ofX. Therefore one can define an action of the cyclic groupCpof orderponXby cyclic permutations of components, in other words in which a chosen generator ofCpsends
As remarked, orbits inXunder this action either have size 1 or sizep. The former happens precisely for those tuples(x,x,…,x){\displaystyle (x,x,\ldots ,x)}for whichxp=e{\displaystyle x^{p}=e}. Counting the elements ofXby orbits, and dividing byp, one sees that the number of elements satisfyingxp=e{\displaystyle x^{p}=e}is divisible byp. Butx=eis one such element, so there must be at leastp− 1other solutions forx, and these solutions are elements of orderp. This completes the proof.
Cauchy's theorem implies a rough classification of allelementary abelian groups(groups whose non-identity elements all have equal, finite order). IfG{\displaystyle G}is such a group, andx∈G{\displaystyle x\in G}has orderp{\displaystyle p}, thenp{\displaystyle p}must be prime, since otherwise Cauchy's theorem applied to the (finite) subgroup generated byx{\displaystyle x}produces an element of order less thanp{\displaystyle p}. Moreover, every finite subgroup ofG{\displaystyle G}has order a power ofp{\displaystyle p}(includingG{\displaystyle G}itself, if it is finite). This argument applies equally top-groups, where every element's order is a power ofp{\displaystyle p}(but not necessarily every order is the same).
One may use the abelian case of Cauchy's Theorem in an inductive proof[5]of the first of Sylow's theorems, similar to the first proof above, although there are also proofs that avoid doing this special case separately.
|
https://en.wikipedia.org/wiki/Cauchy%27s_theorem_(group_theory)
|
Inmathematics, theclassification of finite simple groups(popularly called theenormous theorem[1][2]) is a result ofgroup theorystating that everyfinite simple groupis eithercyclic, oralternating, or belongs to a broad infinite class called thegroups of Lie type, or else it is one of twenty-six exceptions, calledsporadic(theTits groupis sometimes regarded as a sporadic group because it is not strictly agroup of Lie type,[3]in which case there would be 27 sporadic groups). The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.
Simple groups can be seen as the basic building blocks of allfinite groups, reminiscent of the way theprime numbersare the basic building blocks of thenatural numbers. TheJordan–Hölder theoremis a more precise way of stating this fact about finite groups. However, a significant difference frominteger factorizationis that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphicgroups with the samecomposition seriesor, put in another way, theextension problemdoes not have a unique solution.
Daniel Gorenstein(1923–1992),Richard Lyons, andRonald Solomonare gradually publishing a simplified and revised version of the proof.
Theorem—Every finitesimple groupis, up toisomorphism, one of the following groups:
The classification theorem has applications in many branches of mathematics, as questions about the structure offinite groups(and their action on other mathematical objects) can sometimes be reduced to questions about finite simple groups. Thanks to the classification theorem, such questions can sometimes be answered by checking each family of simple groups and each sporadic group.
Daniel Gorensteinannounced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification ofquasithin groups. The completed proof of the classification was announced byAschbacher (2004)after Aschbacher and Smith published a 1221-page proof for the missing quasithin case.
Gorenstein (1982,1983) wrote two volumes outlining the low rank and odd characteristic part of the proof, andMichael Aschbacher, Richard Lyons, and Stephen D. Smith et al. (2011)
wrote a 3rd volume covering the remaining characteristic 2 case. The proof can be broken up into several major pieces as follows:
The simple groups of low2-rankare mostly groups of Lie type of small rank over fields of odd characteristic, together with five alternating and seven characteristic 2 type and nine sporadic groups.
The simple groups of small 2-rank include:
The classification of groups of small 2-rank, especially ranks at most 2, makes heavy use of ordinary and modular character theory, which is almost never directly used elsewhere in the classification.
All groups not of small 2 rank can be split into two major classes: groups of component type and groups of characteristic 2 type. This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and thebalance theoremimplies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type. (For groups of low 2-rank the proof of this breaks down, because theorems such as thesignalizer functortheorem only work for groups with elementary abelian subgroups of rank at least 3.)
A group is said to be of component type if for some centralizerCof an involution,C/O(C) has a component (whereO(C) is the core ofC, the maximal normal subgroup of odd order). These are more or less the groups of Lie type of odd characteristic of large rank, and alternating groups, together with some sporadic groups. A major step in this case is to eliminate the obstruction of the core of an involution. This is accomplished by theB-theorem, which states that every component ofC/O(C) is the image of a component ofC.
The idea is that these groups have a centralizer of an involution with a component that is a smaller quasisimple group, which can be assumed to be already known by induction. So to classify these groups one takes every central extension of every known finite simple group, and finds all simple groups with a centralizer of involution with this as a component. This gives a rather large number of different cases to check: there are not only 26 sporadic groups and 16 families of groups of Lie type and the alternating groups, but also many of the groups of small rank or over small fields behave differently from the general case and have to be treated separately, and the groups of Lie type of even and odd characteristic are also quite different.
A group is of characteristic 2 type if thegeneralized Fitting subgroupF*(Y) of every 2-local subgroupYis a 2-group. As the name suggests these are roughly the groups of Lie type over fields of characteristic 2, plus a handful of others that are alternating or sporadic or of odd characteristic. Their classification is divided into the small and large rank cases, where the rank is the largest rank of an odd abelian subgroup normalizing a nontrivial 2-subgroup, which is often (but not always) the same as the rank of a Cartan subalgebra when the group is a group of Lie type in characteristic 2.
The rank 1 groups are the thin groups, classified by Aschbacher, and the rank 2 ones are the notoriousquasithin groups, classified by Aschbacher and Smith. These correspond roughly to groups of Lie type of ranks 1 or 2 over fields of characteristic 2.
Groups of rank at least 3 are further subdivided into 3 classes by thetrichotomy theorem, proved by Aschbacher for rank 3 and by Gorenstein and Lyons for rank at least 4.
The three classes are groups of GF(2) type (classified mainly by Timmesfeld), groups of "standard type" for some odd prime (classified by theGilman–Griess theoremand work by several others), and groups of uniqueness type, where a result of Aschbacher implies that there are no simple groups.
The general higher rank case consists mostly of the groups of Lie type over fields of characteristic 2 of rank at least 3 or 4.
The main part of the classification produces a characterization of each simple group. It is then necessary to check that there exists a simple group for each characterization and that it is unique. This gives a large number of separate problems; for example, the original proofs of existence and uniqueness of themonster grouptotaled about 200 pages, and the identification of theRee groupsby Thompson and Bombieri was one of the hardest parts of the classification. Many of the existence proofs and some of the uniqueness proofs for the sporadic groups originally used computer calculations, most of which have since been replaced by shorter hand proofs.
In 1972Gorenstein (1979, Appendix) announced a program for completing the classification of finite simple groups, consisting of the following 16 steps:
Many of the items in the table below are taken fromSolomon (2001). The date given is usually the publication date of the complete proof of a result, which is sometimes several years later than the proof or first announcement of the result, so some of the items appear in the "wrong" order.
The proof of the theorem, as it stood around 1985 or so, can be calledfirst generation. Because of the extreme length of the first generation proof, much effort has been devoted to finding a simpler proof, called asecond-generation classification proof. This effort, called "revisionism", was originally led byDaniel Gorenstein, and coauthored withRichard LyonsandRonald Solomon.
As of 2023[update], ten volumes of the second generation proof have been published (Gorenstein, Lyons & Solomon 1994, 1996, 1998, 1999, 2002, 2005, 2018a, 2018b; &Inna Capdeboscq, 2021, 2023). In 2012 Solomon estimated that the project would need another 5 volumes, but said that progress on them was slow. It is estimated that the new proof will eventually fill approximately 5,000 pages. (This length stems in part from the second generation proof being written in a more relaxed style.) However, with the publication of volume 9 of the GLS series, and including the Aschbacher–Smith contribution, this estimate was already reached, with several more volumes still in preparation (the rest of what was originally intended for volume 9, plus projected volumes 10 and 11). Aschbacher and Smith wrote their two volumes devoted to the quasithin case in such a way that those volumes can be part of the second generation proof.
Gorenstein and his collaborators have given several reasons why a simpler proof is possible.
Aschbacher (2004)has called the work on the classification problem by Ulrich Meierfrankenfeld, Bernd Stellmacher, Gernot Stroth, and a few others, athird generation program. One goal of this is to treat all groups in characteristic 2 uniformly using the amalgam method.
Gorenstein has discussed some of the reasons why there might not be a short proof of the classification similar to the classification ofcompact Lie groups.
This section lists some results that have been proved using the classification of finite simple groups.
|
https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
|
Inmathematicsand more precisely ingroup theory, thecommuting probability(also calleddegree of commutativityorcommutativity degree) of afinite groupis theprobabilitythat two randomly chosen elementscommute.[1][2]It can be used to measure how close toabeliana finite group is. It can be generalized to infinitegroupsequipped with a suitableprobability measure,[3]and can also be generalized to otheralgebraic structuressuch asrings.[4]
LetG{\displaystyle G}be afinite group. We definep(G){\displaystyle p(G)}as the averaged number of pairs of elements ofG{\displaystyle G}which commute:
where#X{\displaystyle \#X}denotes thecardinalityof a finite setX{\displaystyle X}.
If one considers theuniform distributiononG2{\displaystyle G^{2}},p(G){\displaystyle p(G)}is the probability that two randomly chosen elements ofG{\displaystyle G}commute. That is whyp(G){\displaystyle p(G)}is called thecommuting probabilityofG{\displaystyle G}.
|
https://en.wikipedia.org/wiki/Commuting_probability
|
Afinite-state machine(FSM) orfinite-state automaton(FSA, plural:automata),finite automaton, or simply astate machine, is a mathematicalmodel of computation. It is anabstract machinethat can be in exactly one of a finite number ofstatesat any given time. The FSM can change from one state to another in response to someinputs; the change from one state to another is called atransition.[1]An FSM is defined by a list of its states, its initial state, and the inputs that trigger each transition. Finite-state machines are of two types—deterministic finite-state machinesandnon-deterministic finite-state machines.[2]For any non-deterministic finite-state machine, an equivalent deterministic one can be constructed.
The behavior of state machines can be observed in many devices in modern society that perform a predetermined sequence of actions depending on a sequence of events with which they are presented. Simple examples are:vending machines, which dispense products when the proper combination of coins is deposited;elevators, whose sequence of stops is determined by the floors requested by riders;traffic lights, which change sequence when cars are waiting;combination locks, which require the input of a sequence of numbers in the proper order.
The finite-state machine has less computational power than some other models of computation such as theTuring machine.[3]The computational power distinction means there are computational tasks that a Turing machine can do but an FSM cannot. This is because an FSM'smemoryis limited by the number of states it has. A finite-state machine has the same computational power as a Turing machine that is restricted such that its head may only perform "read" operations, and always has to move from left to right. FSMs are studied in the more general field ofautomata theory.
An example of a simple mechanism that can be modeled by a state machine is aturnstile.[4][5]A turnstile, used to control access to subways and amusement park rides, is a gate with three rotating arms at waist height, one across the entryway. Initially the arms are locked, blocking the entry, preventing patrons from passing through. Depositing a coin ortokenin a slot on the turnstile unlocks the arms, allowing a single customer to push through. After the customer passes through, the arms are locked again until another coin is inserted.
Considered as a state machine, the turnstile has two possible states:LockedandUnlocked.[4]There are two possible inputs that affect its state: putting a coin in the slot (coin) and pushing the arm (push). In the locked state, pushing on the arm has no effect; no matter how many times the inputpushis given, it stays in the locked state. Putting a coin in – that is, giving the machine acoininput – shifts the state fromLockedtoUnlocked. In the unlocked state, putting additional coins in has no effect; that is, giving additionalcoininputs does not change the state. A customer pushing through the arms gives apushinput and resets the state toLocked.
The turnstile state machine can be represented by astate-transition table, showing for each possible state, the transitions between them (based upon the inputs given to the machine) and the outputs resulting from each input:
The turnstile state machine can also be represented by adirected graphcalled astate diagram(above). Each state is represented by anode(circle). Edges (arrows) show the transitions from one state to another. Each arrow is labeled with the input that triggers that transition. An input that doesn't cause a change of state (such as acoininput in theUnlockedstate) is represented by a circular arrow returning to the original state. The arrow into theLockednode from the black dot indicates it is the initial state.
Astateis a description of the status of a system that is waiting to execute atransition. A transition is a set of actions to be executed when a condition is fulfilled or when an event is received.
For example, when using an audio system to listen to the radio (the system is in the "radio" state), receiving a "next" stimulus results in moving to the next station. When the system is in the "CD" state, the "next" stimulus results in moving to the next track. Identical stimuli trigger different actions depending on the current state.
In some finite-state machine representations, it is also possible to associate actions with a state:
Severalstate-transition tabletypes are used. The most common representation is shown below: the combination of current state (e.g. B) and input (e.g. Y) shows the next state (e.g. C). By itself, the table cannot completely describe the action, so it is common to use footnotes. Other related representations may not have this limitation. For example, an FSM definition including the full action's information is possible usingstate tables(see alsovirtual finite-state machine).
TheUnified Modeling Languagehas a notation for describing state machines.UML state machinesovercome the limitations[citation needed]of traditional finite-state machines while retaining their main benefits. UML state machines introduce the new concepts ofhierarchically nested statesandorthogonal regions, while extending the notion ofactions. UML state machines have the characteristics of bothMealy machinesandMoore machines. They supportactionsthat depend on both the state of the system and the triggeringevent, as in Mealy machines, as well asentry and exit actions, which are associated with states rather than transitions, as in Moore machines.[citation needed]
TheSpecification and Description Languageis a standard fromITUthat includes graphical symbols to describe actions in the transition:
SDL embeds basic data types called "Abstract Data Types", an action language, and an execution semantic in order to make the finite-state machine executable.[citation needed]
There are a large number of variants to represent an FSM such as the one in figure 3.
In addition to their use in modeling reactive systems presented here, finite-state machines are significant in many different areas, includingelectrical engineering,linguistics,computer science,philosophy,biology,mathematics,video game programming, andlogic. Finite-state machines are a class of automata studied inautomata theoryand thetheory of computation.
In computer science, finite-state machines are widely used in modeling of application behavior (control theory), design ofhardware digital systems,software engineering,compilers,network protocols, andcomputational linguistics.
Finite-state machines can be subdivided into acceptors, classifiers, transducers and sequencers.[6]
Acceptors(also calleddetectorsorrecognizers) produce binary output, indicating whether or not the received input is accepted. Each state of an acceptor is eitheracceptingornon accepting. Once all input has been received, if the current state is an accepting state, the input is accepted; otherwise it is rejected. As a rule, input is asequence of symbols(characters); actions are not used. The start state can also be an accepting state, in which case the acceptor accepts the empty string. The example in figure 4 shows an acceptor that accepts the string "nice". In this acceptor, the only accepting state is state 7.
A (possibly infinite) set of symbol sequences, called aformal language, is aregular languageif there is some acceptor that acceptsexactlythat set.[7]For example, the set of binary strings with an even number of zeroes is a regular language (cf. Fig. 5), while the set of all strings whose length is a prime number is not.[8]
An acceptor could also be described as defining a language that would contain every string accepted by the acceptor but none of the rejected ones; that language isacceptedby the acceptor. By definition, the languages accepted by acceptors are theregular languages.
The problem of determining the language accepted by a given acceptor is an instance of thealgebraic path problem—itself a generalization of theshortest path problemto graphs with edges weighted by the elements of an (arbitrary)semiring.[9][10][jargon]
An example of an accepting state appears in Fig. 5: adeterministic finite automaton(DFA) that detects whether thebinaryinput string contains an even number of 0s.
S1(which is also the start state) indicates the state at which an even number of 0s has been input. S1is therefore an accepting state. This acceptor will finish in an accept state, if the binary string contains an even number of 0s (including any binary string containing no 0s). Examples of strings accepted by this acceptor areε(theempty string), 1, 11, 11..., 00, 010, 1010, 10110, etc.
Classifiersare a generalization of acceptors that producen-ary output wherenis strictly greater than two.[11]
Transducersproduce output based on a given input and/or a state using actions. They are used for control applications and in the field ofcomputational linguistics.
In control applications, two types are distinguished:
Sequencers(also calledgenerators) are a subclass of acceptors and transducers that have a single-letter input alphabet. They produce only one sequence, which can be seen as an output sequence of acceptor or transducer outputs.[6]
A further distinction is betweendeterministic(DFA) andnon-deterministic(NFA,GNFA) automata. In a deterministic automaton, every state has exactly one transition for each possible input. In a non-deterministic automaton, an input can lead to one, more than one, or no transition for a given state. Thepowerset constructionalgorithm can transform any nondeterministic automaton into a (usually more complex) deterministic automaton with identical functionality.
A finite-state machine with only one state is called a "combinatorial FSM". It only allows actions upon transitionintoa state. This concept is useful in cases where a number of finite-state machines are required to work together, and when it is convenient to consider a purely combinatorial part as a form of FSM to suit the design tools.[12]
There are other sets of semantics available to represent state machines. For example, there are tools for modeling and designing logic for embedded controllers.[13]They combinehierarchical state machines(which usually have more than one current state), flow graphs, andtruth tablesinto one language, resulting in a different formalism and set of semantics.[14]These charts, likeHarel'soriginal state machines,[15]support hierarchically nested states,orthogonal regions, state actions, and transition actions.[16]
In accordance with the general classification, the following formal definitions are found.
Adeterministic finite-state machineordeterministic finite-state acceptoris aquintuple(Σ,S,s0,δ,F){\displaystyle (\Sigma ,S,s_{0},\delta ,F)}, where:
For both deterministic and non-deterministic FSMs, it is conventional to allowδ{\displaystyle \delta }to be apartial function, i.e.δ(s,x){\displaystyle \delta (s,x)}does not have to be defined for every combination ofs∈S{\displaystyle s\in S}andx∈Σ{\displaystyle x\in \Sigma }. If an FSMM{\displaystyle M}is in a states{\displaystyle s}, the next symbol isx{\displaystyle x}andδ(s,x){\displaystyle \delta (s,x)}is not defined, thenM{\displaystyle M}can announce an error (i.e. reject the input). This is useful in definitions of general state machines, but less useful when transforming the machine. Some algorithms in their default form may require total functions.
A finite-state machine has the same computational power as aTuring machinethat is restricted such that its head may only perform "read" operations, and always has to move from left to right. That is, each formal language accepted by a finite-state machine is accepted by such a kind of restricted Turing machine, and vice versa.[17]
Afinite-state transduceris asextuple(Σ,Γ,S,s0,δ,ω){\displaystyle (\Sigma ,\Gamma ,S,s_{0},\delta ,\omega )}, where:
If the output function depends on the state and input symbol (ω:S×Σ→Γ{\displaystyle \omega :S\times \Sigma \rightarrow \Gamma }) that definition corresponds to theMealy model, and can be modelled as aMealy machine. If the output function depends only on the state (ω:S→Γ{\displaystyle \omega :S\rightarrow \Gamma }) that definition corresponds to theMoore model, and can be modelled as aMoore machine. A finite-state machine with no output function at all is known as asemiautomatonortransition system.
If we disregard the first output symbol of a Moore machine,ω(s0){\displaystyle \omega (s_{0})}, then it can be readily converted to an output-equivalent Mealy machine by setting the output function of every Mealy transition (i.e. labeling every edge) with the output symbol given of the destination Moore state. The converse transformation is less straightforward because a Mealy machine state may have different output labels on its incoming transitions (edges). Every such state needs to be split in multiple Moore machine states, one for every incident output symbol.[18]
Optimizing an FSM means finding a machine with the minimum number of states that performs the same function. The fastest known algorithm doing this is theHopcroft minimization algorithm.[19][20]Other techniques include using animplication table, or the Moore reduction procedure.[21]Additionally, acyclic FSAs can be minimized inlinear time.[22]
In adigital circuit, an FSM may be built using aprogrammable logic device, aprogrammable logic controller,logic gatesandflip flopsorrelays. More specifically, a hardware implementation requires aregisterto store state variables, a block ofcombinational logicthat determines the state transition, and a second block of combinational logic that determines the output of an FSM. One of the classic hardware implementations is theRichards controller.
In aMedvedev machine, the output is directly connected to the state flip-flops minimizing the time delay between flip-flops and output.[23][24]
Throughstate encoding for low powerstate machines may be optimized to minimize power consumption.
The following concepts are commonly used to build software applications with finite-state machines:
Finite automata are often used in thefrontendof programming language compilers. Such a frontend may comprise several finite-state machines that implement alexical analyzerand a parser.
Starting from a sequence of characters, the lexical analyzer builds a sequence of language tokens (such as reserved words, literals, and identifiers) from which the parser builds a syntax tree. The lexical analyzer and the parser handle the regular andcontext-freeparts of the programming language's grammar.[25]
Finite Markov-chain processes are also known assubshifts of finite type.
|
https://en.wikipedia.org/wiki/Finite-state_machine
|
Incommunication complexity, thegap-Hamming problemasks, ifAlice and Bobare each given a (potentially different) string, what is the minimal number of bits that they need to exchange in order for Alice to approximately compute theHamming distancebetween their strings. The solution to the problem roughly states that, if Alice and Bob are each given a string, then anycommunication protocolused to compute the Hamming distance between their strings does (asymptotically) no better than Bob sending his whole string to Alice. More specifically, if Alice and Bob are each givenn{\displaystyle n}-bit strings, there exists no communication protocol that lets Alice compute the hamming distance between their strings to within±n{\displaystyle \pm {\sqrt {n}}}using less thanΩ(n){\displaystyle \Omega (n)}bits.
The gap-Hamming problem has applications to proving lower bounds for many streaming algorithms, including moment frequency estimation[1]and entropy estimation.[2]
In this problem, Alice and Bob each receive a string,x∈{±1}n{\displaystyle x\in \{\pm 1\}^{n}}andy∈{±1}n{\displaystyle y\in \{\pm 1\}^{n}}, respectively, while Alice is required to compute the (partial) function,
using the least amount of communication possible. Here,∗{\displaystyle *}indicates that Alice can return either of±1{\displaystyle \pm 1}andDH(x,y){\displaystyle D_{H}(x,y)}is theHamming distancebetweenx{\displaystyle x}andy{\displaystyle y}. In other words, Alice needs to return whether Bob's string is significantly similar or significantly different from hers while minimizing the number of bits she exchanges with Bob.
The problem's solution states that computingGHD{\displaystyle \operatorname {GHD} }requires at leastΩ(n){\displaystyle \Omega (n)}communication. In particular, it requiresΩ(n){\displaystyle \Omega (n)}communication even whenx{\displaystyle x}andy{\displaystyle y}are chosen uniformly at random from{±1}n{\displaystyle \{\pm 1\}^{n}}.
The gap-Hamming problem was originally proposed by Indyk and Woodruff in the early 2000's, who initially proved a linear lower bound on theone-waycommunication complexity of the problem (where Alice is only allowed to receive data from Bob) and conjectured a linear lower bound in the general case.[3]The question of the infinite-round case (in which Alice and Bob are allowed to exchange as many messages as desired) remained open until Chakrabarti and Regev proved, via ananti-concentrationargument, that the general problem also has linear lower bound complexity, thus settling the original question completely.[4]This result was followed by a series of other papers that sought to simplify or find new approaches to proving the desired lower bound, notably first by Vidick[5]and later by Sherstov,[6]and, recently, with an information-theoretic approach by Hadar, Liu, Polyanskiy, and Shayevitz.[7]
|
https://en.wikipedia.org/wiki/Gap-Hamming_problem
|
Application-Layer Protocol Negotiation(ALPN) is aTransport Layer Security(TLS) extension that allows the application layer to negotiate whichprotocolshould be performed over a secure connection in a manner that avoids additional round trips and which is independent of the application-layer protocols. It is used to establishHTTP/2connections without additional round trips (client and server can communicate over two ports previously assigned to HTTPS withHTTP/1.1and upgrade to use HTTP/2 or continue with HTTP/1.1 without closing the initial connection).
ALPN is supported by these libraries:
In January 2010, Google introduced IETF standard draft describing Next Protocol Negotiation TLS extension.[13]This extension was used to negotiate experimental SPDY connections between Google Chrome and some of Google's servers. As SPDY evolved, NPN was replaced with ALPN.
On July 11, 2014, ALPN was published asRFC7301. ALPN replaces Next Protocol Negotiation (NPN) extension.[14]
TLS False Startwas disabled inGoogle Chromefrom version 20 (2012) onward except for websites with the earlier NPN extension.[15]
ALPN is a TLS extension which is sent on the initialTLS handshake'Client Hello', and it lists the protocols that the client (for example the web browser) supports:
The resulting 'Server Hello' from the web server will also contain the ALPN extension, and it confirms which protocol will be used for the HTTP request:
|
https://en.wikipedia.org/wiki/Application-Layer_Protocol_Negotiation
|
Bullrun(stylizedBULLRUN) is aclandestine, highly classified program to crack encryption of online communications and data, which is run by the United StatesNational Security Agency(NSA).[1][2]The BritishGovernment Communications Headquarters(GCHQ) has a similar program codenamedEdgehill. According to the Bullrun classification guide published byThe Guardian, the program uses multiple methods including computer network exploitation,[3]interdiction, industry relationships, collaboration with otherintelligence communityentities, and advanced mathematical techniques.
Information about the program's existence was leaked in 2013 byEdward Snowden. Although Snowden's documents do not contain technical information on exact cryptanalytic capabilities because Snowden did not have clearance access to such information,[4]they do contain a 2010GCHQpresentation which claims that "vast amounts of encrypted Internet data which have up till now been discarded are now exploitable".[1]A number of technical details regarding the program found in Snowden's documents were additionally censored by the press at the behest of US intelligence officials.[5]Out of all the programs that have been leaked by Snowden, the Bullrun Decryption Program is by far the most expensive. Snowden claims that since 2011, expenses devoted to Bullrun amount to $800 million. The leaked documents reveal that Bullrun seeks to "defeat the encryption used in specific network communication technologies".[6]
According to the NSA's Bullrun Classification Guide, Bullrun is not aSensitive Compartmented Information(SCI) control system or compartment, but the codeword has to be shown in the classification line, after all other classification and dissemination markings. Furthermore, any details about specific cryptographic successes were recommended to be additionally restricted (besides being markedTop Secret//SI) withExceptionally Controlled Informationlabels; a non-exclusive list of possible Bullrun ECI labels was given as: APERIODIC, AMBULANT, AUNTIE, PAINTEDEAGLE, PAWLEYS, PITCHFORD, PENDLETON, PICARESQUE, and PIEDMONT without any details as to what these labels mean.[1][2]
Access to the program is limited to a group of top personnel at theFive Eyes(FVEY), the NSA and thesignals intelligenceagencies of the United Kingdom (GCHQ), Canada (CSE), Australia (ASD), and New Zealand (GCSB). Signals that cannot be decrypted with current technology may be retained indefinitely while the agencies continue to attempt to decrypt them.[2]
Through the NSA-designedClipper chip, which used theSkipjackcipher with an intentional backdoor, and using various specifically designed laws such asCALEA,CESAandrestrictions on export of encryption softwareas evidenced byBernstein v. United States, the U.S. government had publicly attempted in the 1990s to ensure its access to communications and ability to decrypt.[7][8]In particular, technical measures such askey escrow, a euphemism for abackdoor, have met with criticism and little success.
The NSA encourages the manufacturers of security technology to disclose backdoors to their products or encryption keys so that they may access the encrypted data.[9]However, fearing widespread adoption of encryption, the NSA set out to stealthily influence and weaken encryption standards and obtain master keys—either by agreement, by force of law, or by computer network exploitation (hacking).[5]
According to a Bullrun briefing document, the agency had successfully infiltrated both theSecure Sockets Layeras well as somevirtual private networks(VPNs).[1][2]TheNew York Timesreported that: "But by 2006, an N.S.A. document notes, the agency had broken into communications for three foreign airlines, one travel reservation system, one foreign government's nuclear department and another's Internet service by cracking the virtual private networks that protected them. By 2010, the Edgehill program, the British counterencryption effort, was unscrambling VPN traffic for 30 targets and had set a goal of an additional 300."[5]
As part of Bullrun, NSA has also been actively working to "Insert vulnerabilities into commercial encryption systems, IT systems, networks, and endpoint communications devices used by targets".[10]The New York Timeshas reported that the random number generatorDual_EC_DRBGcontains a back door, which would allow the NSA to break encryption keys generated by the random number generator.[11]Even though this random number generator was known to be insecure and slow soon after the standard was published, and a potential NSAkleptographicbackdoor was found in 2007 while alternative random number generators without these flaws were certified and widely available,RSA Securitycontinued using Dual_EC_DRBG in the company'sBSAFE toolkitand Data Protection Manager until September 2013. While RSA Security has denied knowingly inserting a backdoor into BSAFE, it has not yet given an explanation for the continued usage of Dual_EC_DRBG after its flaws became apparent in 2006 and 2007.[12]It was reported on December 20, 2013, that RSA had accepted a payment of $10 million from the NSA to set the random number generator as the default.[13][14]Leaked NSA documents state that their effort was “a challenge in finesse” and that “Eventually, N.S.A. became the sole editor” of the standard.[5]
By 2010, the leaked documents state that the NSA had developed "groundbreaking capabilities" against encrypted Internet traffic. A GCHQ document warned however "These capabilities are among theSIGINTcommunity's most fragile, and the inadvertent disclosure of the simple 'fact of' could alert the adversary and result in immediate loss of the capability."[5]The document later states that "there will be NO 'need to know.'"[5]Several experts, includingBruce SchneierandChristopher Soghoian, had speculated that a successful attack againstRC4, an encryption algorithm used in at least 50 percent of all SSL/TLS traffic at the time, was a plausible avenue, given several publicly knownweaknessesof RC4.[15]Others have speculated that NSA has gained ability to crack 1024-bitRSA/DHkeys.[16]RC4 has since been prohibited for all versions of TLS by RFC 7465 in 2015, due to theRC4 attacksweakening or breaking RC4 used in SSL/TLS.
In the wake of Bullrun revelations, some open source projects, includingFreeBSDandOpenSSL, have seen an increase in their reluctance to (fully) trust hardware-basedcryptographic primitives.[17][18]
Many other software projects, companies and organizations responded with an increase in the evaluation of their security and encryption processes. For example, Google doubled the size of their TLS certificates from 1024 bits to 2048 bits.[19]
Revelations of the NSA backdoors and purposeful complication of standards has led to a backlash in their participation in standards bodies.[20]Prior to the revelations the NSA's presence on these committees was seen as a benefit given their expertise with encryption.[21]
There has been speculation that the NSA was aware of theHeartbleedbug, which caused major websites to be vulnerable to password theft, but did not reveal this information in order to exploit it themselves.[22]
The name "Bullrun" was taken from theFirst Battle of Bull Run, the first major battle of theAmerican Civil War.[1]Its predecessor "Manassas",[2]is both an alternate name for the battle and where the battle took place. "EDGEHILL" is from theBattle of Edgehill, the first battle of theEnglish Civil War.[23]
|
https://en.wikipedia.org/wiki/Bullrun_(decryption_program)
|
Incryptography, acertificate authorityorcertification authority(CA) is an entity that stores, signs, and issuesdigital certificates. A digital certificate certifies the ownership of a public key by the named subject of the certificate. This allows others (relying parties) to rely upon signatures or on assertions made about the private key that corresponds to the certified public key. A CA acts as a trusted third party—trusted both by the subject (owner) of the certificate and by the party relying upon the certificate.[1]The format of these certificates is specified by theX.509orEMVstandard.
One particularly common use for certificate authorities is to sign certificates used inHTTPS, the secure browsing protocol for the World Wide Web. Another common use is in issuing identity cards by national governments for use in electronically signing documents.[2]
Trusted certificates can be used to createsecure connectionsto a server via the Internet. A certificate is essential in order to circumvent a malicious party which happens to be on the route to a target server which acts as if it were the target. Such a scenario is commonly referred to as aman-in-the-middle attack. The client uses the CA certificate to authenticate the CA signature on the server certificate, as part of the authorizations before launching a secure connection.[3]Usually, client software—for example, browsers—include a set of trusted CA certificates. This makes sense, as many users need to trust their client software. A malicious or compromised client can skip any security check and still fool its users into believing otherwise.
The clients of a CA are server supervisors who call for a certificate that their servers will bestow to users. Commercial CAs charge money to issue certificates, and their customers anticipate the CA's certificate to be contained within the majority of web browsers, so that safe connections to the certified servers work efficiently out-of-the-box. The quantity of web browsers, other devices, and applications which trust a particular certificate authority is referred to as ubiquity.Mozilla, which is a non-profit business, issues several commercial CA certificates with its products.[4]While Mozilla developed their own policy, theCA/Browser Forumdeveloped similar guidelines for CA trust. A single CA certificate may be shared among multiple CAs or theirresellers. ArootCA certificate may be the base to issue multipleintermediateCA certificates with varying validation requirements.
In addition to commercial CAs, some non-profits issue publicly-trusted digital certificates without charge, for exampleLet's Encrypt. Some large cloud computing and web hosting companies are also publicly-trusted CAs and issue certificates to services hosted on their infrastructure, for exampleIBM Cloud,Amazon Web Services,Cloudflare, andGoogle Cloud Platform.
Large organizations or government bodies may have their own PKIs (public key infrastructure), each containing their own CAs. Any site usingself-signed certificatesacts as its own CA.
Commercial banks that issueEMVpayment cards are governed by the EMV Certificate Authority,[5]payment schemes that route payment transactions initiated at Point of Sale Terminals (POS) to a Card Issuing Bank to transfer the funds from the card holder's bank account to the payment recipient's bank account. Each payment card presents along with its card data also the Card Issuer Certificate to the POS. The Issuer Certificate is signed by EMV CA Certificate. The POS retrieves the public key of EMV CA from its storage, validates the Issuer Certificate and authenticity of the payment card before sending the payment request to the payment scheme.
Browsers and other clients of sorts characteristically allow users to add or do away with CA certificates at will. While server certificates regularly last for a relatively short period, CA certificates are further extended,[6]so, for repeatedly visited servers, it is less error-prone importing and trusting the CA issued, rather than confirm a security exemption each time the server's certificate is renewed.
Less often, trustworthy certificates are used for encrypting or signing messages. CAs dispense end-user certificates too, which can be used withS/MIME. However, encryption entails the receiver's publickeyand, since authors and receivers of encrypted messages, apparently, know one another, the usefulness of a trusted third party remains confined to the signature verification of messages sent to public mailing lists.
Worldwide, the certificate authority business is fragmented, with national or regional providers dominating their home market. This is because many uses of digital certificates, such as for legally binding digital signatures, are linked to local law, regulations, and accreditation schemes for certificate authorities.
However, the market for globally trustedTLS/SSL server certificatesis largely held by a small number of multinational companies. This market has significantbarriers to entrydue to the technical requirements.[7]While not legally required, new providers may choose to undergo annual security audits (such asWebTrust[8]for certificate authorities in North America andETSIin Europe[9]) to be included as a trusted root by a web browser or operating system.
As of 24 August 2020[update], 147 root certificates, representing 52 organizations, are trusted in theMozilla Firefoxweb browser,[10]168 root certificates, representing 60 organizations, are trusted bymacOS,[11]and 255 root certificates, representing 101 organizations, are trusted byMicrosoft Windows.[12]As of Android 4.2 (Jelly Bean), Android currently contains over 100 CAs that are updated with each release.[13]
On November 18, 2014, a group of companies and nonprofit organizations, including theElectronic Frontier Foundation, Mozilla, Cisco, and Akamai, announcedLet's Encrypt, a nonprofit certificate authority that provides free domain validatedX.509 certificatesas well as software to enable installation and maintenance of certificates.[14]Let's Encrypt is operated by the newly formedInternet Security Research Group, a California nonprofit recognized as federally tax-exempt.[15]
According toNetcraftin May 2015, the industry standard for monitoring active TLS certificates, "Although the global [TLS] ecosystem is competitive, it is dominated by a handful of major CAs — three certificate authorities (Symantec, Comodo, GoDaddy) account for three-quarters of all issued [TLS] certificates on public-facing web servers. The top spot has been held by Symantec (or VeriSign before it was purchased by Symantec) ever since [our] survey began, with it currently accounting for just under a third of all certificates. To illustrate the effect of differing methodologies, amongst the million busiest sites Symantec issued 44% of the valid, trusted certificates in use — significantly more than its overall market share."[16]
As of July 2024[update]the survey company W3Techs, which collects statistics on certificate authority usage among theAlexatop 10 million and the Tranco top 1 million websites, lists the five largest authorities by absolute usage share as below.[17]
The commercial CAs that issue the bulk of certificates for HTTPS servers typically use a technique called "domain validation" to authenticate the recipient of the certificate. The techniques used for domain validation vary between CAs, but in general domain validation techniques are meant to prove that the certificate applicant controls a givendomain name, not any information about the applicant's identity.
Many Certificate Authorities also offerExtended Validation(EV) certificates as a more rigorous alternative to domain validated certificates. Extended validation is intended to verify not only control of a domain name, but additional identity information to be included in the certificate. Some browsers display this additional identity information in a green box in the URL bar. One limitation of EV as a solution to the weaknesses of domain validation is that attackers could still obtain a domain validated certificate for the victim domain, and deploy it during an attack; if that occurred, the difference observable to the victim user would be the absence of a green bar with the company name. There is some question whether users would be likely to recognize this absence as indicative of an attack being in progress: a test usingInternet Explorer 7in 2009 showed that the absence of IE7's EV warnings were not noticed by users, however Microsoft's newer browser,Edge Legacy, shows a significantly greater difference between EV and domain validated certificates, with domain validated certificates having a hollow, gray lock.
Domain validation suffers from certain structural security limitations. In particular, it is always vulnerable to attacks that allow an adversary to observe the domain validation probes that CAs send. These can include attacks against the DNS, TCP, or BGP protocols (which lack the cryptographic protections of TLS/SSL), or the compromise of routers. Such attacks are possible either on the network near a CA, or near the victim domain itself.
One of the most common domain validation techniques involves sending an email containing an authentication token or link to an email address that is likely to be administratively responsible for the domain. This could be the technical contact email address listed in the domain'sWHOISentry, or an administrative email likeadmin@,administrator@,webmaster@,hostmaster@orpostmaster@the domain.[18][19]Some Certificate Authorities may accept confirmation usingroot@,[citation needed]info@, orsupport@in the domain.[20]The theory behind domain validation is that only the legitimate owner of a domain would be able to read emails sent to these administrative addresses.
Domain validation implementations have sometimes been a source of security vulnerabilities. In one instance, security researchers showed that attackers could obtain certificates for webmail sites because a CA was willing to use an email address likessladmin@domain.comfor domain.com, but not all webmail systems had reserved the "ssladmin" username to prevent attackers from registering it.[21]
Prior to 2011, there was no standard list of email addresses that could be used for domain validation, so it was not clear to email administrators which addresses needed to be reserved. The first version of theCA/Browser ForumBaseline Requirements, adopted November 2011, specified a list of such addresses. This allowed mail hosts to reserve those addresses for administrative use, though such precautions are still not universal. In January 2015, a Finnish man registered the username "hostmaster" at the Finnish version ofMicrosoft Liveand was able to obtain a domain-validated certificate for live.fi, despite not being the owner of the domain name.[22]
A CA issuesdigital certificatesthat contain apublic keyand the identity of the owner. The matching private key is not made available publicly, but kept secret by the end user who generated the key pair. The certificate is also a confirmation or validation by the CA that the public key contained in the certificate belongs to the person, organization, server or other entity noted in the certificate. A CA's obligation in such schemes is to verify an applicant's credentials, so that users and relying parties can trust the information in the issued certificate. CAs use a variety of standards and tests to do so. In essence, the certificate authority is responsible for saying "yes, this person is who they say they are, and we, the CA, certify that".[23]
If the user trusts the CA and can verify the CA's signature, then they can also assume that a certain public key does indeed belong to whoever is identified in the certificate.[24]
Public-key cryptographycan be used to encrypt data communicated between two parties. This can typically happen when a user logs on to any site that implements theHTTP Secureprotocol. In this example let us suppose that the user logs on to their bank's homepage www.bank.example to do online banking. When the user opens www.bank.example homepage, they receive a public key along with all the data that their web-browser displays. The public key could be used to encrypt data from the client to the server but the safe procedure is to use it in a protocol that determines a temporary shared symmetric encryption key; messages in such a key exchange protocol can be enciphered with the bank's public key in such a way that only the bank server has the private key to read them.[25]
The rest of the communication then proceeds using the new (disposable) symmetric key, so when the user enters some information to the bank's page and submits the page (sends the information back to the bank) then the data the user has entered to the page will be encrypted by their web browser. Therefore, even if someone can access the (encrypted) data that was communicated from the user to www.bank.example, such eavesdropper cannot read or decipher it.
This mechanism is only safe if the user can be sure that it is the bank that they see in their web browser. If the user types in www.bank.example, but their communication is hijacked and a fake website (that pretends to be the bank website) sends the page information back to the user's browser, the fake web-page can send a fake public key to the user (for which the fake site owns a matching private key). The user will fill the form with their personal data and will submit the page. The fake web-page will then get access to the user's data.
This is what the certificate authority mechanism is intended to prevent. A certificate authority (CA) is an organization that stores public keys and their owners, and every party in a communication trusts this organization (and knows its public key). When the user's web browser receives the public key from www.bank.example it also receives a digital signature of the key (with some more information, in a so-calledX.509certificate). The browser already possesses the public key of the CA and consequently can verify the signature, trust the certificate and the public key in it: since www.bank.example uses a public key that the certification authority certifies, a fake www.bank.example can only use the same public key. Since the fake www.bank.example does not know the corresponding private key, it cannot create the signature needed to verify its authenticity.[26]
It is difficult to assure correctness of match between data and entity when the data are presented to the CA (perhaps over an electronic network), and when the credentials of the person/company/program asking for a certificate are likewise presented. This is why commercial CAs often use a combination of authentication techniques including leveraging government bureaus, the payment infrastructure, third parties' databases and services, and custom heuristics. In some enterprise systems, local forms of authentication such asKerberoscan be used to obtain a certificate which can in turn be used by external relying parties.Notariesare required in some cases to personally know the party whose signature is being notarized; this is a higher standard than is reached by many CAs. According to theAmerican Bar Associationoutline on Online Transaction Management the primary points of US Federal and State statutes enacted regardingdigital signatureshas been to "prevent conflicting and overly burdensome local regulation and to establish that electronic writings satisfy the traditional requirements associated with paper documents." Further the US E-Sign statute and the suggested UETA code[27]help ensure that:
Despite the security measures undertaken to correctly verify the identities of people and companies, there is a risk of a single CA issuing a bogus certificate to an imposter. It is also possible to register individuals and companies with the same or very similar names, which may lead to confusion. To minimize this hazard, thecertificate transparencyinitiativeproposes auditing all certificates in a public unforgeable log, which could help in the prevention ofphishing.[28][29]
In large-scale deployments, Alice may not be familiar with Bob's certificate authority (perhaps they each have a different CA server), so Bob's certificate may also include his CA's public key signed by a different CA2, which is presumably recognizable by Alice. This process typically leads to a hierarchy or mesh of CAs and CA certificates.
A certificate may be revoked before it expires, which signals that it is no longer valid. Without revocation, an attacker would be able to exploit such a compromised or misissued certificate until expiry.[30]Hence, revocation is an important part of apublic key infrastructure.[31]Revocation is performed by the issuing CA, which produces acryptographically authenticatedstatement of revocation.[32]
For distributing revocation information to clients, timeliness of the discovery of revocation (and hence the window for an attacker to exploit a compromised certificate) trades off against resource usage in querying revocation statuses and privacy concerns.[33]If revocation information is unavailable (either due to accident or an attack), clients must decide whether tofail-hardand treat a certificate as if it is revoked (and so degradeavailability) or tofail-softand treat it as unrevoked (and allow attackers to sidestep revocation).[34]
Due to the cost of revocation checks and the availability impact from potentially-unreliable remote services,Web browserslimit the revocation checks they will perform, and will fail-soft where they do.[35]Certificate revocation listsare too bandwidth-costly for routine use, and theOnline Certificate Status Protocolpresents connection latency and privacy issues. Other schemes have been proposed but have not yet been successfully deployed to enable fail-hard checking.[31]
The CA/Browser Forum publishes the Baseline Requirements,[41]a list of policies and technical requirements for CAs to follow. These are a requirement for inclusion in the certificate stores of Firefox[42]and Safari.[43]
On April 14, 2025, the CA/Browser Forum passed a ballot to reduce SSL/TLS certificates to 47 day maximum term by March 15, 2029.[44]
If the CA can be subverted, then the security of the entire system is lost, potentially subverting all the entities that trust the compromised CA.
For example, suppose an attacker, Eve, manages to get a CA to issue to her a certificate that claims to represent Alice. That is, the certificate would publicly state that it represents Alice, and might include other information about Alice. Some of the information about Alice, such as her employer name, might be true, increasing the certificate's credibility. Eve, however, would have the all-important private key associated with the certificate. Eve could then use the certificate to send a digitally signed email to Bob, tricking Bob into believing that the email was from Alice. Bob might even respond with encrypted email, believing that it could only be read by Alice, when Eve is actually able to decrypt it using the private key.
A notable case of CA subversion like this occurred in 2001, when the certificate authorityVeriSignissued two certificates to a person claiming to represent Microsoft. The certificates have the name "Microsoft Corporation", so they could be used to spoof someone into believing that updates to Microsoft software came from Microsoft when they actually did not. The fraud was detected in early 2001. Microsoft and VeriSign took steps to limit the impact of the problem.[45][46]
In 2008, Comodo reseller Certstar sold a certificate for mozilla.com to Eddy Nigg, who had no authority to represent Mozilla.[47]
In 2011 fraudulent certificates were obtained from Comodo andDigiNotar,[48][49]allegedly by Iranian hackers. There is evidence that the fraudulent DigiNotar certificates were used in aman-in-the-middle attackin Iran.[50]
In 2012, it became known that Trustwave issued a subordinate root certificate that was used for transparent traffic management (man-in-the-middle) which effectively permitted an enterprise to sniff SSL internal network traffic using the subordinate certificate.[51]
In 2012, theFlamemalware (also known as SkyWiper) contained modules that had an MD5 collision with a valid certificate issued by a Microsoft Terminal Server licensing certificate that used the broken MD5 hash algorithm. The authors thus was able to conduct acollision attackwith the hash listed in the certificate.[52][53]
In 2015, a Chinese certificate authority named MCS Holdings and affiliated withChina's central domain registryissued unauthorized certificates for Google domains.[54][55]Google thus removed both MCS and the root certificate authority fromChromeand have revoked the certificates.[56]
An attacker who steals a certificate authority's private keys is able to forge certificates as if they were CA, without needed ongoing access to the CA's systems. Key theft is therefore one of the main risks certificate authorities defend against. Publicly trusted CAs almost always store their keys on ahardware security module(HSM), which allows them to sign certificates with a key, but generally prevent extraction of that key with both physical and software controls. CAs typically take the further precaution of keeping the key for their long-termroot certificatesin an HSM that is keptoffline, except when it is needed to sign shorter-lived intermediate certificates. The intermediate certificates, stored in an online HSM, can do the day-to-day work of signing end-entity certificates and keeping revocation information up to date.
CAs sometimes use akey ceremonywhen generating signing keys, in order to ensure that the keys are not tampered with or copied.
The critical weakness in the way that the current X.509 scheme is implemented is that any CA trusted by a particular party can then issue certificates for any domain they choose. Such certificates will be accepted as valid by the trusting party whether they are legitimate and authorized or not.[57]This is a serious shortcoming given that the most commonly encountered technology employing X.509 and trusted third parties is the HTTPS protocol. As all major web browsers are distributed to their end-users pre-configured with a list of trusted CAs that numbers in the dozens this means that any one of these pre-approved trusted CAs can issue a valid certificate for any domain whatsoever.[58]The industry response to this has been muted.[59]Given that the contents of a browser's pre-configured trusted CA list is determined independently by the party that is distributing or causing to be installed the browser application there is really nothing that the CAs themselves can do.
This issue is the driving impetus behind the development of theDNS-based Authentication of Named Entities(DANE) protocol. If adopted in conjunction withDomain Name System Security Extensions(DNSSEC) DANE will greatly reduce if not eliminate the role of trusted third parties in a domain's PKI.
|
https://en.wikipedia.org/wiki/Certificate_authority
|
Certificate Transparency(CT) is anInternet securitystandardfor monitoring and auditing the issuance ofdigital certificates.[1]When an internet user interacts with a website, a trusted third party is needed for assurance that the website is legitimate and that the website's encryption key is valid. This third party, called acertificate authority(CA), will issue a certificate for the website that the user's browser can validate. The security of encrypted internet traffic depends on the trust that certificates are only given out by the certificate authority and that the certificate authority has not been compromised.
Certificate Transparency makes public all issued certificates in the form of adistributed ledger, giving website owners and auditors the ability to detect and expose inappropriately issued certificates.
Work on Certificate Transparency first began in 2011 after the certificate authorityDigiNotarbecame compromised and started issuing malicious certificates. Google engineers submitted a draft to theInternet Engineering Task Force(IETF) in 2012. This effort resulted in IETFRFC6962, a standard defining a system of publiclogsto record all certificates issued by publicly trustedcertificate authorities, allowing efficient identification of mistakenly or maliciously issued certificates.[2]
The certificate transparency system consists of a system ofappend-onlycertificate logs. Logs are operated by many parties, includingbrowservendors andcertificate authorities.[3]Certificates that support certificate transparency must include one or moresigned certificate timestamps(SCTs), which is a promise from a log operator to include the certificate in their log within amaximum merge delay(MMD).[4][3]At some point within the maximum merge delay, the log operator adds the certificate to their log. Each entry in a log references the hash of a previous one, forming aMerkle tree. Thesigned tree head(STH) references the current root of theMerkle tree.
Although anyone can submit a certificate to a CT log, this task is commonly carried out by a CA as follows:[4][5]
Finally, the CA may decide to log the final certificate as well. Let's Encrypt E1 CA, for example, logs both precertificates and final certificates (see CAcrt.sh profile pageunder 'issued certificates' section), whereas Google GTS CA 2A1 does not (seecrt.sh profile page).
Some browsers requireTransport Layer Security(TLS) certificates to have proof of being logged with certificate transparency,[7][8]either through SCTs embedded into the certificate, an extension during the TLS handshake, or throughOCSP:
Due to the large quantities of certificates issued with the WebPKI, certificate transparency logs can grow to contain many certificates. This large quantity of certificates can cause strain on logs. Temporal sharding is a method to reduce the strain on logs by sharding a log into multiple logs, and having each shard only accept precertificates and certificates with an expiration date in a particular time period (usually a calendar year).[15][16][17]Cloudflare's Nimbus series of logs was the first to use temporal sharding.
One of the problems with digital certificate management is that fraudulent certificates take a long time to be spotted, reported andrevoked. An issued certificate not logged using Certificate Transparency may never be spotted at all. The main advantage with Certificate Transparency is the ability for cyber security teams to defend companies and organisations by monitoring for suspicious domains registering certificates. The new certificates for these suspicious domains may have similar names to other legitimate domains and are designed to be used to support malicious activities such as phishing attacks. Certificate Transparency puts cyber security teams in control and enables them to issue domain take down orders for suspicious domains and allows them to apply cyber security controls on web proxies and email gateways for immediate protection.[18]
Domain names that are used on internal networks and have certificates issued by certificate authorities become publicly searchable as their certificates are added to CT logs.
Certificate Transparency depends on verifiable Certificate Transparency logs. A log appends new certificates to an ever-growingMerkle hash tree.[19]: §4To be seen as behaving correctly, a log must:
A log may accept certificates that are not yet fully valid and certificates that have expired.
There are two primary categories of monitors: log integrity monitors (also referred to as log verifiers or log auditors)[19]: §8.3and tracking monitors.[20]Some companies offering monitoring services collect data from all logs and provide paid services for domain tracking. For example, a domain owner can register for Cloudflare's services, which globally monitor all logs and send email updates whenever a certificate is issued for their domain[21], allowing them control over all certificates issued.
Large organizations can maintain their own monitors, which continuously scan for new certificate issued for their domains. If acertificate authorities(CA) attempts to issue a "bad" certificate for one of these domains (intentionally or unintentionally); the monitor will quickly detect it.
Two popular APIs for research and tracking are Sectigo's crt.sh[22]and Cloudflare MerkleTown.[23]These tools facilitate the monitoring of certificate issuance and help organizations stay on top of their domain's security.
While there is an additional consideration of monitoring the monitors themselves, the likelihood of a significant impact on system performance or security due to misbehavior of a single monitor is low [reference needed]. This is because there are numerous log monitors, providing a layered approach to security and minimizing the risk of a single point of failure.
Apple[24]and Google[15]have separate log programs with distinct policies and lists of trusted logs.
Certificate Transparency logs maintain their own root stores and only accept certificates that chain back to the trusted roots.[19]A number of misbehaving logs have been publishing inconsistent root stores in the past.[25]
A new structure for logs is based on dividing the Merkle Tree into tiles. This structure is expected to be faster, easier to operate, and to provide much smaller merge delays (the current Maximum Merge Delay is 24 hours).[26]Chrome has updated its Certificate Transparency (CT) policy to accept SCTs from the new static-CT-API logs only if an SCT from an RFC 6962 log is also present, and it intends to complete the migration to static-CT-API CT logs by the end of 2025.[27]
In 2011, a reseller of the certificate authorityComodowas attacked and the certificate authorityDigiNotarwascompromised,[28]demonstrating existing flaws in the certificate authority ecosystem and prompting work on various mechanisms to prevent or monitor unauthorized certificate issuance.GoogleemployeesBen Laurie, Adam Langley and Emilia Kasper began work on anopen sourceframeworkfor detecting mis-issued certificates the same year. In 2012, they submitted the first draft of the standard toIETFunder the code-name "Sunlight".[29]
In March 2013, Google launched its first certificate transparency log.[30]
In June 2013,RFC6962"Certificate Transparency" was published, based on the 2012 draft.
In September 2013,DigiCertbecame the firstcertificate authorityto implement Certificate Transparency.[31]
In 2015,Google Chromebegan requiring Certificate Transparency for newly issuedExtended Validation Certificates.[32][33]It began requiring Certificate Transparency for all certificates newly issued bySymantecfrom June 1, 2016, after they were found to have issued 187 certificates without the domain owners' knowledge.[34][35]Since April 2018, this requirement has been extended to all certificates.[8]
On March 23, 2018,Cloudflareannounced its own CT log namedNimbus.[36]
In May 2019,certificate authorityLet's Encryptlaunched its own CT log called Oak. Since February 2020, it is included in approved log lists and is usable by all publicly trusted certificate authorities.[37]
In December 2021,RFC9162"Certificate Transparency Version 2.0" was published.[19]Version 2.0 includes major changes to the required structure of the log certificate, as well as support forEd25519as a signature algorithm of SCTs and support for including certificate inclusion proofs with the SCT. However, it has not seen industry adoption and is considereddead on arrival.[38]
In February 2022, Google published an update to their CT policy,[39]which removes the requirement for certificates to include a SCT from their own CT log service, matching all the requirements for certificates to those previously published by Apple.[40]
In February 2025,Mozilla Firefoxdesktop version 135 began requiring Certificate Transparency for all certificates issued by a certificate authority in Mozilla's Root CA Program.[41][42]
In Certificate Transparency Version 2.0, a log must use one of the algorithms in the IANA registry "Signature Algorithms".[19]: 10.2.2[43]
|
https://en.wikipedia.org/wiki/Certificate_Transparency
|
Datagram Transport Layer Security(DTLS) is acommunications protocolprovidingsecuritytodatagram-based applications by allowing them to communicate in a way designed[1][2][3]to preventeavesdropping,tampering, ormessage forgery. The DTLS protocol is based on thestream-orientedTransport Layer Security(TLS) protocol and is intended to provide similar security guarantees. The DTLS protocol datagram preserves the semantics of the underlying transport—the application does not suffer from the delays associated with stream protocols, but because it usesUser Datagram Protocol(UDP) orStream Control Transmission Protocol(SCTP), the application has to deal withpacket reordering, loss of datagram and data larger than the size of a datagramnetwork packet. Because DTLS uses UDP or SCTP rather than TCP it avoids theTCP meltdown problem[4][5]when being used to create a VPN tunnel.
The following documents define DTLS:
DTLS 1.0 is based on TLS 1.1, DTLS 1.2 is based on TLS 1.2, and DTLS 1.3 is based on TLS 1.3. There is no DTLS 1.1 because this version-number was skipped in order to harmonize version numbers with TLS.[2]Like previous DTLS versions, DTLS 1.3 is intended to provide "equivalent security guarantees [to TLS 1.3] with the exception of order protection/non-replayability".[11]
In February 2013 two researchers from Royal Holloway, University of London discovered a timing attack[46]which allowed them to recover (parts of the) plaintext from a DTLS connection using the OpenSSL or GnuTLS implementation of DTLS whenCipher Block Chainingmode encryption was used.
|
https://en.wikipedia.org/wiki/Datagram_Transport_Layer_Security
|
Delegated credentialis a short-livedTLScertificateused to improve security by faster recovery fromprivate keyleakage, without increasing thelatencyof theTLS handshake. It is currently anIETFInternet Draft,[1]and has been in use byCloudflare[2]andFacebook,[3]with browser support byFirefox.[4]
Modern websites and other services usecontent delivery networks(CDNs), which are servers potentially distributed all over the world, in order to respond to a user's request as fast as possible, alongside other services that CDNs provide such asDDoS mitigation. However, in order to establish asecureconnection, the server is required to prove possession of a private key associated with a certificate, which serves as achain of trustlinking the public key and a trusted party. The trusted party is normally acertificate authority(CA).
CAs issue thesedigital certificateswith an expiration time, usually a few months up to a year. It is the server's responsibility to renew the certificate close to its expiration date. Knowledge of a private key associated to a valid certificate is devastating for the site's security, as it allowsMan-in-the-middle attacks, in which a malicious entity can impersonate to a user as a legitimate server. Therefore, these private keys should be kept secure, preferably not distributed over every server in the CDN. Specifically, if a private key is compromised, the corresponding certificate should optimally berevoked, such that browsers will no longer support this certificate. Certificate revocation has two main drawbacks. Firstly, current revocation methods do not work well across all browsers, and put the users at risk; and secondly, upon revocation, the server needs to quickly fetch a new valid certificate from the CA and deploy it across allmirrors.
A delegated credential is a short-lived key (from a few hours to a few days) that the certificate's owner delegates to the server for use in TLS. It is in fact asignature: the certificate's owner uses the certificate's private key to sign a delegated public key, and an expiration time.
Given this delegated credential, a browser can (if it supports it) verify the server's authenticity by verifying the delegated certificate and then verify the certificate itself.
This approach has many advantage over current solutions:
|
https://en.wikipedia.org/wiki/Delegated_credential
|
HTTP Strict Transport Security(HSTS) is a policy mechanism that helps to protect websites againstman-in-the-middle attackssuch asprotocol downgrade attacks[1]andcookie hijacking. It allowsweb serversto declare that web browsers (or other complyinguser agents) should automatically interact with it using onlyHTTPSconnections, which provideTransport Layer Security(TLS/SSL), unlike the insecureHTTPused alone. HSTS is anIETFstandardstrack protocol and is specified inRFC6797.
The HSTS Policy is communicated by the server to the user agent via an HTTP responseheaderfield namedStrict-Transport-Security. HSTS Policy specifies a period of time during which theuser agentshould only access the server in a secure fashion.[2]: §5.2Websites using HSTS often do not accept clear text HTTP, either by rejecting connections over HTTP or systematically redirecting users to HTTPS (though this is not required by the specification). The consequence of this is that a user-agent not capable of doing TLS will not be able to connect to the site.
The protection only applies after a user has visited the site at least once, relying on the principle of "trust on first use". The way this protection works is that when a user entering or selecting an HTTP (not HTTPS) URL to the site, the client, such as a Web browser, will automatically upgrade to HTTPS without making an HTTP request, thereby preventing any HTTP man-in-the-middle attack from occurring.
The HSTS specification was published as RFC 6797 on 19 November 2012 after being approved on 2 October 2012 by theIESGfor publication as aProposed StandardRFC.[3]The authors originally submitted it as anInternet Drafton 17 June 2010. With the conversion to an Internet Draft, the specification name was altered from "Strict Transport Security" (STS) to "HTTP Strict Transport Security", because the specification applies only toHTTP.[4]The HTTP response header field defined in the HSTS specification however remains named "Strict-Transport-Security".
The last so-called "community version" of the then-named "STS" specification was published on 18 December 2009, with revisions based on community feedback.[5]
The original draft specification by Jeff Hodges fromPayPal, Collin Jackson, and Adam Barth was published on 18 September 2009.[6]
The HSTS specification is based on original work by Jackson and Barth as described in their paper "ForceHTTPS: Protecting High-Security Web Sites from Network Attacks".[7]
Additionally, HSTS is the realization of one facet of an overall vision for improving web security, put forward by Jeff Hodges and Andy Steingruebl in their 2010 paperThe Need for Coherent Web Security Policy Framework(s).[8]
A server implements an HSTS policy by supplying a header over an HTTPS connection (HSTS headers over HTTP are ignored).[1]For example, a server could send a header such that future requests to the domain for the next year (max-age is specified in seconds; 31,536,000 is equal to one non-leap year) use only HTTPS:Strict-Transport-Security: max-age=31536000.
When a web application issues HSTS Policy to user agents, conformant user agents behave as follows:[2]: §5
The HSTS Policy helps protect web application users against some passive (eavesdropping) and active networkattacks.[2]: §2.4Aman-in-the-middle attackerhas a greatly reduced ability to intercept requests and responses between a user and a web application server while the user's browser has HSTS Policy in effect for that web application.
The most important security vulnerability that HSTS can fix is SSL-strippingman-in-the-middle attacks, first publicly introduced byMoxie Marlinspikein his 2009 BlackHat Federal talk "New Tricks For Defeating SSL In Practice".[9][10]The SSL (andTLS) stripping attack works by transparently converting a secureHTTPSconnection into a plain HTTP connection. The user can see that the connection is insecure, but crucially there is no way of knowing whether the connectionshouldbe secure. At the time of Marlinspike's talk, many websites did not use TLS/SSL, therefore there was no way of knowing (without prior knowledge) whether the use of plain HTTP was due to an attack, or simply because the website had not implemented TLS/SSL. Additionally, no warnings are presented to the user during the downgrade process, making the attack fairly subtle to all but the most vigilant. Marlinspike's sslstrip tool fully automates the attack.[citation needed]
HSTS addresses this problem[2]: §2.4by informing the browser that connections to the site should always use TLS/SSL. The HSTS header can be stripped by the attacker if this is the user's first visit.Google Chrome,Mozilla Firefox,Internet Explorer, andMicrosoft Edgeattempt to limit this problem by including a "pre-loaded" list of HSTS sites.[11][12][13]Unfortunately this solution cannot scale to include all websites on the internet. Seelimitations, below.
HSTS can also help to prevent having one's cookie-based website login credentials stolen by widely available tools such asFiresheep.[14]
Because HSTS is time limited, it is sensitive to attacks involving shifting the victim's computer time e.g. using falseNTPpackets.[15]
The initial request remains unprotected from active attacks if it uses an insecure protocol such as plain HTTP or if theURIfor the initial request was obtained over aninsecure channel.[2]: §14.6The same applies to the first request after the activity period specified in the advertised HSTS Policymax-age(sites should set a period of several days or months depending on user activity and behavior).
Google Chrome,Mozilla Firefox, andInternet Explorer/Microsoft Edgeaddress this limitation by implementing a "HSTS preloaded list", which is a list that contains known sites supporting HSTS.[16][11][12][13]This list is distributed with the browser so that it uses HTTPS for the initial request to the listed sites as well. As previously mentioned, these pre-loaded lists cannot scale to cover the entire Web. A potential solution might be achieved by usingDNSrecords to declare HSTS Policy, and accessing them securely viaDNSSEC, optionally with certificate fingerprints to ensure validity (which requires running a validating resolver to avoidlast mileissues).[17]
Junade Ali has noted that HSTS is ineffective against the use of phony domains; by using DNS-based attacks, it is possible for a man-in-the-middle interceptor to serve traffic from an artificial domain which is not on the HSTS Preload list,[18]this can be made possible by DNS Spoofing Attacks,[19]or simply a domain name that misleadingly resembles the real domain name such aswww.example.orginstead ofwww.example.com.
Even with an HSTS preloaded list, HSTS cannot prevent advanced attacks against TLS itself, such as theBEASTorCRIMEattacks introduced by Juliano Rizzo and Thai Duong. Attacks against TLS itself areorthogonalto HSTS policy enforcement. Neither can it protect against attacks on the server - if someone compromises it, it will happily serve any content over TLS.
HSTS can be used to near-indelibly tag visiting browsers with recoverable identifying data (supercookies) which can persist in and out of browser "incognito" privacy modes. By creating a web page that makes multiple HTTP requests to selected domains, for example, if twenty browser requests to twenty different domains are used, theoretically over one million visitors can be distinguished (220) due to the resulting requests arriving via HTTP vs. HTTPS; the latter being the previously recorded binary "bits" established earlier via HSTS headers.[20]
Depending on the actual deployment there are certain threats (e.g. cookie injection attacks) that can be avoided by following best practices.
|
https://en.wikipedia.org/wiki/HTTP_Strict_Transport_Security
|
Akey ringis a file which contains multiple public keys ofcertificate authority(CA).
A key ring is a file which is necessary forSecure Sockets Layer(SSL) connection over the web. It is securely stored on theserverwhichhosts the website. It contains thepublic/private key pairfor the particular website. It also contains the public/private key pairs from various certificate authorities and the trusted root certificate for the various certification authorities.
An entity or website administrator has to send acertificate signing request(CSR) to the CA. The CA then returns a signedcertificateto the entity. This certificate received from the CA has to be stored in the key ring.
|
https://en.wikipedia.org/wiki/Key_ring_file
|
Private Communications Technology(PCT) 1.0 was a protocol developed byMicrosoftin the mid-1990s. PCT was designed to address security flaws in version 2.0 ofNetscape'sSecure Sockets Layerprotocol and to force Netscape to hand control of the then-proprietary SSL protocol to an open standards body.[citation needed]
PCT has since been superseded by SSLv3 andTransport Layer Security. For a while it was still supported by Internet Explorer, but PCT 1.0 has been disabled by default sinceIE 5and the option was removed in IE6.[1]It is still found inIISand in the Windows operating system libraries, although inWindows Server 2003it is disabled by default. It is used by old versions ofMSMQas the only choice.
Due to its near disuse, it is arguably a security risk, as it has received less attention in testing than commonly used protocols, and there is little incentive for Microsoft to expend effort on maintaining its implementation of it.
|
https://en.wikipedia.org/wiki/Private_Communications_Technology
|
QUIC(/kwɪk/) is a general-purposetransport layernetwork protocolinitially designed byJim RoskindatGoogle.[1][2][3]It was first implemented and deployed in 2012[4]and was publicly announced in 2013 as experimentation broadened. It was also described at anIETFmeeting.[5][6][7][8]TheChrome web browser,[9]Microsoft Edge,[10][11]Firefox,[12]andSafariall support it.[13]In Chrome, QUIC is used by more than half of all connections to Google's servers.[9]
QUIC improves performance of connection-orientedweb applicationsthat before QUIC usedTransmission Control Protocol(TCP).[2][9]It does this by establishing a number ofmultiplexedconnections between two endpoints usingUser Datagram Protocol(UDP), and is designed to obsolete TCP at the transport layer for many applications. Although its name was initially proposed as an acronym forQuick UDP Internet Connections, in IETF's use of the word, QUIC is not an acronym; it is simply the name of the protocol.[3][8][1]
QUIC works hand-in-hand withHTTP/3's multiplexed connections, allowing multiple streams of data to reach all the endpoints independently, and hence independent ofpacket lossesinvolving other streams. In contrast, HTTP/2 carried over TCP can sufferhead-of-line-blockingdelays if multiple streams are multiplexed on a TCP connection and any of the TCP packets on that connection are delayed or lost.
QUIC's secondary goals include reduced connection and transportlatency, andbandwidthestimation in each direction to avoidcongestion. It also movescongestion controlalgorithms into theuser spaceat both endpoints, rather than thekernel space, which it is claimed[14]will allow these algorithms to improve more rapidly. Additionally, the protocol can be extended withforward error correction(FEC) to further improve performance when errors are expected. It is designed with the intention of avoidingprotocol ossification.
In June 2015, anInternet Draftof a specification for QUIC was submitted to theIETFfor standardization.[15][16]A QUIC working group was established in 2016.[17]In October 2018, the IETF's HTTP and QUIC Working Groups jointly decided to call the HTTP mapping over QUIC "HTTP/3" in advance of making it a worldwide standard.[18]In May 2021, the IETF standardized QUIC inRFC9000, supported byRFC8999,RFC9001andRFC9002.[19]DNS-over-QUICis another application.
Transmission Control Protocol, or TCP, aims to provide an interface for sending streams of data between two endpoints. Data is sent to the TCP system, which ensures it reaches the other end in the exact same form; if any discrepancies occur, the connection will signal an error condition.[20]
To do this, TCP breaks up the data intonetwork packetsand adds small amounts of data to each packet. This additional data includes a sequence number that is used to detect packets that are lost or arrive out of order, and achecksumthat allows the errors within packet data to be detected. When either problem occurs, TCP usesautomatic repeat request(ARQ) to ask the sender to re-send the lost or damaged packet.[20]
In most implementations, TCP will see any error on a connection as a blocking operation, stopping further transfers until the error is resolved or the connection is considered failed. If a single connection is being used to send multiple streams of data, as is the case in theHTTP/2protocol, all of these streams are blocked although only one of them might have a problem. For instance, if a single error occurs while downloading a GIF image used for afavicon, the entire rest of the page will wait while that problem is resolved.[20]This phenomenon is known ashead-of-line blocking.
As the TCP system is designed to look like a "data pipe", or stream, it deliberately has little information regarding the data it transmits. If that data has additional requirements, likeencryptionusingTLS, this must be set up by systems running on top of TCP, using TCP to communicate with similar software on the other end of the connection. Each of these sorts of setup tasks requires its ownhandshakeprocess. This often requires several round-trips of requests and responses until the connection is established. Due to the inherentlatencyof long-distance communications, this can add significant delay to the overall transmission.[20]
TCP has suffered fromprotocol ossification,[21]due to itswire imagebeing incleartextand hence visible to and malleable bymiddleboxes.[22]One measurement found that a third of paths across the Internet encounter at least one intermediary that modifies TCP metadata, and 6.5% of paths encounter harmful ossifying effects from intermediaries.[23]Extensions to TCP have been affected: the design ofMultipath TCP(MPTCP) was constrained by middlebox behaviour,[24][25]and the deployment ofTCP Fast Openhas been likewise hindered.[26][21]
In the context of supportingencryptedHTTPtraffic, QUIC serves a role similar to that of TCP, but with reducedlatencyduring connection setup and more efficient loss recovery when multiple HTTP streams are multiplexed over a single connection. It does this primarily through two changes that rely on the understanding of the behaviour of HTTP traffic.[20]
The first change is to greatly reduce overhead during connection setup. As most HTTP connections will demandTLS, QUIC makes the exchange of setup keys and listing of supported protocols part of the initialhandshake process. When a client opens a connection, the response packet includes the data needed for future packets to use encryption. This eliminates the need to set up an unencryptedpipeand then negotiate the security protocol as separate steps. Other protocols can be serviced in the same way, combining multiple steps into a single request–response pair. This data can then be used both for following requests in the initial setup and future requests that would otherwise be negotiated as separate connections.[20]
The second change is to useUDPrather than TCP as its basis, which does not includelossrecovery. Instead, each QUIC stream is separately flow-controlled, and lost data is retransmitted at the level of QUIC, not UDP. This means that if an error occurs in one stream, like the favicon example above, theprotocol stackcan continue servicing other streams independently. This can be very useful in improving performance on error-prone links, as in most cases considerable additional data may be received before TCP notices a packet is missing or broken, and all of this data is blocked or even flushed while the error is corrected. In QUIC, this data is free to be processed while the single multiplexed stream is repaired.[27]
QUIC includes a number of other changes that improve overall latency and throughput. For instance, the packets are encrypted individually, so that they do not result in the encrypted data waiting for partial packets. This is not generally possible under TCP, where the encryption records are in abytestreamand the protocol stack is unaware of higher-layer boundaries within this stream. These can be negotiated by the layers running on top, but QUIC aims to do all of this in a single handshake process.[8]
Another goal of the QUIC system was to improve performance during network-switching events, like what happens when a user of a mobile device moves from a localWi-Fi hotspotto amobile network. When this occurs on TCP, a lengthy process starts where every existing connection times out one-by-one and is then re-established on demand. To solve this problem, QUIC includes a connection identifier to uniquely identify the connection to the server regardless of source. This allows the connection to be re-established simply by sending a packet, which always contains this ID, as the original connection ID will still be valid even if the user'sIP addresschanges.[28]
QUIC can be implemented in the application space, as opposed to being in theoperating system kernel. This generally invokes additional overhead due tocontext switchesas data is moved between applications. However, in the case of QUIC, the protocol stack is intended to be used by a single application, with each application using QUIC having its own connections hosted on UDP. Ultimately the difference could be very small because much of the overall HTTP/2 stack is already in the applications (or their libraries, more commonly). Placing the remaining parts in those libraries, essentially the error correction, has little effect on the HTTP/2 stack's size or overall complexity.[8]
This organization allows future changes to be made more easily as it does not require changes to thekernelfor updates. One of QUIC's longer-term goals is to add new systems forforward error correction(FEC) and improved congestion control.[28]
One concern about the move from TCP to UDP is that TCP is widely adopted and many of the "middleboxes" in the Internet infrastructure are tuned for TCP and rate-limit or even block UDP. Google carried out a number of exploratory experiments to characterize this and found that only a small number of connections were blocked in this manner.[3]This led to the use of a system for rapid fallback to TCP;Chromium's network stack starts both a QUIC and a conventional TCP connection at the same time, which allows it to fall back with negligible latency.[29]
QUIC has been specifically designed to be deployable and evolvable and to have anti-ossification properties;[30]it is the firstIETFtransport protocol to deliberately minimise its wire image for these ends.[31]Beyond encrypted headers, it is 'greased'[32]and it has protocol invariants explicitly specified.[33]
The security layer of QUIC is based on TLS 1.2 or TLS 1.3.[34]Earlier insecure protocol like TLS 1.0 is not allowed in QUIC stack.
The protocol that was created by Google and taken to the IETF under the name QUIC (already in 2012 around QUIC version 20) is now quite different from the QUIC that has continued to evolve and be refined within the IETF. The original Google QUIC (gQUIC) was designed to be a general purpose web protocol, though it was initially deployed as a protocol to support HTTP(S) in Chromium. The current evolution of the IETF QUIC (iQUIC) protocol is a general purpose transport protocol. Chromium developers continued to track the evolution of IETF QUIC's standardization efforts to adopt and fully comply with the most recent internet standards for QUIC in Chromium.
QUIC was developed with HTTP in mind, and HTTP/3 was its first application.[35][36]DNS-over-QUICis an application of QUIC to name resolution, providing security for data transferred between resolvers similar toDNS-over-TLS.[37]The IETF is developing applications of QUIC for securenetwork tunnelling[36]andstreaming mediadelivery.[38]XMPPhas experimentally been adapted to use QUIC.[39]Another application isSMBover QUIC, which, according to Microsoft, can offer an "SMB VPN" without affecting the user experience.[40]SMB clients use TCP by default and will attempt QUIC if the TCP attempt fails or if intentionally requiring QUIC.
The QUIC code was experimentally developed inGoogle Chromestarting in 2012,[4]and was announced as part of Chromium version 29 (released on August 20, 2013).[18]It is currently enabled by default in Chromium and Chrome.[41]
Support inFirefoxarrived in May 2021.[42][12]
Appleadded experimental support in theWebKit enginethrough the Safari Technology Preview 104 in April 2020.[43]Official support was added inSafari14, included inmacOS Big SurandiOS 14,[44]but the feature needed to be turned on manually.[45]It was later enabled by default in Safari 16.[13]
The cronet library for QUIC and other protocols is available to Android applications as a module loadable viaGoogle Play Services.[46]
cURL7.66, released 11 September 2019, supports HTTP/3 (and thus QUIC).[47][48]
In October 2020, Facebook announced[49]that it has successfully migrated its apps, includingInstagram, and server infrastructure to QUIC, with already 75% of its Internet traffic using QUIC. All mobile apps from Google support QUIC, includingYouTubeandGmail.[50][51]Uber's mobile app also uses QUIC.[51]
As of 2017[update], there are several actively maintained implementations. Google servers support QUIC and Google has published a prototype server.[52]Akamai Technologieshas been supporting QUIC since July 2016.[53][54]AGoimplementation called quic-go[55]is also available, and powers experimental QUIC support in theCaddy server.[56]On July 11, 2017, LiteSpeed Technologies officially began supporting QUIC in their load balancer (WebADC)[57]andLiteSpeed Web Serverproducts.[58]As of October 2019[update], 88.6% of QUIC websites used LiteSpeed and 10.8% usedNginx.[59]Although at first only Google servers supported HTTP-over-QUIC connections,Facebookalso launched the technology in 2018,[18]andCloudflarehas been offering QUIC support on a beta basis since 2018.[60]TheHAProxyload balancer added experimental support for QUIC in March 2022[61]and declared it production-ready in March 2023.[62]As of April 2023[update], 8.9% of all websites use QUIC,[63]up from 5% in March 2021.Microsoft Windows Server 2022supports both HTTP/3[64]and SMB over QUIC[65][10]protocols viaMsQuic. The Application Delivery Controller ofCitrix(Citrix ADC, NetScaler) can function as a QUIC proxy since version 13.[66][67]
In addition, there are several stale community projects: libquic[68]was created by extracting the Chromium implementation of QUIC and modifying it to minimize dependency requirements, and goquic[69]providesGobindings of libquic. Finally, quic-reverse-proxy[70]is aDocker imagethat acts as areverse proxyserver, translating QUIC requests into plain HTTP that can be understood by the origin server.
.NET 5introduces experimental support for QUIC using theMsQuiclibrary.[71]
|
https://en.wikipedia.org/wiki/QUIC
|
Server-Gated Cryptography(SGC), also known asInternational Step-UpbyNetscape, is a defunct mechanism that was used to step up from 40-bit or 56-bit to 128-bit cipher suites withSSL. It was created in response toUnited States federal legislation on the export of strong cryptographyin the 1990s.[1]The legislation had limitedencryptionto weakalgorithmsand shorter key lengths in software exported outside of theUnited States of America. When the legislation added an exception for financial transactions, SGC was created as an extension to SSL with the certificates being restricted to financial organisations. In 1999, this list was expanded to include online merchants, healthcare organizations, and insurance companies.[2]This legislation changed in January 2000, resulting in vendors no longer shipping export-grade browsers and SGC certificates becoming available without restriction.
Internet Explorersupported SGC starting with patched versions ofInternet Explorer 3. SGC becameobsoletewhenInternet Explorer 5.01SP1 andInternet Explorer 5.5started supporting strong encryption without the need for a separate high encryption pack (except onWindows 2000, which needs its own high encryption pack that was included in Service Pack 2 and later).[3]"Export-grade" browsers are unusable on the modern Web due to many servers disabling export cipher suites. Additionally, these browsers are incapable of using SHA-2 family signature hash algorithms like SHA-256. Certification authorities are trying to phase out the new issuance of certificates with the older SHA-1 signature hash algorithm.
The continuing use of SGC facilitates the use of obsolete, insecure Web browsers with HTTPS.[4][5]However, while certificates that use the SHA-1 signature hash algorithm remain available, some certificate authorities continue to issue SGC certificates (often charging a premium for them) although they are obsolete. The reason certificate authorities can charge a premium for SGC certificates is that browsers only allowed a limited number of roots to support SGC.
When an SSL handshake takes place, the software (e.g. aweb browser) would list theciphersthat it supports. Although the weaker exported browsers would only include weaker ciphers in its initial SSL handshake, the browser also contained stronger cryptography algorithms. There are two protocols involved to activate them.Netscape Communicator 4used International Step-Up, which used the now obsolete insecure renegotiation to change to a stronger cipher suite. Microsoft used SGC, which sends a new Client Hello message listing the stronger cipher suites on the same connection after the certificate is determined to be SGC capable, and also supported Netscape Step-Up for compatibility (though this support in the NT 4.0 SP6 and IE 5.01 version had a bug where changing MAC algorithms during Step-Up did not work properly).[citation needed]
|
https://en.wikipedia.org/wiki/Server-Gated_Cryptography
|
Incomputer networking,tcpcryptis atransport layercommunicationencryptionprotocol.[1][2]Unlike prior protocols likeTLS(SSL), tcpcrypt is implemented as aTCPextension. It was designed by a team of six security and networking experts: Andrea Bittau, Mike Hamburg,Mark Handley, David Mazières,Dan Bonehand Quinn Slack.[3]Tcpcrypt has been published as an Internet Draft.[4]Experimentaluser-spaceimplementations are available for Linux, Mac OS X, FreeBSD and Windows. There is also aLinux kernelimplementation.
The TCPINC (TCP Increased Security)working groupwas formed in June 2014 byIETFto work on standardizing security extensions in the TCP protocol.[5]In May 2019 the working group releasedRFC8547andRFC8548as an experimental standard for Tcpcrypt.
Tcpcrypt providesopportunistic encryption— if either side does not support this extension, then the protocol falls back to regular unencrypted TCP. Tcpcrypt also provides encryption to any application using TCP, even ones that do not know about encryption. This enables incremental and seamless deployment.[6]
Unlike TLS, tcpcrypt itself does not do anyauthentication, but passes a unique "session ID" down to the application; the application can then use this token for further authentication. This means that any authentication scheme can be used, including passwords orcertificates. It also does a larger part of the public-key connection initiation on the client side, to reduce load on servers and mitigate DoS attacks.[6]
The first draft of the protocol specification was published in July 2010, withreference implementationsfollowing in August. However, after initial meetings in IETF, proponents of the protocol failed to gain traction for standardization and the project went dormant in 2011.[7]
In 2013 and 2014, followingEdward Snowden'sGlobal surveillance disclosuresabout theNSAand agencies of other governments, IETF took a strong stance for protecting Internet users against surveillance.[8][9]This aligns with tcpcrypt's goals of ubiquitous transparent encryption, which revived interest in standardization of the protocol. An official IETFmailing listwas created for tcpcrypt in March 2014,[10]followed by the formation of the TCPINC (TCP Increased Security)working groupin June[5]and a new version of the draft specification.
Tcpcrypt enforces TCP timestamps and adds its own TCP options to each data packet, amounting to 36 bytes per packet compared to plain TCP. With a mean observed packet size for TCP packets of 471 bytes,[11]this can lead to an overhead of 8% of useful bandwidth. This 36 bytes overhead may not be an issue for internet connections faster than 64kbs but it can be an issue for dial-up internet users.
Compared toTLS/SSL, tcpcrypt is designed to have a lower performance impact. In part this is because tcpcrypt does not have built-in authentication, which can be implemented by the application itself. Cryptography primitives are used in such a way to reduce load on theserverside, because a single server usually has to provide services for far more clients than the reverse.[6]
The current user space implementations are considered experimental and are reportedly unstable on some systems. It also does not supportIPv6yet, which is currently only supported by the Linux kernel version. It is expected that once tcpcrypt becomes a standard, operating systems will come with tcpcrypt support built-in, making the user space solution unnecessary.[citation needed]
|
https://en.wikipedia.org/wiki/Tcpcrypt
|
TLS acceleration(formerly known asSSL acceleration) is a method of offloading processor-intensivepublic-key encryptionforTransport Layer Security(TLS) and its predecessor Secure Sockets Layer (SSL)[1]to a hardware accelerator.
Typically this means having a separate card that plugs into aPCI slotin a computer that contains one or morecoprocessorsable to handle much of the SSL processing.
TLS accelerators may use off-the-shelfCPUs, but most use customASICandRISCchips to do most of the difficult computational work.
The most computationally expensive part of a TLS session is the TLS handshake, where the TLS server (usually a webserver) and the TLS client (usually a web browser) agree on a number of parameters that establish the security of the connection. During the TLS handshake the server and the client establish session keys (symmetric keys, used for the duration of a given session), but the encryption and signature of the TLS handshake messages itself is done using asymmetric keys, which requires more computational power than the symmetric cryptography used for the encryption/decryption of the session data.
Typically a hardware TLS accelerator will offload processing of the TLS handshake while leaving it to the server software to process the less intensesymmetric cryptographyof the actual TLSdata exchange, but some accelerators handle all TLS operations and terminate the TLS connection, thus leaving the server seeing only decrypted connections. Sometimes data centers employ dedicated servers for TLS acceleration in areverse proxyconfiguration.
Modernx86CPUs supportAdvanced Encryption Standard(AES) encoding and decoding in hardware, using theAES instruction setproposed by Intel in March 2008.
Allwinner Technologyprovides a hardware cryptographic accelerator in its A10, A20, A30 and A80ARMsystem-on-chipseries, and all ARM CPUs have acceleration in the later ARMv8 architecture. The accelerator provides theRSApublic-key algorithm, several widely usedsymmetric-key algorithms,cryptographic hash functions, and a cryptographically securepseudo-random number generator.[2]
|
https://en.wikipedia.org/wiki/TLS_acceleration
|
Access codemay refer to:
|
https://en.wikipedia.org/wiki/Access_code_(disambiguation)
|
Acombination lockis a type oflocking devicein which asequenceof symbols, usually numbers, is used to open the lock. The sequence may be entered using a single rotating dial which interacts with several discs orcams, by using a set of several rotating discs with inscribed symbols which directly interact with the locking mechanism, or through an electronic or mechanical keypad. Types range from inexpensive three-digitluggage locksto high-securitysafes. Unlike ordinarypadlocks, combination locks do not use keys.
The earliest known combination lock was excavated in aRomanperiod tomb on theKerameikos,Athens. Attached to a small box, it featured several dials instead of keyholes.[1]In 1206, theMuslimengineerIsmail al-Jazaridocumented a combination lock in his bookal-Ilm Wal-Amal al-Nafi Fi Sina'at al-Hiyal(The Book of Knowledge of Ingenious Mechanical Devices).[2]Muhammad al-Asturlabi (ca. 1200) also made combination locks.[3]
Gerolamo Cardanolater described a combination lock in the 16th century.
U.S. Patents regarding combination padlocks by J.B. Gray in 1841[4]and by J.E. Treat in 1869[5]describe themselves as improvements, suggesting that such mechanisms were already in use.
Joseph Loch was said to have invented the modern combination lock forTiffany's Jewelersin New York City, and from the 1870s to the early 1900s, made many more improvements in the designs and functions of such locks.[6]However, his patent claim states: "I do not claim as my invention a tumbler composed of two disks, one working within the other, such not being my invention.", but there is no reference to prior art of this type of lock.
The first commercially viable single-dial combination lock was patented on 1 February 1910 by John Junkunc, owner of American Lock Company.[7]
One of the simplest types of combination lock, often seen in low-securitybicyclelocks,briefcases, andsuitcases, uses several rotating discs with notches cut into them. The lock is secured by a pin with several teeth on it which hook into the rotating discs. When the notches in the discs align with the teeth on the pin, the lock can be opened.
Therotary combination locksfound onpadlocks, lockers, orsafesmay use a single dial which interacts with several parallel discs orcams. Customarily, a lock of this type is opened by rotating the dial clockwise to the first numeral, counterclockwise to the second, and so on in an alternating fashion until the last numeral is reached. The cams typically have an indentation or notch, and when the correctpermutationis entered, the notches align, allowing the latch to fit into them and open the lock.
The C. L. Gougler Keyless Locks Company manufactured locks for which the combination was a set number of audible clicks to the left and right, allowing them to be unlocked in darkness or by the vision-impaired.[citation needed]
In 1978 a combination lock which could be set by the user to a sequence of his own choosing was invented by Andrew Elliot Rae.[8]At this time the electronic keypad was invented and he was unable to get any manufacturers to back his mechanical lock for lockers, luggage, or brief-cases. The silicon chip locks never became popular due to the need for battery power to maintain their integrity. The patent expired and the original mechanical invention was instantly manufactured and sold worldwide mainly for luggage, lockers, and hotel safes. It is now a standard part of the luggage used by travellers.
Many doors use combination locks which require the user to enter a numeric sequence on akeypadto gain entry. These special locks usually require the additional use of electronic circuitry, although purely mechanical keypad locks have been available since 1936.[9]The chief advantage of this system is that multiple persons can be granted access without having to supply an expensive physical key to each person. Also, in case the key is compromised, "changing" the lock requires only configuring a new key code and informing the users, which will generally be cheaper and quicker than the same process for traditional key locks.
Electronic combination locks, while generally safe from the attacks on their mechanical counterparts, suffer from their own set of flaws. If the arrangement of numbers is fixed, it is easy to determine the lock sequence by viewing several successful accesses. Similarly, the numbers in the combination (but not the actual sequence) may be determined by which keys show signs of recent use. More advanced electronic locks may scramble the numbers' locations randomly to prevent these attacks.
There is a variation of the traditional dial based combination lock wherein the "secret" is encoded in an electronic microcontroller. These are popular for safe andbank vaultdoors where tradition tends towards dial locks rather than keys. They allow many valid combinations, one per authorized user, so changing one person's access has no effect on other users. These locks often have auditing features, recording which combination is used at what time for every opening. Power for the lock may be provided by a battery or by a tiny generator set in operation by spinning the dial.[10][11]
|
https://en.wikipedia.org/wiki/Combination_lock
|
Dicewareis a method for creatingpassphrases,passwords, and other cryptographic variables using ordinarydiceas ahardware random number generator. For each word in the passphrase, five rolls of a six-sided die are required. The numbers from 1 to 6 that come up in the rolls are assembled as a five-digit number, e.g.43146. That number is then used to look up a word in a cryptographic word list. In the original Diceware list43146corresponds tomunch. By generating several words in sequence, a lengthy passphrase can thus be constructed randomly.
A Diceware word list is any list of65=7776unique words, preferably ones the user will find easy to spell and to remember. The contents of the word list do not have to be protected or concealed in any way, as the security of a Diceware passphrase is in the number of words selected, and the number of words each selected word could be taken from. Lists have been compiled for several languages, includingBasque,Bulgarian,Catalan,Chinese,Czech,Danish,Dutch,English,Esperanto,Estonian,Finnish,French,German,Greek,Hebrew,Hungarian,Italian,Japanese,Latin,Māori,Norwegian,Polish,Portuguese,Romanian,Russian,Slovak,Slovenian,Spanish,SwedishandTurkish.[1]
The level of unpredictability of a Diceware passphrase can be easily calculated: each word adds 12.9bitsofentropyto the passphrase (that is,log2(65){\displaystyle \log _{2}(6^{5})}bits). Originally, in 1995, Diceware creator Arnold Reinhold considered five words (64.6 bits) the minimal length needed by average users. However, in 2014 Reinhold started recommending that at least six words (77.5 bits) be used.[2]
This level of unpredictability assumes that potential attackers know three things: that Diceware has been used to generate the passphrase, the particular word list used, and exactly how many words make up the passphrase. If the attacker has less information, the entropy can be greater than12.9 bits/word.[3]
The above calculations of the Diceware algorithm's entropy assume that, as recommended by Diceware's author, each word is separated by a space. If, instead, words are simply concatenated, the calculated entropy is slightly reduced due to redundancy; for example, the three-word Diceware phrases "in put clammy" and "input clam my" become identical if the spaces are removed.
TheElectronic Frontier Foundationpublished three alternative English diceware word lists in 2016, further emphasizing ease-of-memorization with a bias against obscure, abstract or otherwise problematic words; one tradeoff is that typical EFF-style passphrases require typing a larger number of characters.[4][5]
The original diceware word list consists of a line for each of the7,776possible five-die combinations. One excerpt:[6]
Diceware wordlist passphrase examples:[4]
EFF wordlist passphrase examples:[4]
TheXKCD #936 stripshows a password similar to a Diceware generated one, even if the used wordlist is shorter than the regular7,776-words list used for Diceware.[7]
|
https://en.wikipedia.org/wiki/Diceware
|
Anelectronic lock(orelectric lock) is alocking devicewhich operates by means of electric current. Electric locks are sometimes stand-alone with an electronic control assembly mounted directly to the lock. Electric locks may be connected to anaccess controlsystem, the advantages of which include: key control, where keys can be added and removed without re-keying the lock cylinder; fine access control, where time and place are factors; and transaction logging, where activity is recorded. Electronic locks can also be remotely monitored and controlled, both to lock and to unlock.
Electric locks use magnets,solenoids, or motors to actuate the lock by either supplying or removing power. Operating the lock can be as simple as using a switch, for example an apartment intercom door release, or as complex as abiometricbasedaccess controlsystem.
There are two basic types of locks: "preventing mechanism" or operation mechanism.[further explanation needed]
The most basic type of electronic lock is amagnetic lock(informally called a "mag lock"). A large electro-magnet is mounted on the door frame and a corresponding armature is mounted on the door. When the magnet is powered and the door is closed, the armature is held fast to the magnet. Mag locks are simple to install and are very attack-resistant. One drawback is that improperly installed or maintained mag locks can fall on people,[dubious–discuss]and also that one must unlock the mag lock to both enter and to leave. This has causedfire marshalsto imposestrict ruleson the use of mag locks and access control practice in general. Additionally,NFPA101 (Standard for Life Safety and Security), as well as theADA(Americans with Disability Act) require "no prior knowledge" and "one simple movement" to allow "free egress". This means that in an emergency, a person must be able to move to a door and immediately exit with one motion (requiring no push buttons, having another person unlock the door, reading a sign, or "special knowledge").
Other problems include alag time(delay), because the collapsingmagnetic fieldholding the door shut does not release instantaneously. This lag time can cause a user to collide with the still-locked door. Finally, mag locks fail unlocked, in other words, if electrical power is removed they unlock. This could be a problem where security is a primary concern. Additionally, power outages could affect mag locks installed onfire listed doors, which are required to remain latched at all times except when personnel are passing through. Most mag lock designs would not meet current fire codes as the primary means of securing a fire listed door to a frame.[1]Because of this, many commercial doors (this typically does not apply to private residences) are moving over to stand-alone locks, or electric locks installed under aCertified Personnel Program.[further explanation needed]
The first mechanical recodable card lock was invented in 1976 byTor Sørnes, who had worked forVingCardsince the 1950s. The first card lock order was shipped in 1979 toWestin Peachtree Plaza Hotel, Atlanta, US. This product triggered the evolution of electronic locks for the hospitality industry.[further explanation needed]
Electric strikes (also called electric latch release) replace a standard strike mounted on the door frame and receive thelatchand latch bolt. Electric strikes can be simplest to install when they are designed for one-for-one drop-in replacement of a standard strike, but some electric strike designs require that the door frame be heavily modified. Installation of a strike into a fire listed door (for open backed strikes on pairs of doors) or the frame must be done under listing agency authority, if any modifications to the frame are required (mostly for commercial doors and frames). In the US, since there is no current Certified Personnel Program to allow field installation of electric strikes into fire listed door openings, listing agency field evaluations would most likely require the door and frame to be de-listed and replaced.
Electric strikes can allow mechanical free egress: a departing person operates thelocksetin the door, not the electric strike in the door frame. Electric strikes can also be either "fail unlocked" (except in Fire Listed Doors, as they must remain latched when power is not present), or the more-secure "fail locked" design. Electric strikes are easier to attack than a mag lock. It is simple to lever the door open at the strike, as often there is an increased gap between the strike and the door latch.Latch guardplates are often used to cover this gap.
Electric mortise and cylindrical locks are drop-in replacements for door-mounted mechanical locks. An additional hole must be drilled in the door for electric power wires. Also, a power transfer hinge is often used to get the power from the door frame to the door. Electric mortise and cylindrical locks allow mechanical free egress, and can be either fail unlocked or fail locked. In the US, UL rated doors must retain their rating: in new construction doors are cored and then rated. but in retrofits, the doors must be re-rated.
Electrified exit hardware, sometimes called "panic hardware" or "crash bars", are used in fire exit applications. A person wishing to exit pushes against the bar to open the door, making it the easiest of mechanically-free exit methods. Electrified exit hardware can be either fail unlocked or fail locked. A drawback of electrified exit hardware is their complexity, which requires skill to install and maintenance to assure proper function. Only hardware labeled "Fire Exit Hardware" can be installed on fire listed doors and frames and must meet both panic exit listing standards and fire listing standards.
Motor-operated locks are used throughout Europe. A European motor-operated lock has two modes, day mode where only the latch is electrically operated, and night mode where the more securedeadboltis electrically operated.
In South Korea, most homes and apartments have installed electronic locks, which are currently[when?]replacing the lock systems in older homes. South Korea mainly uses a lock system by Gateman.[citation needed]
The "passive" in passive electronic locks means no power supply. Likeelectronic deadbolts, it is a drop-in replacement for mechanical locks. But the difference is that passive electronic locks do not require wiring and are easy to install.
The passive electronic lock integrates a miniature electronic single-chipmicrocomputer. There is no mechanical keyhole, only three metal contacts are retained. When unlocking, insert the electronic key into the keyhole of the passive electronic lock, that is, the three contacts on the head end of the key are in contact with the three contacts on the passive electronic lock. At this time, the key will supply power to the passive electronic lock, and at the same time, read the ID number of the passive electronic lock for verification. When the verification is passed, the key will power the coil in the passive electronic lock. The coil generates a magnetic field and drives the magnet in the passive electronic lock to unlock. At the moment, turn the key to drive the mechanical structure in the passive electronic lock to unlock the lock body. After successful unlocking, the key records the ID number of the passive electronic lock and also records the time of unlocking the passive electronic lock. Passive electronic locks can only be unlocked by a key with unlocking authority, and unlocking will fail if there is no unlocking authority.
Passive electronic locks are currently used in a number of specialized fields, such as power utilities, water utilities, public safety, transportation, data centers, etc.
The programmable electronic lock system is realized by programmable keys, electronic locks and software. When the identification code of the key matches the identification code of the lock, all available keys are operated to unlock. The internal structure of the lock contains a cylinder, which has a contact (lock slot) that is in contact with the key, and a part of it is an electronic control device to store and verify the received identification code and respond (whether it is unlocked). The key contains a power supply device, usually a rechargeable battery or a replaceable battery in the key, used to drive the system to work; it also includes an electronic storage and control device for storing the identification code of the lock.
The software is used to set and modify the data of each key and lock.[2]
Using this type of key and lock control system does not need to change user habits. In addition, compared with the previous mechanical device, its advantage is that only one key can open multiple locks instead of a bunch of keys like the current one. A single key can contain many lock identification codes; which can set the unlock permission for a single user.
A feature of electronic locks is that the locks can deactivated or opened byauthentication, without the use of a traditional physicalkey:
Perhaps the most common form of electronic lock uses a keypad to enter a numerical code orpasswordfor authentication. Some feature an audible response to each press. Combination lengths are usually between four and six digits long.
Another means of authenticating users is to require them to scan or "swipe" asecurity tokensuch as asmart cardor similar, or to interact a token with the lock. For example, some locks can access stored credentials on apersonal digital assistant(PDA) or smartphone, by usinginfrared,Bluetooth, orNFCdata transfer methods.
Asbiometricsbecome more and more prominent as a recognized means of positive identification, their use in security systems increases. Some electronic locks take advantage of technologies such asfingerprintscanning,retinal scanning, iris scanning andvoice printidentification to authenticate users.
Radio-frequency identification(RFID) is the use of an object (typically referred to as an "RFID tag") applied to or incorporated into a product, animal, or person for the purpose of identification and tracking using radio waves. Some tags can be read from several meters away and beyond the line of sight of the reader. This technology is also used in some modern electronic locks. The technology has been approved since before the 1970s, but has become much more prevalent in recent years due to its usages in things like global supply chain management and pet microchipping.[3]
|
https://en.wikipedia.org/wiki/Electronic_lock
|
Kerberos(/ˈkɜːrbərɒs/) is acomputer-networkauthenticationprotocolthat works on the basis ofticketsto allownodescommunicating over a non-secure network to prove their identity to one another in a secure manner. Its designers aimed it primarily at aclient–servermodel, and it providesmutual authentication—both the user and the server verify each other's identity. Kerberos protocol messages are protected againsteavesdroppingandreplay attacks.
Kerberos builds onsymmetric-key cryptographyand requires atrusted third party, and optionally may usepublic-key cryptographyduring certain phases of authentication.[2]Kerberos usesUDP port88 by default.
The protocol was named after the characterKerberos(orCerberus) fromGreek mythology, the ferocious three-headed guard dog ofHades.[3]
TheMassachusetts Institute of Technology(MIT) developed Kerberos in 1988 to protect network services provided byProject Athena.[4][5]Its first version was primarily designed by Steve Miller and Clifford Neuman based on the earlierNeedham–Schroeder symmetric-key protocol.[6][7]Kerberos versions 1 through 3 were experimental and not released outside of MIT.[8]
Kerberos version 4, the first public version, was released on January 24, 1989. Since Kerberos 4 was developed in the United States, and since it used theData Encryption Standard(DES)encryptionalgorithm,U.S. export control restrictionsprevented it from being exported to other countries. MIT created an exportable version of Kerberos 4 with all encryption code removed,[8]called "Bones".[9]Eric Young of Australia'sBond Universityreimplemented DES into Bones, in a version called "eBones", which could be freely used in any country. Sweden'sRoyal Institute of Technologyreleased another reimplementation called KTH-KRB.[10]
Neuman and John Kohl published version 5 in 1993 with the intention of overcoming existing limitations and security problems. Version 5 appeared asRFC 1510, which was then made obsolete byRFC 4120in 2005.
In 2005, theInternet Engineering Task Force(IETF) Kerberos working group updated specifications. Updates included:
MIT makes an implementation of Kerberos freely available, under copyright permissions similar to those used forBSD. In 2007, MIT formed the Kerberos Consortium to foster continued development. Founding sponsors include vendors such asOracle,Apple Inc.,Google,Microsoft, Centrify Corporation and TeamF1 Inc., and academic institutions such as theRoyal Institute of Technologyin Sweden, Stanford University, MIT, and vendors such as CyberSafe offering commercially supported versions.
The client authenticates itself to theAuthentication Server (AS)which is part of thekey distribution center(KDC). The KDC issues aticket-granting ticket (TGT), which is time stamped and encrypts it using theticket-granting service's (TGS)secret key and returns the encrypted result to the user's workstation. This is done infrequently, typically at user logon; the TGT expires at some point although it may be transparently renewed by the user's session manager while they are logged in.
When the client needs to communicate with a service on another node (a "principal", in Kerberos parlance), the client sends the TGT to the TGS, which is another component of the KDC and usually shares the same host as the authentication server. The service must have already been registered with the TGS with aService Principal Name (SPN). The client uses the SPN to request access to this service. After verifying that the TGT is valid and that the user is permitted to access the requested service, the TGS issues aservice ticket (ST)and session keys to the client. The client then sends the ticket to theservice server (SS)along with its service request.
The protocol is described in detail below.
Windows 2000and later versions use Kerberos as their default authentication method.[13]SomeMicrosoftadditions to the Kerberos suite of protocols are documented in RFC 3244 "Microsoft Windows 2000 Kerberos Change Password and Set Password Protocols". RFC 4757 documents Microsoft's use of theRC4cipher. While Microsoft uses and extends the Kerberos protocol, it does not use the MIT software.
Kerberos is used as the preferred authentication method: in general, joining a client to a Windows domain means enabling Kerberos as the default protocol for authentications from that client to services in the Windows domain and all domains with trust relationships to that domain.[13]
In contrast, when either client or server or both are not joined to a domain (or not part of the same trusted domain environment), Windows will instead useNTLMfor authentication between client and server.[13]
Internet web applications can enforce Kerberos as an authentication method for domain-joined clients by using APIs provided underSSPI.
Microsoft Windows and Windows Server includesetspn, acommand-lineutility that can be used to read, modify, or delete the Service Principal Names (SPN) for an Active Directoryservice account.[14][15]
Many Unix-like operating systems, includingFreeBSD, Apple'smacOS,Red Hat Enterprise Linux,Oracle'sSolaris, IBM'sAIX,HP-UXand others, include software for Kerberos authentication of users or services. A variety of non-Unix like operating systems such asz/OS,IBM iandOpenVMSalso feature Kerberos support. Embedded implementation of the Kerberos V authentication protocol for client agents and network services running on embedded platforms is also available from companies[which?].
TheData Encryption Standard(DES) cipher can be used in combination with Kerberos, but is no longer an Internet standard because it is weak.[16]Security vulnerabilities exist in products that implement legacy versions of Kerberos which lack support for newer encryption ciphers like AES.
|
https://en.wikipedia.org/wiki/Kerberos_(protocol)
|
Akeyfile(orkey-file) is a file on a computer which containsencryptionor license keys.
A common use is web server software running secure socket layer (SSL) protocols. Server-specific keys issued by trusted authorities are merged into the keyfile along with the trusted root certificates. By this method keys can be updated without recompiling software or rebooting the server.
A keyfile is often part of apublic key infrastructure(PKI).
Some applications use a keyfile to hold licensing information, which is periodically reviewed to ensure currency and compliance. Other applications allow users to merge multiple service-specific security settings into a single common store (for example,Apple Computer's Keychainin laterMac OS Xversions,GNOME KeyringandKWalletin theGNOMEandKDEenvironments inLinux, respectively).
|
https://en.wikipedia.org/wiki/Keyfile
|
PassMap/ˈpæsmæp/is a map-based graphical password method ofauthentication, similar topasswords, proposed byNational Tsing Hua Universityresearchers. The wordPassMaporiginates from the wordpasswordby substitutingwordwithmap.
PassMap was proposed byNational Tsing Hua Universityresearchers Hung-Min Sun, Yao-Hsin Chen, Chiung-Cheng Fang, and Shih-Ying Chang at the 7thAssociation for Computing MachinerySymposium on Information, Computer and Communications Security. They defined PassMap as letting a consumer get authenticated by choosing a series of points on a bigworld map. Their study showed that for people, PassMap passwords are more user-friendly and memorable.[1]
Users are shownGoogle Mapson their screen, through which they can zoom in to choose any two points they want to become their PassMap password. Since PassMap uses Google Maps, it cannot be used in applications that lack Internet access or Google Maps integration.[2]By default, PassMap's screen is set to the eighth zoom level and is centered onTaiwan. PassMap has no constraints on the zoom level, so consumers are allowed to select dots at unsafer, lower levels, like level 8. It does notnormalizeerror tolerancebased on a screen's zoom position.[3]PassMap's effective login percentage is 92.59%.[4]
Ritika Sachdev wrote in theInternational Journal of Pure and Applied Research in Engineering and Technologythat based on psychological studies, people can effortlessly recall the milestones they have visited. Sachdev called PassMap a "highly subjective or customized based password to ensure security".[5]
S. Rajarajan, M. Prabhu, and S. Palanivel praised PassMap for having "good memorability due to the usage of map for the password mechanism". But they noted that, like manygraphical passwords, PassMap is susceptible to ashoulder surfingintrusion.[2]
|
https://en.wikipedia.org/wiki/PassMap
|
Password fatigueis the feeling experienced by many people who are required to remember an excessive number ofpasswordsas part of their daily routine, such as tolog into a computer at work, undo abicycle lockor conduct banking from anautomated teller machine. The concept is also known aspassword chaos,or more broadly asidentity chaos.[1]
The increasing prominence ofinformation technologyand theInternetin employment,finance, recreation and other aspects of people's lives, and the ensuing introduction ofsecure transactiontechnology, has led to people accumulating a proliferation of accounts and passwords.
According to a survey conducted in February 2020 bypassword managerNordpass, a typical user has 100 passwords.[2]
Some factors causing password fatigue are:
Some companies are well organized in this respect and have implemented alternative authentication methods,[3]or have adopted technologies so that a user's credentials are entered automatically. However, others may not focus onease of use, or even worsen the situation, by constantly implementing new applications with their own authentication system.
As password fatigue continues to challenge users, notable advances in password management techniques have emerged to alleviate this burden. These innovative approaches provide alternatives to traditional password-based authentication systems. Here are some notable strategies:
Biometric authentication methods offer a seamless and secure alternative to traditional passwords, including fingerprint recognition, facial recognition, and iris scanning. Users can authenticate their identities without remembering complex passwords by leveraging unique biological characteristics. Companies like Okta and Transmit Security have developed robust biometric authentication solutions, reducing reliance on traditional passwords.[5]
Security tokens, also referred to as hardware tokens or authentication tokens, add an extra layer of security beyond passwords. These physical devices generate a one-time passcode or cryptographic key that users input alongside their passwords for authentication. This two-factor authentication (2FA) method enhances security while reducing the cognitive load of managing multiple passwords. Secret Double Octopus is a notable provider of security token solutions.[5]
Passwordless authentication services represent a significant shift in authentication methods by eliminating the need for passwords. Instead, these services utilize alternative verification methods, such as biometric authentication, security keys, or magic email links. By removing passwords from the equation, passwordless authentication significantly simplifies the user experience and reduces the risk of password-related security breaches. Okta, Transmit Security, and Secret Double Octopus are pioneering providers of passwordless authentication solutions.[5]
Emerging technologies in behavioral biometrics analyze unique behavioral patterns, such as typing speed, mouse movements, and touchscreen interactions, for user authentication. By continuously monitoring these behavioral signals, the system can accurately verify a user's identity without requiring an explicit authentication action. Behavioral biometrics provide a seamless authentication experience while minimizing the cognitive load associated with traditional password-based systems.[5]
These innovative approaches offer promising alternatives to traditional password management techniques, delivering enhancements in security, usability, and user convenience. As technology advances, further progress in authentication methods will effectively address the ongoing challenge of password fatigue.[5]
|
https://en.wikipedia.org/wiki/Password_fatigue
|
Password notification emailor password recovery email is a commonpassword recoverytechnique used bywebsites. If a user forgets theirpassword, a password recoveryemailis sent which contains enough information for the user to access theiraccountagain. This method of password retrieval relies on the assumption that only the legitimate owner of the account has access to the inbox for that particular email address.
The process is often initiated by the user clicking on a forgotten password link on the website where, after entering theirusernameor email address, the password notification email would be automatically sent to the inbox of the account holder. This email may contain a temporary password or aURLthat can be followed to enter a new password for that account. The new password or the URL often contain a randomly generatedstringof text that can only be obtained by reading that particular email.[1]
Another method used is to send all or part of the original password in the email. Sending only a few characters of the password can help the user to remember their original password without having to reveal the whole password to them.
The main issue is that the contents of the password notification email can be easily discovered by anyone with access to the inbox of the account owner.[2]This could be as a result ofshoulder surfingor if the inbox itself is not password protected. The contents could then be used to compromise the security of the account. The user would therefore have the responsibility of either securely deleting the email or ensuring that its contents are not revealed to anyone else. A partial solution to this problem, is to cause any links contained within the email to expire after a period of time, making the email useless if it is not used quickly after it is sent.
Any method that sends part of the original password means that the password is stored in plain text and leaves the password open to an attack from hackers.[3]This is why it is typical for newer sites to create a new password generate a token. If the site gets hacked, the password contained within could be used to access other accounts used by the user, if that user had chosen to use the same password for two or more accounts. Additionally, emails areoften not secure. Unless an email had beenencryptedprior to being sent, its contents could be read by anyone whoeavesdropson the email.
|
https://en.wikipedia.org/wiki/Password_notification_e-mail
|
Living in the intersection ofcryptographyandpsychology,password psychologyis the study of what makespasswordsorcryptographic keyseasy to remember or guess.
In order for apasswordto work successfully and provide security to its user, it must be kept secret and un-guessable; this also requires the user to memorize their password. The psychology behind choosing a password is a unique balance between memorization,securityand convenience. Password security involves many psychological and social issues including; whether or not to share a password, the feeling of security, and the eventual choice of whether or not to change a password. Passwords may also be reflective of personality. Those who are more uptight or security-oriented may choose longer or more complicated passwords. Those who are lax or who feel more secure in their everyday lives may never change their password.[1]The most common password is Password1, which may point to convenience over security as the main concern forinternetusers.[2][3]
The use and memorization of both nonsense and meaningful alphanumeric material has had a long history in psychology beginning withHermann Ebbinghaus. Since then, numerous studies have established that not only are both meaningful and nonsense "words" easily forgotten, but that both their forgetting curves are exponential with time.[4]Chomskyadvocates meaning as arising fromsemanticfeatures, leading to the idea of "concept formation" in the 1930s.[4]
Research is being done to find new ways of enhancing and creating new techniques for cognitive ability and memorization when it comes to password selection.[5]A study from 2004 indicates that the typical college student creates about 4 different passwords for use with about 8 different items, such as computers, cell phones, and email accounts, and the typical password is used for about two items.[6]Information about the type of passwords points to an approximate even split between linguistic and numeric passwords with about a quarter using a mix of linguistic/numeric information. Names (proper, nicknames) are the most common information used for passwords, and dates are the second most common type of information used in passwords.[6]Research is also being done regarding the effect of policies that force users to create more secure and effective passwords.[7]The results of this study show that a password composition policy reduces the similarity of passwords to dictionary words. However, such a policy did not reduce the use of meaningful information in passwords such as names and birth dates, nor did it reduce password recycling.[7]
Password psychology is directly linked to memorization and the use ofmnemonics. Mnemonic devices are often used as passwords but many choose to use simpler passwords. It has been shown that mnemonic devices and simple passwords are equally easy to remember and that the choice of convenience plays a key role in password creation.[8]
In order to address the issues presented by memorization and security, many businesses and internet sites have turned to accepting differentauthentication protocols. This authentication could be abiometric, a 2D key, amulti-factor authentication, apasswordless authentication, orcognitive passwordsthat are question based. Many of these options are more expensive, time-consuming, or still require some form of memorization. Thus, most businesses and individuals still use the common format of single word and text-based passwords as security protection.
The most common alternative to tradition passwords and PIN codes has been biometric authentication.[9]Biometric authentication is a method where systems use physical and/or behavioral traits unique to a specific individual to authorize access.[9]Some of the most popular forms of biometric passwords are as follows: fingerprint, palm prints, iris, retina, voice, and facial structure.[10]The appeal of biometrics as a form of passwords is that they increase security.[11]Only one person has access to a set of fingerprints orretinal patterns, which means the likelihood of hacking decreases significantly. Biometric authentication has 4 important factors, or modules, that keep systems and accounts from being compromised: sensor module, feature extraction module, template database, and matching module.[9]These 4 sections of biometric authentication, while more involved, create a layer of protection that a tradition password option cannot. The sensor module is responsible for getting a hold of a user’s method of protection whether it be fingerprint scan, facial scan, or voice.[11][9]The second module, feature extraction, is where all the raw data acquired from the previous module is broken down into the key components. The template, or database module, takes the key components gathered previously and saves them virtually. Lastly, the matching module is employed in order to verify if the inputted biometric method is legitimate.[11][9][10]The modules that record, process, and verify biometrics, need to be run in 2 different stages: enrollment and recognition. These stages contain more substages. In the enrollment stage, the four modules work at once as a digital version of the biometric data is generated and stored.[11]The recognition stage has two subsections called verification and identification.[11]During verification process, the system must ensure that the individual trying to gain access is who they are stating they are. The identification process fully identifies the individual.
Though biometric authentication is seen increasingly more often, it is not without its issues. A biometric system is affected by similar issues that a tradition password system has. When a user inputs their biometric information, one of four things can happen. A user may be truly be who they say they are and are granted access to the system. Conversely, a user may be impersonating someone and will be rejected access. The two other scenarios are when an authentic user is rejected access and an impersonator is granted access.[11]This type of fraud can occur as there are certain individuals that may share virtually identical voices.[11]In other instances, the initial attempt to record the biometric data may have been compromised. During the 4 modules, a user may have inputted corrupted data. An example of this is most commonly seen in fingerprints, where an individual may use a wet finger or a scarred finger to record their data.[11]These errors introduce insecurity.[9]These issues can occur for facial recognition. If a pair of twins or even two people who look similar try to access a system, they may be granted access.
|
https://en.wikipedia.org/wiki/Password_psychology
|
Password synchronizationis a process, usually supported by software such aspassword managers, through which a user maintains a single password across multipleIT systems.[1]
Provided that all the systems enforce mutually-compatible password standards (e.g. concerning minimum and maximum password length, supported characters, etc.), the user can choose a new password at any time and deploy the same password on his or her own login accounts across multiple, linked systems.
Where different systems have mutually incompatible standards regarding what can be stored in a password field, the user may be forced to choose more than one (but still fewer than the number of systems) passwords. This may happen, for example, where the maximum password length on one system is shorter than the minimum length in another, or where one system requires use of a punctuation mark but another forbids it.
Password synchronization is a function of certainidentity managementsystems and it is considered easier to implement thanenterprise single sign-on (SSO), as there is normally no client software deployment or need for active user enrollment.[1]
Password synchronization makes it easier for IT users to recall passwords and so manage their access to multiple systems, for example on an enterprise network.[1]Since they only have to remember one or at most a few passwords, users are less likely to forget them or write them down, resulting in fewer calls to the IT Help Desk and less opportunity for coworkers, intruders or thieves to gain improper access. Through suitable security awareness, automated policy enforcement and training activities, users can be encouraged or forced to choosestronger passwordsas they have fewer to remember.
If the single, synchronized password is compromised (for example, if it is guessed, disclosed, determined bycryptanalysisfrom one of the systems, intercepted on an insecure communications path, or if the user is socially engineered into resetting it to a known value), all the systems that share that password are vulnerable to improper access. In mostsingle sign-onand password vault solutions, compromise of the primary or master password (in other words, the password used to unlock access to the individual unique passwords used on other systems) also compromises all the associated systems, so the two approaches are similar.
Depending on the software used, password synchronization may be triggered by a password change on any one of the synchronized systems (whether initiated by the user or an administrator) and/or by the user initiating the change centrally through the software, perhaps through a web interface.
Some password synchronization systems may copy password hashes from one system to another, where the hashing algorithm is the same. In general, this is not the case and access to a plaintext password is required.
Two processes which yields synchronized passwords are shown in the following animations, hosted by software vendor Hitachi ID Systems:1
|
https://en.wikipedia.org/wiki/Password_synchronization
|
Arandom password generatoris asoftwareprogram orhardwaredevice that takes input from arandomorpseudo-randomnumber generator and automatically generates apassword. Random passwords can be generated manually, using simple sources of randomness such asdiceorcoins, or they can be generated using a computer.
While there are many examples of "random" password generator programs available on the Internet, generating randomness can be tricky, and many programs do not generate random characters in a way that ensures strong security. A common recommendation is to useopen sourcesecurity tools where possible, since they allow independent checks on the quality of the methods used. Simply generating a password at random does not ensure the password is a strong password, because it is possible, although highly unlikely, to generate an easily guessed or cracked password. In fact, there is no need at all for a password to have been produced by a perfectly random process: it just needs to be sufficiently difficult to guess.
A password generator can be part of apassword manager. When apassword policyenforces complex rules, it can be easier to use a password generator based on that set of rules than to manually create passwords.
Long strings of random characters are difficult for most people to memorize.Mnemonichashes, which reversibly convert random strings into more memorable passwords, can substantially improve the ease of memorization. As thehashcan be processed by a computer to recover the original 60-bit string, it has at least as much information content as the original string.[1]Similar techniques are used inmemory sport.
Random password generators normally output a string of symbols of specified length. These can be individual characters from some character set, syllables designed to form pronounceable passwords, or words from some word list to form apassphrase. The program can be customized to ensure the resulting password complies with the local password policy, say by always producing a mix of letters, numbers and special characters. Such policies typically reduce strength slightly below the formula that follows, because symbols are no longer independently produced.[citation needed]
ThePassword strengthof a random password against a particular attack (brute-force search), can be calculated by computing theinformation entropyof the random process that produced it. If each symbol in the password is produced independently and with uniform probability, the entropy in bits is given by the formulaH=Llog2N{\textstyle H=L\,\log _{2}N}, whereNis the number of possible symbols andLis the number of symbols in the password. The function log2is thebase-2 logarithm.His typically measured inbits.[2][3]
Any password generator is limited by the state space of the pseudo-random number generator used if it is based on one. Thus a password generated using a 32-bit generator is limited to 32 bits entropy, regardless of the number of characters the password contains.[citation needed]
A large number of password generator programs and websites are available on the Internet. Their quality varies and can be hard to assess if there is no clear description of the source of randomness that is used and if source code is not provided to allow claims to be checked. Furthermore, and probably most importantly, transmitting candidate passwords over the Internet raises obvious security concerns, particularly if the connection to the password generation site's program is not properly secured or if the site is compromised in some way. Without asecure channel, it is not possible to prevent eavesdropping, especially over public networks such as theInternet. A possible solution to this issue is to generate the password using a client-side programming language such as JavaScript. The advantage of this approach is that the generated password stays in the client computer and is not transmitted to or from an external server.[original research?]
TheWeb Cryptography APIis theWorld Wide Web Consortium’s (W3C) recommendation for a low-level interface that would increase the security ofweb applicationsby allowing them to performcryptographic functionswithout having to access raw keying material. The Web Crypto API provides a reliable way to generate passwords using thecrypto.getRandomValues()method. Here is the simple Javascript code that generate the strong password using web crypto API.[4][5]
Many computer systems already have an application (typically named "apg") to implement the password generator standard FIPS 181.[6]FIPS 181—Automated Password Generator—describes a standard process for converting random bits (from a hardware random number generator) into somewhat pronounceable "words" suitable for a passphrase.[7]However, in 1994 an attack on the FIPS 181 algorithm was discovered, such that an attacker can expect, on average, to break into 1% of accounts that have passwords based on the algorithm, after searching just 1.6 million passwords. This is due to the non-uniformity in the distribution of passwords generated, which can be addressed by using longer passwords or by modifying the algorithm.[8][9]
Yet another method is to use physical devices such asdiceto generate the randomness. One simple way to do this uses a 6 by 6 table of characters. The first die roll selects a row in the table and the second a column. So, for example, a roll of 2 followed by a roll of 4 would select the letter"j"from thefractionationtable below.[10]To generate upper/lower case characters or some symbols a coin flip can be used, heads capital, tails lower case. If a digit was selected in the dice rolls, a heads coin flip might select the symbol above it on a standard keyboard, such as the '$' above the '4' instead of '4'.
|
https://en.wikipedia.org/wiki/Random_password_generator
|
Ashibboleth(/ˈʃɪbəlɛθ,-ɪθ/ⓘ;[1][2]Hebrew:שִׁבֹּלֶת,romanized:šībbōleṯ) is anycustomor tradition, usually a choice of phrasing or single word, that distinguishes one group of people from another.[3][2][4]Historically, shibboleths have been used aspasswords, ways of self-identification, signals of loyalty and affinity, ways of maintaining traditional segregation, or protection from threats. It has also come to mean a moral formula held tenaciously and unreflectingly, or ataboo.[5]
The term originates from the Hebrew wordshibbóleth(שִׁבֹּלֶת), which means the part of a plant containinggrain, such as theearof a stalk ofwheatorrye;[6][7][2][8]or less commonly (but arguably more appropriately)[a]'flood,torrent'.[9]: 10[10]: 69
The modern use derives from an account in theHebrew Bible, in which pronunciation of this word was used to distinguishEphraimites, whose dialect used a different first consonant. The difference concerns the Hebrew lettershin, which is now pronounced as[ʃ](as inshoe).[11]In theBook of Judgeschapter 12, after the inhabitants ofGileadunder the command ofJephthahinflicted a military defeat upon the invadingtribe of Ephraim(around 1370–1070 BC), the surviving Ephraimites tried to cross theriver Jordanback into their home territory, but the Gileadites secured the river's fords to stop them. To identify and kill these Ephraimites, the Gileadites told each suspected survivor to say the wordshibboleth. The Ephraimite dialect resulted in a pronunciation that, to Gileadites, sounded likesibboleth.[11]In Judges 12:5–6 in theKing James Bible, the anecdote appears thus (with the word already in its current English spelling):
And the Gileadites took the passages of Jordan before the Ephraimites: and it was so, that when those Ephraimites which were escaped said, Let me go over; that the men of Gilead said unto him, Art thou an Ephraimite? If he said, Nay;
Then said they unto him, Say now Shibboleth: and he said Sibboleth: for he could not frame to pronounce it right. Then they took him, and slew him at the passages of Jordan: and there fell at that time of the Ephraimites forty and two thousand.
Shibbolethhas been described as the first "password" in Western literature[13]: 93but exactly how it worked is not known; it has long been debated by scholars of Semitic languages.[14][15]It may have been quite subtle: the men of Ephraim were unlikely to be "caught totally napping by any test that involved some gross and readily detectable difference of pronunciation";[16]: 274On a superficial reading the fleeing Ephraimites were betrayed by their dialect: they saidsibbōleth. But it has been asked why they did not simply repeat what the Gileadite sentries told them to say[14]: 250— "they surely would have used the required sound to save their necks",[17]since peoples in the region could say both "sh" and "s".[18][19]"We have yet to learn how the suspects were caught by the catchword".[17]A related problem (akin tofalse positives) is how the test spared neutral tribes with whom the Gileadite guards had no quarrel, yet pinpointed the Ephraimite enemy.[20]: 98
Ephraim Avigdor Speisertherefore proposed that the test involved a more challenging sound than could be written down in the later biblical Hebrew narrative, namely thephoneme⟨θ⟩ (≈ English "th"). Present in archaic Hebrew (said Speiser) but later lost in most dialects, the Gileadites, who lived across a dialect boundary (the river Jordan), had retained it in theirs. Thus, what the Gileadite guards would have demanded was the passwordthibbōlet. The phoneme is difficult for naive users — to this day, wrote Speiser, most non-Arab Muslims cannot pronounce the classical Arabic equivalent — hence the best the Ephraimite refugees could manage wassibbōlet.[17]Speiser's solution has had a mixed reception,[21]but has been revived byGary A. Rendsburg.[22]
John Emertonargued that "Perhaps [the Ephraimites] could pronounceš, but they articulated the consonant in a different way from the Gileadites, and their pronunciation sounded to the men of Gilead likes". There is a range of ways of pronouncing the two phonemes. "An old clergyman of my acquaintance used to say 'O Lord, save the Queen' in such a way that it sounded [to me] like 'O Lord, shave the Queen'", and analogies could be found amongst Hebrew users in modern Lithuania and Morocco.[15]: 256Berkeleyscholar Ronald Hendel agreed, saying the theory was supported by a document recently dug up near modernAmman. It tended to show that, across the Jordan, the pronunciation of the phoneme "sh" was heard as "s" by Hebrew speakers from the opposite side of the river. "This is why Gileaditešibbōletis repeated by the Ephraimites assibbōlet: they simply repeated the word as they heard it".[14]Other solutions have been proposed.[23]
David Marcus has contended that linguistic scholars have missed the point of the biblical anecdote: The purpose of the laterJudeannarrator was not to record some phonetic detail, but to satirise the incompetence of "thehigh and mighty northern Ephraimites". "The shibboleth episode ridicules the Ephraimites who are portrayed as incompetent nincompoops who cannot even repeat a test-word spoken by the Gileadite guards".[20]
In modernEnglish, a shibboleth can have asociologicalmeaning, referring to anyin-groupword or phrase that can distinguish members from outsiders.[24]It is also sometimes used in a broader sense to meanjargon, the proper use of which identifies speakers as members of a particular group orsubculture.
Ininformation technology,Shibbolethis a community-wide password that enables members of that community to access an online resource without revealing their individual identities. The origin server can vouch for the identity of the individual user without giving the target server any further identifying information.[25]Hence the individual user does not know the password that is actually employed – it is generated internally by the origin server – and so cannot betray it to outsiders.
The term can also be used pejoratively, suggesting that the original meaning of a symbol has in effect been lost and that the symbol now serves merely to identify allegiance, being described as "nothing more than a shibboleth". In 1956,economistPaul Samuelsonapplied the termshibbolethin works includingFoundations of Economic Analysisto mean an idea for which "the means becomes the end, and the letter of the law takes precedence over the spirit."[26]Samuelson admitted thatshibbolethis an imperfect term for this phenomenon.[27]
Shibboleths have been used by different subcultures throughout the world at different times. Regional differences, level of expertise, and computer coding techniques are several forms that shibboleths have taken.
There is a legend that before theBattle of the Golden Spursin May 1302, theFlemishslaughtered every Frenchman they could find in the city ofBruges, an act known as theMatins of Bruges.[28]They identified Frenchmen based on their inability to pronounce the Flemish phraseschild en vriend, 'shield and friend', or possiblygilden vriend, 'friend of the Guilds'. However, many Medieval Flemish dialects did not contain the clustersch-either (even today'sKortrijkdialect hassk-), and Medieval French rolled the r just as Flemish did.[b]
There is an anecdote inSicilythat, during the rebellion of theSicilian Vespersin 1282, the inhabitants of the island killed theFrench occupierswho, when questioned, could not correctly pronounce the Sicilian wordcìciri'chickpeas'.[29]
FollowingMayor Albert's Rebellionin 1312Kraków, Poles used thePolish languageshibbolethSoczewica, koło, miele, młyn('Lentil, wheel, grinds (verb), mill') to distinguish the German-speaking burghers. Those who could not properly pronounce this phrase were executed.[30]
Bûter, brea, en griene tsiis; wa't dat net sizze kin, is gjin oprjochte Fries('Butter, rye bread and green cheese, whoever cannot say that is not a genuine Frisian') was a phrase used by theFrisianPier Gerlofs Doniaduring aFrisian rebellion(1515–1523). Ships whose crew could not pronounce this properly were usually plundered and soldiers who could not were beheaded by Donia.[31]
Newspaper advertisements in 18th-century America seeking absconding servants or apprentices frequently used the shibboleth method to identify them. Since most runaways were from the British Isles originally, they were identified by their distinctive regional accents, e.g. "speaks broad Yorkshire". Studying a large number of these advertisements,Allen Walker Readnoticed an exception: runaways were never advertised as having London or eastern counties accents. From this he inferred that their speech did not differ from the bulk of the American population. "Thus in the colonial period American English had a consistency of its own, most closely approximating the type of the region around London".[32]
In Japan during the 1923Kantō Massacre, in which ethnicKoreans in Japanwere hunted down and killed by vigilantes after rumors spread that they were committing crimes,[33]shibboleths were attested to having been used to identify Koreans. The Japanese poetShigeji Tsuboiwrote that he overheard vigilantes asking people to pronounce the phrasejūgoen gojissen(Japanese:15円50銭,lit.'fifteenyen, fifty sen').[34]If the person pronounced it aschūkoen kochissen, he was reportedly dragged away for punishment.[34][35]Both Korean and Japanese people recalled similar shibboleths being used, includingichien gojissen(lit.'one yen, fifty sen').[33]Other strings attested to werega-gi-gu-ge-go(Japanese:がぎぐげご) andka-ki-ku-ke-ko(Japanese:かきくけこ), which were thought difficult for Koreans to pronounce.[34]
In October 1937, the Spanish word for parsley,perejil, was used as a shibboleth to identify Haitian immigrants living along the border in the Dominican Republic. The Dominican dictator,Rafael Trujillo, ordered the execution of these people. It is alleged that between 20,000 and 30,000 individuals were murdered within a few days in theParsley Massacre, although more recent scholarship and the lack of evidence such as mass graves puts the actual estimate closer to between 1,000 and 12,168.[36]
During theGerman occupation of the NetherlandsinWorld War II, the Dutch used the name of the seaside town ofScheveningenas a shibboleth to tell Germans from Dutch ("Sch" inDutchis analyzed as the letter "s" combined with thedigraph"ch", producing theconsonant cluster[sx], while inGerman"Sch" is read as thetrigraph"sch", pronounced[ʃ], closer to "sh" sound in English).[37][38][24]
Some American soldiers in the Pacific theater in World War II used the wordlollapaloozaas a shibboleth tochallengeunidentified persons, on the premise that Japanese peoplewould often pronounce both letters L and Ras rolled Rs.[39]In Oliver Gramling'sFree Men Are Fighting: The Story of World War II(1942) the author notes that, in the war, Japanese spies would often approach checkpoints posing as American orFilipinomilitary personnel. A shibboleth such aslollapaloozawould be used by the sentry, who, if the first two syllables come back asrorra,would "open fire without waiting to hear the remainder".[40]Another sign/countersign used by the Allied forces: the challenge/sign was "flash", thepassword"thunder", and the countersign "Welcome".[41]This was used duringD-DayduringWorld War IIdue to the rarity of thevoiceless dental fricative(th-sound) andvoiced labial–velar approximant(w-sound) in German.[citation needed]
DuringThe Troublesin Northern Ireland, use of the nameDerry or Londonderryfor the province's second-largest city was often taken as an indication of the speaker's political stance, and as such frequently implied more than simply naming the location.[42]The pronunciation of the name of the letterHis a related shibboleth, with Catholics pronouncing it as "haitch" and Protestants oftenpronouncing the letter differently.[43]
During theBlack Julyriots of Sri Lanka in 1983, many Tamils were massacred by Sinhalese youths. In many cases these massacres took the form of boarding buses and getting the passengers to pronounce words that had[b]at the beginning (likebaldiya'bucket') and executing the people who found it difficult.[44][45]
In Australia and New Zealand, the words "fish and chips" are often used to highlight the difference in each country's short-i vowel sound [ɪ] and asking someone to say the phrase can identify which country they are from. Australian English has a higher forward sound [i], close to the y in happy and city, while New Zealand English has a lower backward sound [ɘ], a slightly higher version of the a in about and comma. Thus, New Zealanders hear Australians say "feesh and cheeps", while Australians hear New Zealanders say "fush and chups".[46]A long drawn out pronunciation of the names of the citiesBrisbaneandMelbournerather than the typically Australian rapid "bun" ending is a common way for someone to be exposed as new to the country. Within Australia, what someone calls "devon", or how he names the size of beer he orders can often pinpoint what state he is from, as both of these have varied names across the country.[citation needed]
In Canada, the name of Canada's second largest city,Montreal, is pronounced/ˌmʌntriˈɔːl/by English-speaking locals. This contrasts with the typical American pronunciation of the city as/ˌmɒntriˈɔːl/.[47]
In the United States, the name of the stateNevadacomes from the Spanishnevada[neˈβaða], meaning 'snow-covered'.[48]Nevadans pronounce the second syllable with the "a" as in "trap" (/nɪˈvædə/) while some people from outside of the state can pronounce it with the "a" as in "palm" (/nɪˈvɑːdə/).[49]Although many Americans interpret the latter back vowel as being closer to the Spanish pronunciation, it is not the pronunciation used by Nevadans. Likewise, the same test can be used to identify someone unfamiliar with southwestMissouri, as the city ofNevada, Missouriis pronounced with the "a" as in "cape" (/nɪˈveɪdə/).
During theRusso-Ukrainian War(2014–present), Ukrainians have used the wordpalianytsia(a type of Ukrainian bread) to distinguish between Ukrainians and Russians.[50]
Afurtive shibbolethis a type of a shibboleth that identifies individuals as being part of a group, not based on their ability to pronounce one or more words, but on their ability to recognize a seemingly innocuous phrase as a secret message. For example, members ofAlcoholics Anonymoussometimes refer to themselves as "a friend of Bill W.", which is a reference to AA's founder,William Griffith Wilson. To the uninitiated, this would seem like a casual – if off-topic – remark, but other AA members would understand its meaning.[51]
Similarly, duringWorld War II, a homosexualUS sailormight call himself a "friend of Dorothy", a tongue-in-cheek acknowledgment of a stereotypicalaffinity for Judy GarlandinThe Wizard of Oz. This code was so effective that theNaval Investigative Service, upon learning that the phrase was a way for gay sailors to identify each other, undertook a search for this "Dorothy", whom they believed to be an actual woman with connections to homosexual servicemen in the Chicago area.[52][53]Many cruise lines still host "Friends of Dorothy" meetings for LGBT guests to gather.[54]
Likewise, homosexuals in Britain might use thecant languagePolari.[55]
Mark Twainused an explicit shibboleth to conceal a furtive shibboleth. InThe Innocents Abroadhe told the Shibboleth story in seemingly "inept and uninteresting" detail. To the initiated, however, the wording revealed that Twain was afreemason.[56]
"Fourteen Words", "14", or "14/88"are furtive shibboleths used amongwhite supremacistsin the Anglosphere.[57]
Colombian conceptual artistDoris Salcedocreated a work titledShibbolethatTate Modern, London, in 2007–2008. The piece consisted of a 548-foot-long crack that bisected the floor of the Tate's lobby space.
Salcedo said of the work:
It represents borders, the experience of immigrants, the experience of segregation, the experience of racial hatred. It is the experience of a Third World person coming into the heart of Europe. For example, the space which illegal immigrants occupy is a negative space. And so this piece is a negative space.[58]
|
https://en.wikipedia.org/wiki/Shibboleth
|
Usability of web authentication systemsrefers to the efficiency and user acceptance of online authentication systems.[1]Examples of web authentication systems arepasswords,federated identity systems(e.g.GoogleOAuth2.0,Facebook Connect,Sign in with Apple), email-basedsingle sign-on(SSO) systems (e.g. SAW, Hatchet),QR code-based systems (e.g. Snap2Pass, WebTicket) or any other system used to authenticate a user's identity on the web. Even though theusabilityof webauthenticationsystems should be a key consideration in selecting a system, very few web authentication systems (other than passwords) have been subjected to formalusabilitystudies or analysis.[2]
A web authentication system needs to be as usable as possible whilst not compromising thesecuritythat it needs to ensure.[1]The system needs to restrict access by malicious users whilst allowing access toauthorisedusers. If the authentication system does not have sufficient security, malicious users could easily gain access to the system. On the other hand, if the authentication system is too complicated and restrictive, an authorised user would not be able to (or want to) use it.[3]Strong security is achievable in any system, but even the most secure authentication system can be undermined by the users of the system, often referred to as the "weak links" in computer security.[4]
Users tend to inadvertently increase or decrease security of a system. If a system is not usable, security could suffer as users will try to minimize the effort required to provide input for authentication, such as writing down their passwords on paper. A more usable system could prevent this from happening. Users are more likely to oblige to authentication requests from systems that are important (e.g. online banking), as opposed to less important systems (e.g. a forum that the user visits infrequently) where these mechanisms might just be ignored. Users accept the security measures only up to a certain point before becoming annoyed by complicated authentication mechanisms.[4]An important factor in the usability of a web authentication system is thus the convenience factor for the user around it.
The preferred web authentication system for web applications is the password,[4]despite its poor usability and several security concerns.[5]This widely used system usually contains mechanisms that were intended to increase security (e.g. requiring users to have high entropy passwords) but lead to password systems being less usable and inadvertently less secure.[6]This is because users find these high entropy passwords harder to remember.[7]Application creators need to make a paradigm shift to develop more usable authentication systems that take the user's needs into account.[5]Replacing the ubiquitous password based systems with more usable (and possibly more secure) systems could lead to major benefits for both the owners of the application and its users.
To measure the usability of a web authentication system, one can use the "usability–deployability–security" or "UDS" framework[5]or a standard metric, such as thesystem usability scale.[2]The UDS framework looks at three broad categories, namely usability deployability and security of a web authentication system and then rates the tested system as either offering or not offering a specific benefit linked to one (or more) of the categories. An authentication system is then classified as either offering or not offering a specific benefit within the categories of usability deployability and security.[5]
Measuring usability of web authentication systems will allow for formal evaluation of a web authentication system and determine the ranking of the system relative to others. While a lot of research regarding web authentication system is currently being done, it tends to focus on security and not usability.[1]Future research should be evaluated formally for usability using a comparable metric or technique. This will enable the comparison of various authentication systems, as well as determining whether an authentication system meets a minimum usability benchmark.[2]
It has been found that security experts tend to focus more onsecurityand less on the usability aspects of web authentication systems.[5]This is problematic as there needs to be a balance between thesecurityof a system and itsease-of-use.
A study conducted in 2015[2]found that users tend to prefer Single sign-on (like those provided by Google and Facebook) based systems. Users preferred these systems because they found them fast and convenient to use.[2]Single sign-on based systems have resulted in substantial improvements in both usability and security.[5]SSO reduces the need for users to remember many usernames and passwords as well as the time needed to authenticate themselves, thereby improving the usability of the system.
Usability will become more and more important as more applications move online and require robust and reliable authentication systems that are both usable and secure. The use of brainwaves in authentication systems[8]have been proposed as a possible way to achieve this. However more research and usability studies are required.
|
https://en.wikipedia.org/wiki/Usability_of_web_authentication_systems
|
Inmathematics,Light's associativity testis a procedure invented by F. W. Light for testing whether abinary operationdefined in afinite setby aCayley multiplication tableisassociative. The naive procedure for verification of the associativity of a binary operation specified by a Cayley table, which compares the two products that can be formed from each triple of elements, is cumbersome. Light's associativity test simplifies the task in some instances (although it does not improve the worst-case runtime of the naive algorithm, namelyO(n3){\displaystyle {\mathcal {O}}\left(n^{3}\right)}for sets of sizen{\displaystyle n}).
Let a binary operation ' · ' be defined in a finite setAby a Cayley table. Choosing some elementainA, two new binary operations are defined inAas follows:
The Cayley tables of these operations are constructed and compared. If the tables coincide thenx· (a·y) = (x·a) ·yfor allxandy. This is repeated for every element of the setA.
The example below illustrates a further simplification in the procedure for the construction and comparison of the Cayley tables of the operations '⋆{\displaystyle \star }' and '∘{\displaystyle \circ }'.
It is not even necessary to construct the Cayley tables of '⋆{\displaystyle \star }' and '∘{\displaystyle \circ }' forallelements ofA. It is enough to compare Cayley tables of '⋆{\displaystyle \star }' and '∘{\displaystyle \circ }' corresponding to the elements in a proper generating subset ofA.
When the operation ' . ' iscommutative, then x⋆{\displaystyle \star }y = y∘{\displaystyle \circ }x. As a result, only part of each Cayley table must be computed, because x⋆{\displaystyle \star }x = x∘{\displaystyle \circ }x always holds, and x⋆{\displaystyle \star }y = x∘{\displaystyle \circ }y implies y⋆{\displaystyle \star }x = y∘{\displaystyle \circ }x.
When there is anidentity elemente, it does not need to be included in the Cayley tables because x⋆{\displaystyle \star }y = x∘{\displaystyle \circ }y always holds if at least one of x and y are equal to e.
Consider the binary operation ' · ' in the setA= {a,b,c,d,e} defined by the following Cayley table (Table 1):
The set {c,e} is a generating set for the setAunder the binary operation defined by the above table, for,a=e·e,b=c·c,d=c·e. Thus it is enough to verify that the binary operations '⋆{\displaystyle \star }' and '∘{\displaystyle \circ }' corresponding toccoincide and also that the binary operations '⋆{\displaystyle \star }' and '∘{\displaystyle \circ }' corresponding toecoincide.
To verify that the binary operations '⋆{\displaystyle \star }' and '∘{\displaystyle \circ }' corresponding toccoincide, choose the row in Table 1 corresponding to the elementc:
This row is copied as the header row of a new table (Table 3):
Under the headeracopy the corresponding column in Table 1, under the headerbcopy the corresponding column in Table 1, etc., and construct Table 4.
The column headers of Table 4 are now deleted to get Table 5:
The Cayley table of the binary operation '⋆{\displaystyle \star }' corresponding to the elementcis given by Table 6.
Next choose theccolumn of Table 1:
Copy this column to the index column to get Table 8:
Against the index entryain Table 8 copy the corresponding row in Table 1, against the index entrybcopy the corresponding row in Table 1, etc., and construct Table 9.
The index entries in the first column of Table 9 are now deleted to get Table 10:
The Cayley table of the binary operation '∘{\displaystyle \circ }' corresponding to the elementcis given by Table 11.
One can verify that the entries in the various cells in Table 6 agrees with the entries in the corresponding cells of Table 11. This shows thatx· (c·y) = (x·c) ·yfor allxandyinA. If there were some discrepancy then it would not be true thatx· (c·y) = (x·c) ·yfor allxandyinA.
Thatx· (e·y) = (x·e) ·yfor allxandyinAcan be verified in a similar way by constructing the following tables (Table 12 and Table 13):
It is not necessary to construct the Cayley tables (Table 6 and table 11) of the binary operations '⋆{\displaystyle \star }' and '∘{\displaystyle \circ }'. It is enough to copy the column corresponding to the headercin Table 1 to the index column in Table 5 and form the following table (Table 14) and verify that thea-row of Table 14 is identical with thea-row of Table 1, theb-row of Table 14 is identical with theb-row of Table 1, etc. This is to be repeatedmutatis mutandisfor all the elements of the generating set ofA.
Computer softwarecan be written to carry out Light's associativity test. Kehayopulu and Argyris have developed such a program forMathematica.[1]
Light's associativity test can be extended to test associativity in a more general context.[2][3]
LetT= {t1,t2,…{\displaystyle \ldots },tm} be amagmain which the operation is denoted byjuxtaposition. LetX= {x1,x2,…{\displaystyle \ldots },xn} be a set. Let there be a mapping from theCartesian productT×XtoXdenoted by (t,x) ↦txand let it be required to test whether this map has the property
A generalization of Light's associativity test can be applied to verify whether the above property holds or not. In mathematical notations, the generalization runs as follows: For eachtinT, letL(t) be them×nmatrix of elements ofXwhosei- th row is
and letR(t) be them×nmatrix of elements ofX, the elements of whosej- th column are
According to the generalised test (due to Bednarek), that the property to be verified holds if and only ifL(t) =R(t) for alltinT. WhenX=T, Bednarek's test reduces to Light's test.
There is a randomized algorithm by Rajagopalan andSchulmanto test associativity in time proportional to the input size. (The method also works for testing certain other identities.) Specifically, the runtime isO(n2log1δ){\displaystyle O(n^{2}\log {\frac {1}{\delta }})}for ann×n{\displaystyle n\times n}table and error probabilityδ{\displaystyle \delta }.
The algorithm can be modified to produce a triple⟨a,b,c⟩{\displaystyle \langle a,b,c\rangle }for which(ab)c≠a(bc){\displaystyle (ab)c\neq a(bc)}, if there is one, in timeO(n2logn⋅log1δ){\displaystyle O(n^{2}\log n\cdot \log {\frac {1}{\delta }})}.[4]
|
https://en.wikipedia.org/wiki/Light%27s_associativity_test
|
Inmathematics, atelescoping seriesis aserieswhose general termtn{\displaystyle t_{n}}is of the formtn=an+1−an{\displaystyle t_{n}=a_{n+1}-a_{n}}, i.e. the difference of two consecutive terms of asequence(an){\displaystyle (a_{n})}. As a consequence the partial sums of the series only consists of two terms of(an){\displaystyle (a_{n})}after cancellation.[1][2]
The cancellation technique, with part of each term cancelling with part of the next term, is known as themethod of differences.
An early statement of the formula for the sum or partial sums of a telescoping series can be found in a 1644 work byEvangelista Torricelli,De dimensione parabolae.[3]
Telescopingsumsare finite sums in which pairs of consecutive terms partly cancel each other, leaving only parts of the initial and final terms.[1][4]Letan{\displaystyle a_{n}}be the elements of a sequence of numbers. Then∑n=1N(an−an−1)=aN−a0.{\displaystyle \sum _{n=1}^{N}\left(a_{n}-a_{n-1}\right)=a_{N}-a_{0}.}Ifan{\displaystyle a_{n}}converges to a limitL{\displaystyle L}, the telescopingseriesgives:∑n=1∞(an−an−1)=L−a0.{\displaystyle \sum _{n=1}^{\infty }\left(a_{n}-a_{n-1}\right)=L-a_{0}.}
Every series is a telescoping series of its own partial sums.[5]
Inprobability theory, aPoisson processis a stochastic process of which the simplest case involves "occurrences" at random times, the waiting time until the next occurrence having amemorylessexponential distribution, and the number of "occurrences" in any time interval having aPoisson distributionwhose expected value is proportional to the length of the time interval. LetXtbe the number of "occurrences" before timet, and letTxbe the waiting time until thexth "occurrence". We seek theprobability density functionof therandom variableTx. We use theprobability mass functionfor the Poisson distribution, which tells us that
where λ is the average number of occurrences in any time interval of length 1. Observe that the event {Xt≥ x} is the same as the event {Tx≤t}, and thus they have the same probability. Intuitively, if something occurs at leastx{\displaystyle x}times before timet{\displaystyle t}, we have to wait at mostt{\displaystyle t}for thexth{\displaystyle xth}occurrence. The density function we seek is therefore
The sum telescopes, leaving
For other applications, see:
Atelescoping productis a finiteproduct(or the partial product of an infinite product) that can be canceled by the method of quotients to be eventually only a finite number of factors.[7][8]It is the finite products in which consecutive terms cancel denominator with numerator, leaving only the initial and final terms. Letan{\displaystyle a_{n}}be a sequence of numbers. Then,∏n=1Nan−1an=a0aN.{\displaystyle \prod _{n=1}^{N}{\frac {a_{n-1}}{a_{n}}}={\frac {a_{0}}{a_{N}}}.}Ifan{\displaystyle a_{n}}converges to 1, the resulting product gives:∏n=1∞an−1an=a0{\displaystyle \prod _{n=1}^{\infty }{\frac {a_{n-1}}{a_{n}}}=a_{0}}
For example, the infinite product[7]∏n=2∞(1−1n2){\displaystyle \prod _{n=2}^{\infty }\left(1-{\frac {1}{n^{2}}}\right)}simplifies as∏n=2∞(1−1n2)=∏n=2∞(n−1)(n+1)n2=limN→∞∏n=2Nn−1n×∏n=2Nn+1n=limN→∞[12×23×34×⋯×N−1N]×[32×43×54×⋯×NN−1×N+1N]=limN→∞[12]×[N+1N]=12×limN→∞[N+1N]=12.{\displaystyle {\begin{aligned}\prod _{n=2}^{\infty }\left(1-{\frac {1}{n^{2}}}\right)&=\prod _{n=2}^{\infty }{\frac {(n-1)(n+1)}{n^{2}}}\\&=\lim _{N\to \infty }\prod _{n=2}^{N}{\frac {n-1}{n}}\times \prod _{n=2}^{N}{\frac {n+1}{n}}\\&=\lim _{N\to \infty }\left\lbrack {{\frac {1}{2}}\times {\frac {2}{3}}\times {\frac {3}{4}}\times \cdots \times {\frac {N-1}{N}}}\right\rbrack \times \left\lbrack {{\frac {3}{2}}\times {\frac {4}{3}}\times {\frac {5}{4}}\times \cdots \times {\frac {N}{N-1}}\times {\frac {N+1}{N}}}\right\rbrack \\&=\lim _{N\to \infty }\left\lbrack {\frac {1}{2}}\right\rbrack \times \left\lbrack {\frac {N+1}{N}}\right\rbrack \\&={\frac {1}{2}}\times \lim _{N\to \infty }\left\lbrack {\frac {N+1}{N}}\right\rbrack \\&={\frac {1}{2}}.\end{aligned}}}
|
https://en.wikipedia.org/wiki/Telescoping_series
|
Inmathematics, aseriesis, roughly speaking, anadditionofinfinitelymanyterms, one after the other.[1]The study of series is a major part ofcalculusand its generalization,mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures incombinatoricsthroughgenerating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such asphysics,computer science,statisticsandfinance.
Among theAncient Greeks, the idea that apotentially infinitesummationcould produce a finite result was consideredparadoxical, most famously inZeno's paradoxes.[2][3]Nonetheless, infinite series were applied practically by Ancient Greek mathematicians includingArchimedes, for instance in thequadrature of the parabola.[4][5]The mathematical side of Zeno's paradoxes was resolved using the concept of alimitduring the 17th century, especially through the early calculus ofIsaac Newton.[6]The resolution was made more rigorous and further improved in the 19th century through the work ofCarl Friedrich GaussandAugustin-Louis Cauchy,[7]among others, answering questions about which of these sums exist via thecompleteness of the real numbersand whether series terms can be rearranged or not without changing their sums usingabsolute convergenceandconditional convergenceof series.
In modern terminology, any orderedinfinite sequence(a1,a2,a3,…){\displaystyle (a_{1},a_{2},a_{3},\ldots )}of terms, whether those terms are numbers,functions,matrices, or anything else that can be added, defines a series, which is the addition of theai{\displaystyle a_{i}}one after the other. To emphasize that there are an infinite number of terms, series are often also calledinfinite seriesto contrast withfinite series, a term sometimes used forfinite sums. Series are represented by anexpressionlikea1+a2+a3+⋯,{\displaystyle a_{1}+a_{2}+a_{3}+\cdots ,}or, usingcapital-sigma summation notation,[8]∑i=1∞ai.{\displaystyle \sum _{i=1}^{\infty }a_{i}.}
The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in a finite amount of time. However, if the terms and their finite sums belong to asetthat haslimits, it may be possible to assign a value to a series, called thesum of the series. This value is the limit asn{\displaystyle n}tends toinfinityof the finite sums of then{\displaystyle n}first terms of the series if the limit exists.[9][10][11]These finite sums are called thepartial sumsof the series. Using summation notation,∑i=1∞ai=limn→∞∑i=1nai,{\displaystyle \sum _{i=1}^{\infty }a_{i}=\lim _{n\to \infty }\,\sum _{i=1}^{n}a_{i},}if it exists.[9][10][11]When the limit exists, the series isconvergentorsummableand also the sequence(a1,a2,a3,…){\displaystyle (a_{1},a_{2},a_{3},\ldots )}issummable, and otherwise, when the limit does not exist, the series isdivergent.[9][10][11]
The expression∑i=1∞ai{\textstyle \sum _{i=1}^{\infty }a_{i}}denotes both the series—the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the explicit limit of the process. This is a generalization of the similar convention of denoting bya+b{\displaystyle a+b}both theaddition—the process of adding—and its result—thesumofa{\displaystyle a}andb{\displaystyle b}.
Commonly, the terms of a series come from aring, often thefieldR{\displaystyle \mathbb {R} }of thereal numbersor the fieldC{\displaystyle \mathbb {C} }of thecomplex numbers. If so, the set of all series is also itself a ring, one in which the addition consists of adding series terms together term by term and the multiplication is theCauchy product.[12][13][14]
Aseriesor, redundantly, aninfinite series, is an infinite sum. It is often represented as[8][15][16]a0+a1+a2+⋯ora1+a2+a3+⋯,{\displaystyle a_{0}+a_{1}+a_{2}+\cdots \quad {\text{or}}\quad a_{1}+a_{2}+a_{3}+\cdots ,}where thetermsak{\displaystyle a_{k}}are the members of asequenceofnumbers,functions, or anything else that can beadded. A series may also be represented withcapital-sigma notation:[8][16]∑k=0∞akor∑k=1∞ak.{\displaystyle \sum _{k=0}^{\infty }a_{k}\qquad {\text{or}}\qquad \sum _{k=1}^{\infty }a_{k}.}
It is also common to express series using a few first terms, an ellipsis, a general term, and then a final ellipsis, the general term being an expression of then{\displaystyle n}th term as afunctionofn{\displaystyle n}:a0+a1+a2+⋯+an+⋯orf(0)+f(1)+f(2)+⋯+f(n)+⋯.{\displaystyle a_{0}+a_{1}+a_{2}+\cdots +a_{n}+\cdots \quad {\text{ or }}\quad f(0)+f(1)+f(2)+\cdots +f(n)+\cdots .}For example,Euler's numbercan be defined with the series∑n=0∞1n!=1+1+12+16+⋯+1n!+⋯,{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!}}=1+1+{\frac {1}{2}}+{\frac {1}{6}}+\cdots +{\frac {1}{n!}}+\cdots ,}wheren!{\displaystyle n!}denotes the product of then{\displaystyle n}firstpositive integers, and0!{\displaystyle 0!}is conventionally equal to1.{\displaystyle 1.}[17][18][19]
Given a seriess=∑k=0∞ak{\textstyle s=\sum _{k=0}^{\infty }a_{k}}, itsn{\displaystyle n}thpartial sumis[9][10][11][16]sn=∑k=0nak=a0+a1+⋯+an.{\displaystyle s_{n}=\sum _{k=0}^{n}a_{k}=a_{0}+a_{1}+\cdots +a_{n}.}
Some authors directly identify a series with its sequence of partial sums.[9][11]Either the sequence of partial sums or the sequence of terms completely characterizes the series, and the sequence of terms can be recovered from the sequence of partial sums by taking the differences between consecutive elements,an=sn−sn−1.{\displaystyle a_{n}=s_{n}-s_{n-1}.}
Partial summation of a sequence is an example of a linearsequence transformation, and it is also known as theprefix sumincomputer science. The inverse transformation for recovering a sequence from its partial sums is thefinite difference, another linear sequence transformation.
Partial sums of series sometimes have simpler closed form expressions, for instance anarithmetic serieshas partial sumssn=∑k=0n(a+kd)=a+(a+d)+(a+2d)+⋯+(a+nd)=(n+1)(a+12nd),{\displaystyle s_{n}=\sum _{k=0}^{n}\left(a+kd\right)=a+(a+d)+(a+2d)+\cdots +(a+nd)=(n+1){\bigl (}a+{\tfrac {1}{2}}nd{\bigr )},}and ageometric serieshas partial sums[20][21][22]sn=∑k=0nark=a+ar+ar2+⋯+arn=a1−rn+11−r{\displaystyle s_{n}=\sum _{k=0}^{n}ar^{k}=a+ar+ar^{2}+\cdots +ar^{n}=a{\frac {1-r^{n+1}}{1-r}}}ifr≠1{\displaystyle r\neq 1}or simplysn=a(n+1){\displaystyle s_{n}=a(n+1)}ifr=1{\displaystyle r=1}.
Strictly speaking, a series is said toconverge, to beconvergent, or to besummablewhen the sequence of its partial sums has alimit. When the limit of the sequence of partial sums does not exist, the seriesdivergesor isdivergent.[23]When the limit of the partial sums exists, it is called thesum of the seriesorvalue of the series:[9][10][11][16]∑k=0∞ak=limn→∞∑k=0nak=limn→∞sn.{\displaystyle \sum _{k=0}^{\infty }a_{k}=\lim _{n\to \infty }\sum _{k=0}^{n}a_{k}=\lim _{n\to \infty }s_{n}.}A series with only a finite number of nonzero terms is always convergent. Such series are useful for considering finite sums without taking care of the numbers of terms.[24]When the sum exists, the difference between the sum of a series and itsn{\displaystyle n}th partial sum,s−sn=∑k=n+1∞ak,{\textstyle s-s_{n}=\sum _{k=n+1}^{\infty }a_{k},}is known as then{\displaystyle n}thtruncation errorof the infinite series.[25][26]
An example of a convergent series is the geometric series1+12+14+18+⋯+12k+⋯.{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{k}}}+\cdots .}
It can be shown by algebraic computation that each partial sumsn{\displaystyle s_{n}}is∑k=0n12k=2−12n.{\displaystyle \sum _{k=0}^{n}{\frac {1}{2^{k}}}=2-{\frac {1}{2^{n}}}.}As one haslimn→∞(2−12n)=2,{\displaystyle \lim _{n\to \infty }\left(2-{\frac {1}{2^{n}}}\right)=2,}the series is convergent and converges to2{\displaystyle 2}with truncation errors1/2n{\textstyle 1/2^{n}}.[20][21][22]
By contrast, the geometric series∑k=0∞2k{\displaystyle \sum _{k=0}^{\infty }2^{k}}is divergent in thereal numbers.[20][21][22]However, it is convergent in theextended real number line, with+∞{\displaystyle +\infty }as its limit and+∞{\displaystyle +\infty }as its truncation error at every step.[27]
When a series's sequence of partial sums is not easily calculated and evaluated for convergence directly,convergence testscan be used to prove that the series converges or diverges.
In ordinaryfinite summations, terms of the summation can be grouped and ungrouped freely without changing the result of the summation as a consequence of theassociativityof addition.a0+a1+a2={\displaystyle a_{0}+a_{1}+a_{2}={}}a0+(a1+a2)={\displaystyle a_{0}+(a_{1}+a_{2})={}}(a0+a1)+a2.{\displaystyle (a_{0}+a_{1})+a_{2}.}Similarly, in a series, any finite groupings of terms of the series will not change the limit of the partial sums of the series and thus will not change the sum of the series. However, if an infinite number of groupings is performed in an infinite series, then the partial sums of the grouped series may have a different limit than the original series and different groupings may have different limits from one another; the sum ofa0+a1+a2+⋯{\displaystyle a_{0}+a_{1}+a_{2}+\cdots }may not equal the sum ofa0+(a1+a2)+{\displaystyle a_{0}+(a_{1}+a_{2})+{}}(a3+a4)+⋯.{\displaystyle (a_{3}+a_{4})+\cdots .}
For example,Grandi's series1−1+1−1+⋯{\displaystyle 1-1+1-1+\cdots }has a sequence of partial sums that alternates back and forth between1{\displaystyle 1}and0{\displaystyle 0}and does not converge. Grouping its elements in pairs creates the series(1−1)+(1−1)+(1−1)+⋯={\displaystyle (1-1)+(1-1)+(1-1)+\cdots ={}}0+0+0+⋯,{\displaystyle 0+0+0+\cdots ,}which has partial sums equal to zero at every term and thus sums to zero. Grouping its elements in pairs starting after the first creates the series1+(−1+1)+{\displaystyle 1+(-1+1)+{}}(−1+1)+⋯={\displaystyle (-1+1)+\cdots ={}}1+0+0+⋯,{\displaystyle 1+0+0+\cdots ,}which has partial sums equal to one for every term and thus sums to one, a different result.
In general, grouping the terms of a series creates a new series with a sequence of partial sums that is asubsequenceof the partial sums of the original series. This means that if the original series converges, so does the new series after grouping: all infinite subsequences of a convergent sequence also converge to the same limit. However, if the original series diverges, then the grouped series do not necessarily diverge, as in this example of Grandi's series above. However, divergence of a grouped series does imply the original series must be divergent, since it proves there is a subsequence of the partial sums of the original series which is not convergent, which would be impossible if it were convergent. This reasoning was applied inOresme's proof of the divergence of the harmonic series,[28]and it is the basis for the generalCauchy condensation test.[29][30]
In ordinary finite summations, terms of the summation can be rearranged freely without changing the result of the summation as a consequence of thecommutativityof addition.a0+a1+a2={\displaystyle a_{0}+a_{1}+a_{2}={}}a0+a2+a1={\displaystyle a_{0}+a_{2}+a_{1}={}}a2+a1+a0.{\displaystyle a_{2}+a_{1}+a_{0}.}Similarly, in a series, any finite rearrangements of terms of a series does not change the limit of the partial sums of the series and thus does not change the sum of the series: for any finite rearrangement, there will be some term after which the rearrangement did not affect any further terms: any effects of rearrangement can be isolated to the finite summation up to that term, and finite summations do not change under rearrangement.
However, as for grouping, an infinitary rearrangement of terms of a series can sometimes lead to a change in the limit of the partial sums of the series. Series with sequences of partial sums that converge to a value but whose terms could be rearranged to a form a series with partial sums that converge to some other value are calledconditionally convergentseries. Those that converge to the same value regardless of rearrangement are calledunconditionally convergentseries.
For series of real numbers and complex numbers, a seriesa0+a1+a2+⋯{\displaystyle a_{0}+a_{1}+a_{2}+\cdots }is unconditionally convergentif and only ifthe series summing theabsolute valuesof its terms,|a0|+|a1|+|a2|+⋯,{\displaystyle |a_{0}|+|a_{1}|+|a_{2}|+\cdots ,}is also convergent, a property calledabsolute convergence. Otherwise, any series of real numbers or complex numbers that converges but does not converge absolutely is conditionally convergent. Any conditionally convergent sum of real numbers can be rearranged to yield any other real number as a limit, or to diverge. These claims are the content of theRiemann series theorem.[31][32][33]
A historically important example of conditional convergence is thealternating harmonic series,
∑n=1∞(−1)n+1n=1−12+13−14+15−⋯,{\displaystyle \sum \limits _{n=1}^{\infty }{(-1)^{n+1} \over n}=1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots ,}which has a sum of thenatural logarithm of 2, while the sum of the absolute values of the terms is theharmonic series,∑n=1∞1n=1+12+13+14+15+⋯,{\displaystyle \sum \limits _{n=1}^{\infty }{1 \over n}=1+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 5}+\cdots ,}which diverges per the divergence of the harmonic series,[28]so the alternating harmonic series is conditionally convergent. For instance, rearranging the terms of the alternating harmonic series so that each positive term of the original series is followed by two negative terms of the original series rather than just one yields[34]1−12−14+13−16−18+15−110−112+⋯=(1−12)−14+(13−16)−18+(15−110)−112+⋯=12−14+16−18+110−112+⋯=12(1−12+13−14+15−16+⋯),{\displaystyle {\begin{aligned}&1-{\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{3}}-{\frac {1}{6}}-{\frac {1}{8}}+{\frac {1}{5}}-{\frac {1}{10}}-{\frac {1}{12}}+\cdots \\[3mu]&\quad =\left(1-{\frac {1}{2}}\right)-{\frac {1}{4}}+\left({\frac {1}{3}}-{\frac {1}{6}}\right)-{\frac {1}{8}}+\left({\frac {1}{5}}-{\frac {1}{10}}\right)-{\frac {1}{12}}+\cdots \\[3mu]&\quad ={\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{6}}-{\frac {1}{8}}+{\frac {1}{10}}-{\frac {1}{12}}+\cdots \\[3mu]&\quad ={\frac {1}{2}}\left(1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}+\cdots \right),\end{aligned}}}which is12{\displaystyle {\tfrac {1}{2}}}times the original series, so it would have a sum of half of the natural logarithm of 2. By the Riemann series theorem, rearrangements of the alternating harmonic series to yield any other real number are also possible.
The addition of two seriesa0+a1+a2+⋯{\textstyle a_{0}+a_{1}+a_{2}+\cdots }andb0+b1+b2+⋯{\textstyle b_{0}+b_{1}+b_{2}+\cdots }is given by the termwise sum[13][35][36][37](a0+b0)+(a1+b1)+(a2+b2)+⋯{\textstyle (a_{0}+b_{0})+(a_{1}+b_{1})+(a_{2}+b_{2})+\cdots \,}, or, in summation notation,∑k=0∞ak+∑k=0∞bk=∑k=0∞ak+bk.{\displaystyle \sum _{k=0}^{\infty }a_{k}+\sum _{k=0}^{\infty }b_{k}=\sum _{k=0}^{\infty }a_{k}+b_{k}.}
Using the symbolssa,n{\displaystyle s_{a,n}}andsb,n{\displaystyle s_{b,n}}for the partial sums of the added series andsa+b,n{\displaystyle s_{a+b,n}}for the partial sums of the resulting series, this definition implies the partial sums of the resulting series followsa+b,n=sa,n+sb,n.{\displaystyle s_{a+b,n}=s_{a,n}+s_{b,n}.}Then the sum of the resulting series, i.e., the limit of the sequence of partial sums of the resulting series, satisfieslimn→∞sa+b,n=limn→∞(sa,n+sb,n)=limn→∞sa,n+limn→∞sb,n,{\displaystyle \lim _{n\rightarrow \infty }s_{a+b,n}=\lim _{n\rightarrow \infty }(s_{a,n}+s_{b,n})=\lim _{n\rightarrow \infty }s_{a,n}+\lim _{n\rightarrow \infty }s_{b,n},}when the limits exist. Therefore, first, the series resulting from addition is summable if the series added were summable, and, second, the sum of the resulting series is the addition of the sums of the added series. The addition of two divergent series may yield a convergent series: for instance, the addition of a divergent series with a series of its terms times−1{\displaystyle -1}will yield a series of all zeros that converges to zero. However, for any two series where one converges and the other diverges, the result of their addition diverges.[35]
For series of real numbers or complex numbers, series addition isassociative,commutative, andinvertible. Therefore series addition gives the sets of convergent series of real numbers or complex numbers the structure of anabelian groupand also gives the sets of all series of real numbers or complex numbers (regardless of convergence properties) the structure of an abelian group.
The product of a seriesa0+a1+a2+⋯{\textstyle a_{0}+a_{1}+a_{2}+\cdots }with a constant numberc{\displaystyle c}, called ascalarin this context, is given by the termwise product[35]ca0+ca1+ca2+⋯{\textstyle ca_{0}+ca_{1}+ca_{2}+\cdots }, or, in summation notation,
c∑k=0∞ak=∑k=0∞cak.{\displaystyle c\sum _{k=0}^{\infty }a_{k}=\sum _{k=0}^{\infty }ca_{k}.}
Using the symbolssa,n{\displaystyle s_{a,n}}for the partial sums of the original series andsca,n{\displaystyle s_{ca,n}}for the partial sums of the series after multiplication byc{\displaystyle c}, this definition implies thatsca,n=csa,n{\displaystyle s_{ca,n}=cs_{a,n}}for alln,{\displaystyle n,}and therefore alsolimn→∞sca,n=climn→∞sa,n,{\textstyle \lim _{n\rightarrow \infty }s_{ca,n}=c\lim _{n\rightarrow \infty }s_{a,n},}when the limits exist. Therefore if a series is summable, any nonzero scalar multiple of the series is also summable and vice versa: if a series is divergent, then any nonzero scalar multiple of it is also divergent.
Scalar multiplication of real numbers and complex numbers is associative, commutative, invertible, and itdistributes overseries addition.
In summary, series addition and scalar multiplication gives the set of convergent series and the set of series of real numbers the structure of areal vector space. Similarly, one getscomplex vector spacesfor series and convergent series of complex numbers. All these vector spaces are infinite dimensional.
The multiplication of two seriesa0+a1+a2+⋯{\displaystyle a_{0}+a_{1}+a_{2}+\cdots }andb0+b1+b2+⋯{\displaystyle b_{0}+b_{1}+b_{2}+\cdots }to generate a third seriesc0+c1+c2+⋯{\displaystyle c_{0}+c_{1}+c_{2}+\cdots }, called the Cauchy product,[12][13][14][36][38]can be written in summation notation(∑k=0∞ak)⋅(∑k=0∞bk)=∑k=0∞ck=∑k=0∞∑j=0kajbk−j,{\displaystyle {\biggl (}\sum _{k=0}^{\infty }a_{k}{\biggr )}\cdot {\biggl (}\sum _{k=0}^{\infty }b_{k}{\biggr )}=\sum _{k=0}^{\infty }c_{k}=\sum _{k=0}^{\infty }\sum _{j=0}^{k}a_{j}b_{k-j},}with eachck=∑j=0kajbk−j={\textstyle c_{k}=\sum _{j=0}^{k}a_{j}b_{k-j}={}\!}a0bk+a1bk−1+⋯+ak−1b1+akb0.{\displaystyle \!a_{0}b_{k}+a_{1}b_{k-1}+\cdots +a_{k-1}b_{1}+a_{k}b_{0}.}Here, the convergence of the partial sums of the seriesc0+c1+c2+⋯{\displaystyle c_{0}+c_{1}+c_{2}+\cdots }is not as simple to establish as for addition. However, if both seriesa0+a1+a2+⋯{\displaystyle a_{0}+a_{1}+a_{2}+\cdots }andb0+b1+b2+⋯{\displaystyle b_{0}+b_{1}+b_{2}+\cdots }areabsolutely convergentseries, then the series resulting from multiplying them also converges absolutely with a sum equal to the product of the two sums of the multiplied series,[13][36][39]limn→∞sc,n=(limn→∞sa,n)⋅(limn→∞sb,n).{\displaystyle \lim _{n\rightarrow \infty }s_{c,n}=\left(\,\lim _{n\rightarrow \infty }s_{a,n}\right)\cdot \left(\,\lim _{n\rightarrow \infty }s_{b,n}\right).}
Series multiplication of absolutely convergent series of real numbers and complex numbers is associative, commutative, and distributes over series addition. Together with series addition, series multiplication gives the sets of absolutely convergent series of real numbers or complex numbers the structure of acommutativering, and together with scalar multiplication as well, the structure of acommutative algebra; these operations also give the sets of all series of real numbers or complex numbers the structure of anassociative algebra.
∑n=1∞1n2=112+122+132+142+⋯=π26{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}
∑n=1∞(−1)n+1(4)2n−1=41−43+45−47+49−411+413−⋯=π{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}(4)}{2n-1}}={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}-\cdots =\pi }
∑n=1∞(−1)n+1n=ln2{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=\ln 2}
∑n=1∞12nn=ln2{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2^{n}n}}=\ln 2}
∑n=0∞(−1)nn!=1−11!+12!−13!+⋯=1e{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}=1-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+\cdots ={\frac {1}{e}}}
∑n=0∞1n!=10!+11!+12!+13!+14!+⋯=e{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots =e}
One of the simplest tests for convergence of a series, applicable to all series, is thevanishing conditionorn{\displaystyle n}th-term test: Iflimn→∞an≠0{\textstyle \lim _{n\to \infty }a_{n}\neq 0}, then the series diverges; iflimn→∞an=0{\textstyle \lim _{n\to \infty }a_{n}=0}, then the test is inconclusive.[46][47]
When every term of a series is a non-negative real number, for instance when the terms are theabsolute valuesof another series of real numbers or complex numbers, the sequence of partial sums is non-decreasing. Therefore a series with non-negative terms converges if and only if the sequence of partial sums is bounded, and so finding a bound for a series or for the absolute values of its terms is an effective way to prove convergence or absolute convergence of a series.[48][49][47][50]
For example, the series1+14+19+⋯+1n2+⋯{\textstyle 1+{\frac {1}{4}}+{\frac {1}{9}}+\cdots +{\frac {1}{n^{2}}}+\cdots \,}is convergent and absolutely convergent because1n2≤1n−1−1n{\textstyle {\frac {1}{n^{2}}}\leq {\frac {1}{n-1}}-{\frac {1}{n}}}for alln≥2{\displaystyle n\geq 2}and atelescoping sumargument implies that the partial sums of the series of those non-negative bounding terms are themselves bounded above by 2.[43]The exact value of this series is16π2{\textstyle {\frac {1}{6}}\pi ^{2}}; seeBasel problem.
This type of bounding strategy is the basis for general series comparison tests. First is the generaldirect comparison test:[51][52][47]For any series∑an{\textstyle \sum a_{n}}, If∑bn{\textstyle \sum b_{n}}is anabsolutely convergentseries such that|an|≤C|bn|{\displaystyle \left\vert a_{n}\right\vert \leq C\left\vert b_{n}\right\vert }for some positive real numberC{\displaystyle C}and for sufficiently largen{\displaystyle n}, then∑an{\textstyle \sum a_{n}}converges absolutely as well. If∑|bn|{\textstyle \sum \left\vert b_{n}\right\vert }diverges, and|an|≥|bn|{\displaystyle \left\vert a_{n}\right\vert \geq \left\vert b_{n}\right\vert }for all sufficiently largen{\displaystyle n}, then∑an{\textstyle \sum a_{n}}also fails to converge absolutely, although it could still be conditionally convergent, for example, if thean{\displaystyle a_{n}}alternate in sign. Second is the generallimit comparison test:[53][54]If∑bn{\textstyle \sum b_{n}}is an absolutely convergent series such that|an+1an|≤|bn+1bn|{\displaystyle \left\vert {\tfrac {a_{n+1}}{a_{n}}}\right\vert \leq \left\vert {\tfrac {b_{n+1}}{b_{n}}}\right\vert }for sufficiently largen{\displaystyle n}, then∑an{\textstyle \sum a_{n}}converges absolutely as well. If∑|bn|{\textstyle \sum \left|b_{n}\right|}diverges, and|an+1an|≥|bn+1bn|{\displaystyle \left\vert {\tfrac {a_{n+1}}{a_{n}}}\right\vert \geq \left\vert {\tfrac {b_{n+1}}{b_{n}}}\right\vert }for all sufficiently largen{\displaystyle n}, then∑an{\textstyle \sum a_{n}}also fails to converge absolutely, though it could still be conditionally convergent if thean{\displaystyle a_{n}}vary in sign.
Using comparisons togeometric seriesspecifically,[20][21]those two general comparison tests imply two further common and generally useful tests for convergence of series with non-negative terms or for absolute convergence of series with general terms. First is theratio test:[55][56][57]if there exists a constantC<1{\displaystyle C<1}such that|an+1an|<C{\displaystyle \left\vert {\tfrac {a_{n+1}}{a_{n}}}\right\vert <C}for all sufficiently largen{\displaystyle n}, then∑an{\textstyle \sum a_{n}}converges absolutely. When the ratio is less than1{\displaystyle 1}, but not less than a constant less than1{\displaystyle 1}, convergence is possible but this test does not establish it. Second is theroot test:[55][58][59]if there exists a constantC<1{\displaystyle C<1}such that|an|1/n≤C{\displaystyle \textstyle \left\vert a_{n}\right\vert ^{1/n}\leq C}for all sufficiently largen{\displaystyle n}, then∑an{\textstyle \sum a_{n}}converges absolutely.
Alternatively, using comparisons to series representations ofintegralsspecifically, one derives theintegral test:[60][61]iff(x){\displaystyle f(x)}is a positivemonotone decreasingfunction defined on theinterval[1,∞){\displaystyle [1,\infty )}then for a series with termsan=f(n){\displaystyle a_{n}=f(n)}for alln{\displaystyle n},∑an{\textstyle \sum a_{n}}converges if and only if theintegral∫1∞f(x)dx{\textstyle \int _{1}^{\infty }f(x)\,dx}is finite. Using comparisons to flattened-out versions of a series leads toCauchy's condensation test:[29][30]if the sequence of termsan{\displaystyle a_{n}}is non-negative and non-increasing, then the two series∑an{\textstyle \sum a_{n}}and∑2ka(2k){\textstyle \sum 2^{k}a_{(2^{k})}}are either both convergent or both divergent.
A series of real or complex numbers is said to beconditionally convergent(orsemi-convergent) if it is convergent but not absolutely convergent. Conditional convergence is tested for differently than absolute convergence.
One important example of a test for conditional convergence is thealternating series testorLeibniz test:[62][63][64]A series of the form∑(−1)nan{\textstyle \sum (-1)^{n}a_{n}}with allan>0{\displaystyle a_{n}>0}is calledalternating. Such a series converges if the non-negativesequencean{\displaystyle a_{n}}ismonotone decreasingand converges to0{\displaystyle 0}. The converse is in general not true. A famous example of an application of this test is thealternating harmonic series∑n=1∞(−1)n+1n=1−12+13−14+15−⋯,{\displaystyle \sum \limits _{n=1}^{\infty }{(-1)^{n+1} \over n}=1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots ,}which is convergent per the alternating series test (and its sum is equal toln2{\displaystyle \ln 2}), though the series formed by taking the absolute value of each term is the ordinaryharmonic series, which is divergent.[65][66]
The alternating series test can be viewed as a special case of the more generalDirichlet's test:[67][68][69]if(an){\displaystyle (a_{n})}is a sequence of terms of decreasing nonnegative real numbers that converges to zero, and(λn){\displaystyle (\lambda _{n})}is a sequence of terms with bounded partial sums, then the series∑λnan{\textstyle \sum \lambda _{n}a_{n}}converges. Takingλn=(−1)n{\displaystyle \lambda _{n}=(-1)^{n}}recovers the alternating series test.
Abel's testis another important technique for handling semi-convergent series.[67][29]If a series has the form∑an=∑λnbn{\textstyle \sum a_{n}=\sum \lambda _{n}b_{n}}where the partial sums of the series with termsbn{\displaystyle b_{n}},sb,n=b0+⋯+bn{\displaystyle s_{b,n}=b_{0}+\cdots +b_{n}}are bounded,λn{\displaystyle \lambda _{n}}hasbounded variation, andlimλnbn{\displaystyle \lim \lambda _{n}b_{n}}exists: ifsupn|sb,n|<∞,{\textstyle \sup _{n}|s_{b,n}|<\infty ,}∑|λn+1−λn|<∞,{\textstyle \sum \left|\lambda _{n+1}-\lambda _{n}\right|<\infty ,}andλnsb,n{\displaystyle \lambda _{n}s_{b,n}}converges, then the series∑an{\textstyle \sum a_{n}}is convergent.
Other specialized convergence tests for specific types of series include theDini test[70]forFourier series.
The evaluation of truncation errors of series is important innumerical analysis(especiallyvalidated numericsandcomputer-assisted proof). It can be used to prove convergence and to analyzerates of convergence.
When conditions of thealternating series testare satisfied byS:=∑m=0∞(−1)mum{\textstyle S:=\sum _{m=0}^{\infty }(-1)^{m}u_{m}}, there is an exact error evaluation.[71]Setsn{\displaystyle s_{n}}to be the partial sumsn:=∑m=0n(−1)mum{\textstyle s_{n}:=\sum _{m=0}^{n}(-1)^{m}u_{m}}of the given alternating seriesS{\displaystyle S}. Then the next inequality holds:|S−sn|≤un+1.{\displaystyle |S-s_{n}|\leq u_{n+1}.}
By using theratio, we can obtain the evaluation of the error term when thehypergeometric seriesis truncated.[72]
For thematrix exponential:
exp(X):=∑k=0∞1k!Xk,X∈Cn×n,{\displaystyle \exp(X):=\sum _{k=0}^{\infty }{\frac {1}{k!}}X^{k},\quad X\in \mathbb {C} ^{n\times n},}
the following error evaluation holds (scaling and squaring method):[73][74][75]
Tr,s(X):=(∑j=0r1j!(X/s)j)s,‖exp(X)−Tr,s(X)‖≤‖X‖r+1sr(r+1)!exp(‖X‖).{\displaystyle T_{r,s}(X):={\biggl (}\sum _{j=0}^{r}{\frac {1}{j!}}(X/s)^{j}{\biggr )}^{s},\quad {\bigl \|}\exp(X)-T_{r,s}(X){\bigr \|}\leq {\frac {\|X\|^{r+1}}{s^{r}(r+1)!}}\exp(\|X\|).}
Under many circumstances, it is desirable to assign generalized sums to series which fail to converge in the strict sense that their sequences of partial sums do not converge. Asummation methodis any method for assigning sums to divergent series in a way that systematically extends the classical notion of the sum of a series. Summation methods includeCesàro summation,generalized Cesàro(C,α){\displaystyle (C,\alpha )}summation,Abel summation, andBorel summation, in order of applicability to increasingly divergent series. These methods are all based onsequence transformationsof the original series of terms or of its sequence of partial sums. An alternative family of summation methods are based onanalytic continuationrather than sequence transformation.
A variety of general results concerning possible summability methods are known. TheSilverman–Toeplitz theoremcharacterizesmatrix summation methods, which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients. The most general methods for summing a divergent series arenon-constructiveand concernBanach limits.
A series of real- or complex-valued functions
∑n=0∞fn(x){\displaystyle \sum _{n=0}^{\infty }f_{n}(x)}
ispointwise convergentto a limitf(x){\displaystyle f(x)}on a setE{\displaystyle E}if the series converges for eachx{\displaystyle x}inE{\displaystyle E}as a series of real or complex numbers. Equivalently, the partial sums
sN(x)=∑n=0Nfn(x){\displaystyle s_{N}(x)=\sum _{n=0}^{N}f_{n}(x)}
converge tof(x){\displaystyle f(x)}asN{\displaystyle N}goes to infinity for eachx{\displaystyle x}inE{\displaystyle E}.
A stronger notion of convergence of a series of functions isuniform convergence. A series converges uniformly in a setE{\displaystyle E}if it converges pointwise to the functionf(x){\displaystyle f(x)}at every point ofE{\displaystyle E}and the supremum of these pointwise errors in approximating the limit by theN{\displaystyle N}th partial sum,
supx∈E|sN(x)−f(x)|{\displaystyle \sup _{x\in E}{\bigl |}s_{N}(x)-f(x){\bigr |}}
converges to zero with increasingN{\displaystyle N},independentlyofx{\displaystyle x}.
Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit. For example, if a series of continuous functions converges uniformly, then the limit function is also continuous. Similarly, if thefn{\displaystyle f_{n}}areintegrableon a closed and bounded intervalI{\displaystyle I}and converge uniformly, then the series is also integrable onI{\displaystyle I}and can be integrated term by term. Tests for uniform convergence includeWeierstrass' M-test,Abel's uniform convergence test,Dini's test, and theCauchy criterion.
More sophisticated types of convergence of a series of functions can also be defined. Inmeasure theory, for instance, a series of functions convergesalmost everywhereif it converges pointwise except on a set ofmeasure zero. Othermodes of convergencedepend on a differentmetric spacestructure on thespace of functionsunder consideration. For instance, a series of functionsconverges in meanto a limit functionf{\displaystyle f}on a setE{\displaystyle E}if
limN→∞∫E|sN(x)−f(x)|2dx=0.{\displaystyle \lim _{N\rightarrow \infty }\int _{E}{\bigl |}s_{N}(x)-f(x){\bigr |}^{2}\,dx=0.}
Apower seriesis a series of the form
∑n=0∞an(x−c)n.{\displaystyle \sum _{n=0}^{\infty }a_{n}(x-c)^{n}.}
TheTaylor seriesat a pointc{\displaystyle c}of a function is a power series that, in many cases, converges to the function in a neighborhood ofc{\displaystyle c}. For example, the series
∑n=0∞xnn!{\displaystyle \sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}
is the Taylor series ofex{\displaystyle e^{x}}at the origin and converges to it for everyx{\displaystyle x}.
Unless it converges only atx=c{\displaystyle x=c}, such a series converges on a certain open disc of convergence centered at the pointc{\displaystyle c}in the complex plane, and may also converge at some of the points of the boundary of the disc. The radius of this disc is known as theradius of convergence, and can in principle be determined from the asymptotics of the coefficientsan{\displaystyle a_{n}}. The convergence is uniform onclosedandbounded(that is,compact) subsets of the interior of the disc of convergence: to wit, it isuniformly convergent on compact sets.
Historically, mathematicians such asLeonhard Euleroperated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.
While many uses of power series refer to their sums, it is also possible to treat power series asformal sums, meaning that no addition operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition. In this setting, the sequence of coefficients itself is of interest, rather than the convergence of the series. Formal power series are used incombinatoricsto describe and studysequencesthat are otherwise difficult to handle, for example, using the method ofgenerating functions. TheHilbert–Poincaré seriesis a formal power series used to studygraded algebras.
Even if the limit of the power series is not considered, if the terms support appropriate structure then it is possible to define operations such asaddition,multiplication,derivative,antiderivativefor power series "formally", treating the symbol "+" as if it corresponded to addition. In the most common setting, the terms come from acommutative ring, so that the formal power series can be added term-by-term and multiplied via theCauchy product. In this case the algebra of formal power series is thetotal algebraof themonoidofnatural numbersover the underlying term ring.[76]If the underlying term ring is adifferential algebra, then the algebra of formal power series is also a differential algebra, with differentiation performed term-by-term.
Laurent series generalize power series by admitting terms into the series with negative as well as positive exponents. A Laurent series is thus any series of the form
∑n=−∞∞anxn.{\displaystyle \sum _{n=-\infty }^{\infty }a_{n}x^{n}.}
If such a series converges, then in general it does so in anannulusrather than a disc, and possibly some boundary points. The series converges uniformly on compact subsets of the interior of the annulus of convergence.
ADirichlet seriesis one of the form
∑n=1∞anns,{\displaystyle \sum _{n=1}^{\infty }{a_{n} \over n^{s}},}
wheres{\displaystyle s}is acomplex number. For example, if allan{\displaystyle a_{n}}are equal to1{\displaystyle 1}, then the Dirichlet series is theRiemann zeta function
ζ(s)=∑n=1∞1ns.{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.}
Like the zeta function, Dirichlet series in general play an important role inanalytic number theory. Generally a Dirichlet series converges if the real part ofs{\displaystyle s}is greater than a number called the abscissa of convergence. In many cases, a Dirichlet series can be extended to ananalytic functionoutside the domain of convergence byanalytic continuation. For example, the Dirichlet series for the zeta function converges absolutely whenRe(s)>1{\displaystyle \operatorname {Re} (s)>1}, but the zeta function can be extended to a holomorphic function defined onC∖{1}{\displaystyle \mathbb {C} \setminus \{1\}}with a simplepoleat1{\displaystyle 1}.
This series can be directly generalized togeneral Dirichlet series.
A series of functions in which the terms aretrigonometric functionsis called atrigonometric series:
A0+∑n=1∞(Ancosnx+Bnsinnx).{\displaystyle A_{0}+\sum _{n=1}^{\infty }\left(A_{n}\cos nx+B_{n}\sin nx\right).}
The most important example of a trigonometric series is theFourier seriesof a function.
Asymptotic series, typically calledasymptotic expansions, are infinite series whose terms are functions of a sequence of differentasymptotic ordersand whose partial sums are approximations of some other function in anasymptotic limit. In general they do not converge, but they are still useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. They are crucial tools inperturbation theoryand in theanalysis of algorithms.
An asymptotic series cannot necessarily be made to produce an answer as exactly as desired away from the asymptotic limit, the way that an ordinary convergent series of functions can. In fact, a typical asymptotic series reaches its best practical approximation away from the asymptotic limit after a finite number of terms; if more terms are included, the series will produce less accurate approximations.
Infinite series play an important role in modern analysis ofAncient Greekphilosophy of motion, particularly inZeno's paradoxes.[77]The paradox ofAchilles and the tortoisedemonstrates that continuous motion would require anactual infinityof temporal instants, which was arguably anabsurdity: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on.Zenois said to have argued that therefore Achilles couldneverreach the tortoise, and thus that continuous movement must be an illusion. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the purely mathematical and imaginative side of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise. However, in modern philosophy of motion the physical side of the problem remains open, with both philosophers and physicists doubting, like Zeno, that spatial motions are infinitely divisible: hypothetical reconciliations ofquantum mechanicsandgeneral relativityin theories ofquantum gravityoften introducequantizationsofspacetimeat thePlanck scale.[78][79]
GreekmathematicianArchimedesproduced the first known summation of an infinite series with a
method that is still used in the area of calculus today. He used themethod of exhaustionto calculate theareaunder the arc of aparabolawith the summation of an infinite series,[5]and gave a remarkably accurate approximation ofπ.[80][81]
Mathematicians from theKerala schoolwere studying infinite seriesc.1350 CE.[82]
In the 17th century,James Gregoryworked in the newdecimalsystem on infinite series and published severalMaclaurin series. In 1715, a general method for constructing theTaylor seriesfor all functions for which they exist was provided byBrook Taylor.Leonhard Eulerin the 18th century, developed the theory ofhypergeometric seriesandq-series.
The investigation of the validity of infinite series is considered to begin withGaussin the 19th century. Euler had already considered the hypergeometric series
1+αβ1⋅γx+α(α+1)β(β+1)1⋅2⋅γ(γ+1)x2+⋯{\displaystyle 1+{\frac {\alpha \beta }{1\cdot \gamma }}x+{\frac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}x^{2}+\cdots }
on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.
Cauchy(1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The termsconvergenceanddivergencehad been introduced long before byGregory(1668).Leonhard EulerandGausshad given various criteria, andColin Maclaurinhad anticipated some of Cauchy's discoveries. Cauchy advanced the theory ofpower seriesby his expansion of a complexfunctionin such a form.
Abel(1826) in his memoir on thebinomial series
1+m1!x+m(m−1)2!x2+⋯{\displaystyle 1+{\frac {m}{1!}}x+{\frac {m(m-1)}{2!}}x^{2}+\cdots }
corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values ofm{\displaystyle m}andx{\displaystyle x}. He showed the necessity of considering the subject of continuity in questions of convergence.
Cauchy's methods led to special rather than general criteria, and
the same may be said ofRaabe(1832), who made the first elaborate investigation of the subject, ofDe Morgan(from 1842), whose
logarithmic testDuBois-Reymond(1873) andPringsheim(1889) have
shown to fail within a certain region; ofBertrand(1842),Bonnet(1843),Malmsten(1846, 1847, the latter without integration);Stokes(1847),Paucker(1852),Chebyshev(1852), andArndt(1853).
General criteria began withKummer(1835), and have been studied byEisenstein(1847),Weierstrassin his various
contributions to the theory of functions,Dini(1867),
DuBois-Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.
The theory ofuniform convergencewas treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it
successfully wereSeidelandStokes(1847–48). Cauchy took up the
problem again (1853), acknowledging Abel's criticism, and reaching
the same conclusions which Stokes had already found. Thomae used the
doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform
convergence, in spite of the demands of the theory of functions.
A series is said to be semi-convergent (or conditionally convergent) if it is convergent but notabsolutely convergent.
Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, byMalmsten(1847).Schlömilch(Zeitschrift, Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder andBernoulli's function
F(x)=1n+2n+⋯+(x−1)n.{\displaystyle F(x)=1^{n}+2^{n}+\cdots +(x-1)^{n}.}
Genocchi(1852) has further contributed to the theory.
Among the early writers wasWronski, whose "loi suprême" (1815) was hardly recognized untilCayley(1873) brought it into
prominence.
Fourier serieswere being investigated
as the result of physical considerations at the same time that
Gauss, Abel, and Cauchy were working out the theory of infinite
series. Series for the expansion of sines and cosines, of multiple
arcs in powers of the sine and cosine of the arc had been treated byJacob Bernoulli(1702) and his brotherJohann Bernoulli(1701) and still
earlier byVieta. Euler andLagrangesimplified the subject,
as didPoinsot,Schröter,Glaisher, andKummer.
Fourier (1807) set for himself a different problem, to
expand a given function ofx{\displaystyle x}in terms of the sines or cosines of
multiples ofx{\displaystyle x}, a problem which he embodied in hisThéorie analytique de la chaleur(1822). Euler had already given the formulas for determining the coefficients in the series;
Fourier was the first to assert and attempt to prove the general
theorem.Poisson(1820–23) also attacked the problem from a
different standpoint. Fourier did not, however, settle the question
of convergence of his series, a matter left forCauchy(1826) to
attempt and for Dirichlet (1829) to handle in a thoroughly
scientific manner (seeconvergence of Fourier series). Dirichlet's treatment (Crelle, 1829), of trigonometric series was the subject of criticism and improvement by
Riemann (1854), Heine,Lipschitz,Schläfli, anddu Bois-Reymond. Among other prominent contributors to the theory of
trigonometric and Fourier series wereDini,Hermite,Halphen,
Krause, Byerly andAppell.
Definitions may be given for infinitary sums over an arbitrary index setI.{\displaystyle I.}[83]This generalization introduces two main differences from the usual notion of series: first, there may be no specific order given on the setI{\displaystyle I}; second, the setI{\displaystyle I}may be uncountable. The notions of convergence need to be reconsidered for these, then, because for instance the concept ofconditional convergencedepends on the ordering of the index set.
Ifa:I↦G{\displaystyle a:I\mapsto G}is afunctionfrom anindex setI{\displaystyle I}to a setG,{\displaystyle G,}then the "series" associated toa{\displaystyle a}is theformal sumof the elementsa(x)∈G{\displaystyle a(x)\in G}over the index elementsx∈I{\displaystyle x\in I}denoted by the
∑x∈Ia(x).{\displaystyle \sum _{x\in I}a(x).}
When the index set is the natural numbersI=N,{\displaystyle I=\mathbb {N} ,}the functiona:N↦G{\displaystyle a:\mathbb {N} \mapsto G}is asequencedenoted bya(n)=an.{\displaystyle a(n)=a_{n}.}A series indexed on the natural numbers is an ordered formal sum and so we rewrite∑n∈N{\textstyle \sum _{n\in \mathbb {N} }}as∑n=0∞{\textstyle \sum _{n=0}^{\infty }}in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers
∑n=0∞an=a0+a1+a2+⋯.{\displaystyle \sum _{n=0}^{\infty }a_{n}=a_{0}+a_{1}+a_{2}+\cdots .}
When summing a family{ai:i∈I}{\displaystyle \left\{a_{i}:i\in I\right\}}of non-negative real numbers over the index setI{\displaystyle I}, define
∑i∈Iai=sup{∑i∈Aai:A⊆I,Afinite}∈[0,+∞].{\displaystyle \sum _{i\in I}a_{i}=\sup {\biggl \{}\sum _{i\in A}a_{i}\,:A\subseteq I,A{\text{ finite}}{\biggr \}}\in [0,+\infty ].}
Any sum over non-negative reals can be understood as the integral of a non-negative function with respect to thecounting measure, which accounts for the many similarities between the two constructions.
When the supremum is finite then the set ofi∈I{\displaystyle i\in I}such thatai>0{\displaystyle a_{i}>0}is countable. Indeed, for everyn≥1,{\displaystyle n\geq 1,}thecardinality|An|{\displaystyle \left|A_{n}\right|}of the setAn={i∈I:ai>1/n}{\displaystyle A_{n}=\left\{i\in I:a_{i}>1/n\right\}}is finite because
1n|An|=∑i∈An1n≤∑i∈Anai≤∑i∈Iai<∞.{\displaystyle {\frac {1}{n}}\,\left|A_{n}\right|=\sum _{i\in A_{n}}{\frac {1}{n}}\leq \sum _{i\in A_{n}}a_{i}\leq \sum _{i\in I}a_{i}<\infty .}
Hence the setA={i∈I:ai>0}=⋃n=1∞An{\displaystyle A=\left\{i\in I:a_{i}>0\right\}=\bigcup _{n=1}^{\infty }A_{n}}iscountable.
IfI{\displaystyle I}is countably infinite and enumerated asI={i0,i1,…}{\displaystyle I=\left\{i_{0},i_{1},\ldots \right\}}then the above defined sum satisfies
∑i∈Iai=∑k=0∞aik,{\displaystyle \sum _{i\in I}a_{i}=\sum _{k=0}^{\infty }a_{i_{k}},}provided the value∞{\displaystyle \infty }is allowed for the sum of the series.
Leta:I→X{\displaystyle a:I\to X}be a map, also denoted by(ai)i∈I,{\displaystyle \left(a_{i}\right)_{i\in I},}from some non-empty setI{\displaystyle I}into aHausdorffabeliantopological groupX.{\displaystyle X.}LetFinite(I){\displaystyle \operatorname {Finite} (I)}be the collection of allfinitesubsetsofI,{\displaystyle I,}withFinite(I){\displaystyle \operatorname {Finite} (I)}viewed as adirected set,orderedunderinclusion⊆{\displaystyle \,\subseteq \,}withunionasjoin.
The family(ai)i∈I,{\displaystyle \left(a_{i}\right)_{i\in I},}is said to beunconditionally summableif the followinglimit, which is denoted by∑i∈Iai{\displaystyle \textstyle \sum _{i\in I}a_{i}}and is called thesumof(ai)i∈I,{\displaystyle \left(a_{i}\right)_{i\in I},}exists inX:{\displaystyle X:}
∑i∈Iai:=limA∈Finite(I)∑i∈Aai=lim{∑i∈Aai:A⊆I,Afinite}{\displaystyle \sum _{i\in I}a_{i}:=\lim _{A\in \operatorname {Finite} (I)}\ \sum _{i\in A}a_{i}=\lim {\biggl \{}\sum _{i\in A}a_{i}\,:A\subseteq I,A{\text{ finite }}{\biggr \}}}Saying that the sumS:=∑i∈Iai{\displaystyle \textstyle S:=\sum _{i\in I}a_{i}}is the limit of finite partial sums means that for every neighborhoodV{\displaystyle V}of the origin inX,{\displaystyle X,}there exists a finite subsetA0{\displaystyle A_{0}}ofI{\displaystyle I}such that
S−∑i∈Aai∈Vfor every finite supersetA⊇A0.{\displaystyle S-\sum _{i\in A}a_{i}\in V\qquad {\text{ for every finite superset}}\;A\supseteq A_{0}.}
BecauseFinite(I){\displaystyle \operatorname {Finite} (I)}is nottotally ordered, this is not alimit of a sequenceof partial sums, but rather of anet.[84][85]
For every neighborhoodW{\displaystyle W}of the origin inX,{\displaystyle X,}there is a smaller neighborhoodV{\displaystyle V}such thatV−V⊆W.{\displaystyle V-V\subseteq W.}It follows that the finite partial sums of an unconditionally summable family(ai)i∈I,{\displaystyle \left(a_{i}\right)_{i\in I},}form aCauchy net, that is, for every neighborhoodW{\displaystyle W}of the origin inX,{\displaystyle X,}there exists a finite subsetA0{\displaystyle A_{0}}ofI{\displaystyle I}such that
∑i∈A1ai−∑i∈A2ai∈Wfor all finite supersetsA1,A2⊇A0,{\displaystyle \sum _{i\in A_{1}}a_{i}-\sum _{i\in A_{2}}a_{i}\in W\qquad {\text{ for all finite supersets }}\;A_{1},A_{2}\supseteq A_{0},}which implies thatai∈W{\displaystyle a_{i}\in W}for everyi∈I∖A0{\displaystyle i\in I\setminus A_{0}}(by takingA1:=A0∪{i}{\displaystyle A_{1}:=A_{0}\cup \{i\}}andA2:=A0{\displaystyle A_{2}:=A_{0}}).
WhenX{\displaystyle X}iscomplete, a family(ai)i∈I{\displaystyle \left(a_{i}\right)_{i\in I}}is unconditionally summable inX{\displaystyle X}if and only if the finite sums satisfy the latter Cauchy net condition. WhenX{\displaystyle X}is complete and(ai)i∈I,{\displaystyle \left(a_{i}\right)_{i\in I},}is unconditionally summable inX,{\displaystyle X,}then for every subsetJ⊆I,{\displaystyle J\subseteq I,}the corresponding subfamily(aj)j∈J,{\displaystyle \left(a_{j}\right)_{j\in J},}is also unconditionally summable inX.{\displaystyle X.}
When the sum of a family of non-negative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological groupX=R.{\displaystyle X=\mathbb {R} .}
If a family(ai)i∈I{\displaystyle \left(a_{i}\right)_{i\in I}}inX{\displaystyle X}is unconditionally summable then for every neighborhoodW{\displaystyle W}of the origin inX,{\displaystyle X,}there is a finite subsetA0⊆I{\displaystyle A_{0}\subseteq I}such thatai∈W{\displaystyle a_{i}\in W}for every indexi{\displaystyle i}not inA0.{\displaystyle A_{0}.}IfX{\displaystyle X}is afirst-countable spacethen it follows that the set ofi∈I{\displaystyle i\in I}such thatai≠0{\displaystyle a_{i}\neq 0}is countable. This need not be true in a general abelian topological group (see examples below).
Suppose thatI=N.{\displaystyle I=\mathbb {N} .}If a familyan,n∈N,{\displaystyle a_{n},n\in \mathbb {N} ,}is unconditionally summable in a Hausdorffabelian topological groupX,{\displaystyle X,}then the series in the usual sense converges and has the same sum,
∑n=0∞an=∑n∈Nan.{\displaystyle \sum _{n=0}^{\infty }a_{n}=\sum _{n\in \mathbb {N} }a_{n}.}
By nature, the definition of unconditional summability is insensitive to the order of the summation. When∑an{\displaystyle \textstyle \sum a_{n}}is unconditionally summable, then the series remains convergent after anypermutationσ:N→N{\displaystyle \sigma :\mathbb {N} \to \mathbb {N} }of the setN{\displaystyle \mathbb {N} }of indices, with the same sum,
∑n=0∞aσ(n)=∑n=0∞an.{\displaystyle \sum _{n=0}^{\infty }a_{\sigma (n)}=\sum _{n=0}^{\infty }a_{n}.}
Conversely, if every permutation of a series∑an{\displaystyle \textstyle \sum a_{n}}converges, then the series is unconditionally convergent. WhenX{\displaystyle X}iscompletethen unconditional convergence is also equivalent to the fact that all subseries are convergent; ifX{\displaystyle X}is aBanach space, this is equivalent to say that for every sequence of signsεn=±1{\displaystyle \varepsilon _{n}=\pm 1}, the series
∑n=0∞εnan{\displaystyle \sum _{n=0}^{\infty }\varepsilon _{n}a_{n}}
converges inX.{\displaystyle X.}
IfX{\displaystyle X}is atopological vector space(TVS) and(xi)i∈I{\displaystyle \left(x_{i}\right)_{i\in I}}is a (possiblyuncountable) family inX{\displaystyle X}then this family issummable[86]if the limitlimA∈Finite(I)xA{\displaystyle \textstyle \lim _{A\in \operatorname {Finite} (I)}x_{A}}of thenet(xA)A∈Finite(I){\displaystyle \left(x_{A}\right)_{A\in \operatorname {Finite} (I)}}exists inX,{\displaystyle X,}whereFinite(I){\displaystyle \operatorname {Finite} (I)}is thedirected setof all finite subsets ofI{\displaystyle I}directed by inclusion⊆{\displaystyle \,\subseteq \,}andxA:=∑i∈Axi.{\textstyle x_{A}:=\sum _{i\in A}x_{i}.}
It is calledabsolutely summableif in addition, for every continuous seminormp{\displaystyle p}onX,{\displaystyle X,}the family(p(xi))i∈I{\displaystyle \left(p\left(x_{i}\right)\right)_{i\in I}}is summable.
IfX{\displaystyle X}is a normable space and if(xi)i∈I{\displaystyle \left(x_{i}\right)_{i\in I}}is an absolutely summable family inX,{\displaystyle X,}then necessarily all but a countable collection ofxi{\displaystyle x_{i}}’s are zero. Hence, in normed spaces, it is usually only ever necessary to consider series with countably many terms.
Summable families play an important role in the theory ofnuclear spaces.
The notion of series can be easily extended to the case of aseminormed space.
Ifxn{\displaystyle x_{n}}is a sequence of elements of a normed spaceX{\displaystyle X}and ifx∈X{\displaystyle x\in X}then the series∑xn{\displaystyle \textstyle \sum x_{n}}converges tox{\displaystyle x}inX{\displaystyle X}if the sequence of partial sums of the series(∑n=0Nxn)N=1∞{\textstyle {\bigl (}\!\!~\sum _{n=0}^{N}x_{n}{\bigr )}_{N=1}^{\infty }}converges tox{\displaystyle x}inX{\displaystyle X}; to wit,
‖x−∑n=0Nxn‖→0asN→∞.{\displaystyle {\Biggl \|}x-\sum _{n=0}^{N}x_{n}{\Biggr \|}\to 0\quad {\text{ as }}N\to \infty .}
More generally, convergence of series can be defined in anyabelianHausdorfftopological group.
Specifically, in this case,∑xn{\displaystyle \textstyle \sum x_{n}}converges tox{\displaystyle x}if the sequence of partial sums converges tox.{\displaystyle x.}
If(X,|⋅|){\displaystyle (X,|\cdot |)}is aseminormed space, then the notion of absolute convergence becomes:
A series∑i∈Ixi{\textstyle \sum _{i\in I}x_{i}}of vectors inX{\displaystyle X}converges absolutelyif
∑i∈I|xi|<+∞{\displaystyle \sum _{i\in I}\left|x_{i}\right|<+\infty }
in which case all but at most countably many of the values|xi|{\displaystyle \left|x_{i}\right|}are necessarily zero.
If a countable series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite-dimensional Banach spaces (theorem ofDvoretzky & Rogers (1950)).
Conditionally convergent series can be considered ifI{\displaystyle I}is awell-orderedset, for example, anordinal numberα0.{\displaystyle \alpha _{0}.}In this case, define bytransfinite recursion:
∑β<α+1aβ=aα+∑β<αaβ{\displaystyle \sum _{\beta <\alpha +1}\!a_{\beta }=a_{\alpha }+\sum _{\beta <\alpha }a_{\beta }}
and for a limit ordinalα,{\displaystyle \alpha ,}
∑β<αaβ=limγ→α∑β<γaβ{\displaystyle \sum _{\beta <\alpha }a_{\beta }=\lim _{\gamma \to \alpha }\,\sum _{\beta <\gamma }a_{\beta }}
if this limit exists. If all limits exist up toα0,{\displaystyle \alpha _{0},}then the series converges.
|
https://en.wikipedia.org/wiki/Series_(mathematics)
|
Inmathematics, abinary operationiscommutativeif changing the order of theoperandsdoes not change the result. It is a fundamental property of many binary operations, and manymathematical proofsdepend on it. Perhaps most familiar as a property of arithmetic, e.g."3 + 4 = 4 + 3"or"2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such asdivisionandsubtraction, that do not have it (for example,"3 − 5 ≠ 5 − 3"); such operations arenotcommutative, and so are referred to asnoncommutative operations.
The idea that simple operations, such as themultiplicationandadditionof numbers, are commutative was for many centuries implicitly assumed. Thus, this property was not named until the 19th century, when newalgebraic structuresstarted to be studied.[1]
Abinary operation∗{\displaystyle *}on asetSiscommutativeifx∗y=y∗x{\displaystyle x*y=y*x}for allx,y∈S{\displaystyle x,y\in S}.[2]An operation that is not commutative is said to benoncommutative.[3]
One says thatxcommuteswithyor thatxandycommuteunder∗{\displaystyle *}if[4]x∗y=y∗x.{\displaystyle x*y=y*x.}
So, an operation is commutative if every two elements commute.[4]An operation is noncommutative if there are two elements such thatx∗y≠y∗x.{\displaystyle x*y\neq y*x.}This does not exclude the possibility that some pairs of elements commute.[3]
Some types ofalgebraic structuresinvolve an operation that does not require commutativity. If this operation is commutative for a specific structure, the structure is often said to becommutative. So,
However, in the case ofalgebras, the phrase "commutative algebra" refers only toassociative algebrasthat have a commutative multiplication.[18]
Records of the implicit use of the commutative property go back to ancient times. TheEgyptiansused the commutative property ofmultiplicationto simplify computingproducts.[19]Euclidis known to have assumed the commutative property of multiplication in his bookElements.[20]Formal uses of the commutative property arose in the late 18th and early 19th centuries when mathematicians began to work on a theory of functions. Nowadays, the commutative property is a well-known and basic property used in most branches of mathematics.[2]
The first recorded use of the termcommutativewas in a memoir byFrançois Servoisin 1814, which used the wordcommutativeswhen describing functions that have what is now called the commutative property.[21]Commutativeis the feminine form of the French adjectivecommutatif, which is derived from the French nouncommutationand the French verbcommuter, meaning "to exchange" or "to switch", a cognate ofto commute. The term then appeared in English in 1838. inDuncan Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in theTransactions of the Royal Society of Edinburgh.[22]
|
https://en.wikipedia.org/wiki/Commutativity
|
Inmathematics, thedistributive propertyofbinary operationsis a generalization of thedistributive law, which asserts that the equalityx⋅(y+z)=x⋅y+x⋅z{\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z}is always true inelementary algebra.
For example, inelementary arithmetic, one has2⋅(1+3)=(2⋅1)+(2⋅3).{\displaystyle 2\cdot (1+3)=(2\cdot 1)+(2\cdot 3).}Therefore, one would say thatmultiplicationdistributesoveraddition.
This basic property of numbers is part of the definition of mostalgebraic structuresthat have two operations called addition and multiplication, such ascomplex numbers,polynomials,matrices,rings, andfields. It is also encountered inBoolean algebraandmathematical logic, where each of thelogical and(denoted∧{\displaystyle \,\land \,}) and thelogical or(denoted∨{\displaystyle \,\lor \,}) distributes over the other.
Given asetS{\displaystyle S}and twobinary operators∗{\displaystyle \,*\,}and+{\displaystyle \,+\,}onS,{\displaystyle S,}
x∗(y+z)=(x∗y)+(x∗z);{\displaystyle x*(y+z)=(x*y)+(x*z);}
(y+z)∗x=(y∗x)+(z∗x);{\displaystyle (y+z)*x=(y*x)+(z*x);}
When∗{\displaystyle \,*\,}iscommutative, the three conditions above arelogically equivalent.
The operators used for examples in this section are those of the usualaddition+{\displaystyle \,+\,}andmultiplication⋅.{\displaystyle \,\cdot .\,}
If the operation denoted⋅{\displaystyle \cdot }is not commutative, there is a distinction between left-distributivity and right-distributivity:
a⋅(b±c)=a⋅b±a⋅c(left-distributive){\displaystyle a\cdot \left(b\pm c\right)=a\cdot b\pm a\cdot c\qquad {\text{ (left-distributive) }}}(a±b)⋅c=a⋅c±b⋅c(right-distributive).{\displaystyle (a\pm b)\cdot c=a\cdot c\pm b\cdot c\qquad {\text{ (right-distributive) }}.}
In either case, the distributive property can be described in words as:
To multiply asum(ordifference) by a factor, each summand (orminuendandsubtrahend) is multiplied by this factor and the resulting products are added (or subtracted).
If the operation outside the parentheses (in this case, the multiplication) is commutative, then left-distributivity implies right-distributivity and vice versa, and one talks simply ofdistributivity.
One example of an operation that is "only" right-distributive is division, which is not commutative:(a±b)÷c=a÷c±b÷c.{\displaystyle (a\pm b)\div c=a\div c\pm b\div c.}In this case, left-distributivity does not apply:a÷(b±c)≠a÷b±a÷c{\displaystyle a\div (b\pm c)\neq a\div b\pm a\div c}
The distributive laws are among the axioms forrings(like the ring ofintegers) andfields(like the field ofrational numbers). Here multiplication is distributive over addition, but addition is not distributive over multiplication. Examples of structures with two operations that are each distributive over the other areBoolean algebrassuch as thealgebra of setsor theswitching algebra.
Multiplying sums can be put into words as follows: When a sum is multiplied by a sum, multiply each summand of a sum with each summand of the other sum (keeping track of signs) then add up all of the resulting products.
In the following examples, the use of the distributive law on the set of real numbersR{\displaystyle \mathbb {R} }is illustrated. When multiplication is mentioned in elementary mathematics, it usually refers to this kind of multiplication. From the point of view of algebra, the real numbers form afield, which ensures the validity of the distributive law.
The distributive law is valid formatrix multiplication. More precisely,(A+B)⋅C=A⋅C+B⋅C{\displaystyle (A+B)\cdot C=A\cdot C+B\cdot C}for alll×m{\displaystyle l\times m}-matricesA,B{\displaystyle A,B}andm×n{\displaystyle m\times n}-matricesC,{\displaystyle C,}as well asA⋅(B+C)=A⋅B+A⋅C{\displaystyle A\cdot (B+C)=A\cdot B+A\cdot C}for alll×m{\displaystyle l\times m}-matricesA{\displaystyle A}andm×n{\displaystyle m\times n}-matricesB,C.{\displaystyle B,C.}Because the commutative property does not hold for matrix multiplication, the second law does not follow from the first law. In this case, they are two different laws.
In standard truth-functional propositional logic,distribution[3][4]in logical proofs uses two validrules of replacementto expand individual occurrences of certainlogical connectives, within someformula, into separate applications of those connectives across subformulas of the given formula. The rules are(P∧(Q∨R))⇔((P∧Q)∨(P∧R))and(P∨(Q∧R))⇔((P∨Q)∧(P∨R)){\displaystyle (P\land (Q\lor R))\Leftrightarrow ((P\land Q)\lor (P\land R))\qquad {\text{ and }}\qquad (P\lor (Q\land R))\Leftrightarrow ((P\lor Q)\land (P\lor R))}where "⇔{\displaystyle \Leftrightarrow }", also written≡,{\displaystyle \,\equiv ,\,}is ametalogicalsymbolrepresenting "can be replaced in a proof with" or "islogically equivalentto".
Distributivityis a property of some logical connectives of truth-functionalpropositional logic. The following logical equivalences demonstrate that distributivity is a property of particular connectives. The following are truth-functionaltautologies.(P∧(Q∨R))⇔((P∧Q)∨(P∧R))Distribution ofconjunctionoverdisjunction(P∨(Q∧R))⇔((P∨Q)∧(P∨R))Distribution ofdisjunctionoverconjunction(P∧(Q∧R))⇔((P∧Q)∧(P∧R))Distribution ofconjunctionoverconjunction(P∨(Q∨R))⇔((P∨Q)∨(P∨R))Distribution ofdisjunctionoverdisjunction(P→(Q→R))⇔((P→Q)→(P→R))Distribution ofimplication(P→(Q↔R))⇔((P→Q)↔(P→R))Distribution ofimplicationoverequivalence(P→(Q∧R))⇔((P→Q)∧(P→R))Distribution ofimplicationoverconjunction(P∨(Q↔R))⇔((P∨Q)↔(P∨R))Distribution ofdisjunctionoverequivalence{\displaystyle {\begin{alignedat}{13}&(P&&\;\land &&(Q\lor R))&&\;\Leftrightarrow \;&&((P\land Q)&&\;\lor (P\land R))&&\quad {\text{ Distribution of }}&&{\text{ conjunction }}&&{\text{ over }}&&{\text{ disjunction }}\\&(P&&\;\lor &&(Q\land R))&&\;\Leftrightarrow \;&&((P\lor Q)&&\;\land (P\lor R))&&\quad {\text{ Distribution of }}&&{\text{ disjunction }}&&{\text{ over }}&&{\text{ conjunction }}\\&(P&&\;\land &&(Q\land R))&&\;\Leftrightarrow \;&&((P\land Q)&&\;\land (P\land R))&&\quad {\text{ Distribution of }}&&{\text{ conjunction }}&&{\text{ over }}&&{\text{ conjunction }}\\&(P&&\;\lor &&(Q\lor R))&&\;\Leftrightarrow \;&&((P\lor Q)&&\;\lor (P\lor R))&&\quad {\text{ Distribution of }}&&{\text{ disjunction }}&&{\text{ over }}&&{\text{ disjunction }}\\&(P&&\to &&(Q\to R))&&\;\Leftrightarrow \;&&((P\to Q)&&\to (P\to R))&&\quad {\text{ Distribution of }}&&{\text{ implication }}&&{\text{ }}&&{\text{ }}\\&(P&&\to &&(Q\leftrightarrow R))&&\;\Leftrightarrow \;&&((P\to Q)&&\leftrightarrow (P\to R))&&\quad {\text{ Distribution of }}&&{\text{ implication }}&&{\text{ over }}&&{\text{ equivalence }}\\&(P&&\to &&(Q\land R))&&\;\Leftrightarrow \;&&((P\to Q)&&\;\land (P\to R))&&\quad {\text{ Distribution of }}&&{\text{ implication }}&&{\text{ over }}&&{\text{ conjunction }}\\&(P&&\;\lor &&(Q\leftrightarrow R))&&\;\Leftrightarrow \;&&((P\lor Q)&&\leftrightarrow (P\lor R))&&\quad {\text{ Distribution of }}&&{\text{ disjunction }}&&{\text{ over }}&&{\text{ equivalence }}\\\end{alignedat}}}
((P∧Q)∨(R∧S))⇔(((P∨R)∧(P∨S))∧((Q∨R)∧(Q∨S)))((P∨Q)∧(R∨S))⇔(((P∧R)∨(P∧S))∨((Q∧R)∨(Q∧S))){\displaystyle {\begin{alignedat}{13}&((P\land Q)&&\;\lor (R\land S))&&\;\Leftrightarrow \;&&(((P\lor R)\land (P\lor S))&&\;\land ((Q\lor R)\land (Q\lor S)))&&\\&((P\lor Q)&&\;\land (R\lor S))&&\;\Leftrightarrow \;&&(((P\land R)\lor (P\land S))&&\;\lor ((Q\land R)\lor (Q\land S)))&&\\\end{alignedat}}}
In approximate arithmetic, such asfloating-point arithmetic, the distributive property of multiplication (and division) over addition may fail because of the limitations ofarithmetic precision. For example, the identity1/3+1/3+1/3=(1+1+1)/3{\displaystyle 1/3+1/3+1/3=(1+1+1)/3}fails indecimal arithmetic, regardless of the number ofsignificant digits. Methods such asbanker's roundingmay help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.
Distributivity is most commonly found insemirings, notably the particular cases ofringsanddistributive lattices.
A semiring has two binary operations, commonly denoted+{\displaystyle \,+\,}and∗,{\displaystyle \,*,}and requires that∗{\displaystyle \,*\,}must distribute over+.{\displaystyle \,+.}
A ring is a semiring with additive inverses.
Alatticeis another kind ofalgebraic structurewith two binary operations,∧and∨.{\displaystyle \,\land {\text{ and }}\lor .}If either of these operations distributes over the other (say∧{\displaystyle \,\land \,}distributes over∨{\displaystyle \,\lor }), then the reverse also holds (∨{\displaystyle \,\lor \,}distributes over∧{\displaystyle \,\land \,}), and the lattice is called distributive. See alsoDistributivity (order theory).
ABoolean algebracan be interpreted either as a special kind of ring (aBoolean ring) or a special kind of distributive lattice (aBoolean lattice). Each interpretation is responsible for different distributive laws in the Boolean algebra.
Similar structures without distributive laws arenear-ringsandnear-fieldsinstead of rings anddivision rings. The operations are usually defined to be distributive on the right but not on the left.
In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of the above conditions or the extension to infinitary operations. Especially inorder theoryone finds numerous important variants of distributivity, some of which include infinitary operations, such as theinfinite distributive law; others being defined in the presence of onlyonebinary operation, such as the according definitions and their relations are given in the articledistributivity (order theory). This also includes the notion of acompletely distributive lattice.
In the presence of an ordering relation, one can also weaken the above equalities by replacing={\displaystyle \,=\,}by either≤{\displaystyle \,\leq \,}or≥.{\displaystyle \,\geq .}Naturally, this will lead to meaningful concepts only in some situations. An application of this principle is the notion ofsub-distributivityas explained in the article oninterval arithmetic.
Incategory theory, if(S,μ,ν){\displaystyle (S,\mu ,\nu )}and(S′,μ′,ν′){\displaystyle \left(S^{\prime },\mu ^{\prime },\nu ^{\prime }\right)}aremonadson acategoryC,{\displaystyle C,}adistributive lawS.S′→S′.S{\displaystyle S.S^{\prime }\to S^{\prime }.S}is anatural transformationλ:S.S′→S′.S{\displaystyle \lambda :S.S^{\prime }\to S^{\prime }.S}such that(S′,λ){\displaystyle \left(S^{\prime },\lambda \right)}is alax map of monadsS→S{\displaystyle S\to S}and(S,λ){\displaystyle (S,\lambda )}is acolax map of monadsS′→S′.{\displaystyle S^{\prime }\to S^{\prime }.}This is exactly the data needed to define a monad structure onS′.S{\displaystyle S^{\prime }.S}: the multiplication map isS′μ.μ′S2.S′λS{\displaystyle S^{\prime }\mu .\mu ^{\prime }S^{2}.S^{\prime }\lambda S}and the unit map isη′S.η.{\displaystyle \eta ^{\prime }S.\eta .}See:distributive law between monads.
Ageneralized distributive lawhas also been proposed in the area ofinformation theory.
The ubiquitousidentitythat relates inverses to the binary operation in anygroup, namely(xy)−1=y−1x−1,{\displaystyle (xy)^{-1}=y^{-1}x^{-1},}which is taken as an axiom in the more general context of asemigroup with involution, has sometimes been called anantidistributive property(of inversion as aunary operation).[5]
In the context of anear-ring, which removes the commutativity of the additively written group and assumes only one-sided distributivity, one can speak of (two-sided)distributive elementsbut also ofantidistributive elements. The latter reverse the order of (the non-commutative) addition; assuming a left-nearring (i.e. one which all elements distribute when multiplied on the left), then an antidistributive elementa{\displaystyle a}reverses the order of addition when multiplied to the right:(x+y)a=ya+xa.{\displaystyle (x+y)a=ya+xa.}[6]
In the study ofpropositional logicandBoolean algebra, the termantidistributive lawis sometimes used to denote the interchange between conjunction and disjunction when implication factors over them:[7](a∨b)⇒c≡(a⇒c)∧(b⇒c){\displaystyle (a\lor b)\Rightarrow c\equiv (a\Rightarrow c)\land (b\Rightarrow c)}(a∧b)⇒c≡(a⇒c)∨(b⇒c).{\displaystyle (a\land b)\Rightarrow c\equiv (a\Rightarrow c)\lor (b\Rightarrow c).}
These twotautologiesare a direct consequence of the duality inDe Morgan's laws.
|
https://en.wikipedia.org/wiki/Distributivity
|
Inmathematics, specifically inabstract algebra,power associativityis a property of abinary operationthat is a weak form ofassociativity.
Analgebra(or more generally amagma) is said to be power-associative if thesubalgebragenerated by any element is associative. Concretely, this means that if an elementx{\displaystyle x}is performed an operation∗{\displaystyle *}by itself several times, it doesn't matter in which order the operations are carried out, so for instancex∗(x∗(x∗x))=(x∗(x∗x))∗x=(x∗x)∗(x∗x){\displaystyle x*(x*(x*x))=(x*(x*x))*x=(x*x)*(x*x)}.
Everyassociative algebrais power-associative, but so are all otheralternative algebras(like theoctonions, which are non-associative) and even non-alternativeflexible algebraslike thesedenions,trigintaduonions, andOkubo algebras. Any algebra whose elements areidempotentis also power-associative.
Exponentiationto the power of anypositive integercan be defined consistently whenever multiplication is power-associative. For example, there is no need to distinguish whetherx3should be defined as (xx)xor asx(xx), since these are equal. Exponentiation to the power of zero can also be defined if the operation has anidentity element, so the existence of identity elements is useful in power-associative contexts.
Over afieldofcharacteristic0, an algebra is power-associative if and only if it satisfies[x,x,x]=0{\displaystyle [x,x,x]=0}and[x2,x,x]=0{\displaystyle [x^{2},x,x]=0}, where[x,y,z]:=(xy)z−x(yz){\displaystyle [x,y,z]:=(xy)z-x(yz)}is theassociator(Albert 1948).
Over an infinite field ofprimecharacteristicp>0{\displaystyle p>0}there is no finite set of identities that characterizes power-associativity, but there are infinite independent sets, as described by Gainov (1970):
A substitution law holds forrealpower-associative algebras with unit, which basically asserts that multiplication ofpolynomialsworks as expected. Forfa real polynomial inx, and for anyain such an algebra definef(a) to be the element of the algebra resulting from the obvious substitution ofaintof. Then for any two such polynomialsfandg, we have that(fg)(a) =f(a)g(a).
|
https://en.wikipedia.org/wiki/Power_associativity
|
Inabstract algebra,alternativityis a property of abinary operation. AmagmaGis said to beleft alternativeif(xx)y=x(xy){\displaystyle (xx)y=x(xy)}for allx,y∈G{\displaystyle x,y\in G}andright alternativeify(xx)=(yx)x{\displaystyle y(xx)=(yx)x}for allx,y∈G{\displaystyle x,y\in G}. A magma that is both left and right alternative is said to bealternative(flexible).[1]
Anyassociativemagma (that is, asemigroup) is alternative. More generally, a magma in which every pair of elements generates an associative submagma must be alternative. Theconverse, however, is not true, in contrast to the situation inalternative algebras.
Examples of alternative algebras include:
Thisalgebra-related article is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/Alternativity
|
Inmathematics, particularlyabstract algebra, abinary operation• on asetisflexibleif it satisfies theflexible identity:
for any two elementsaandbof the set. Amagma(that is, a set equipped with a binary operation) is flexible if the binary operation with which it is equipped is flexible. Similarly, anonassociative algebrais flexible if its multiplication operator is flexible.
Everycommutativeorassociativeoperation is flexible, so flexibility becomes important for binary operations that are neither commutative nor associative, e.g. for themultiplicationofsedenions, which are not evenalternative.
In 1954,Richard D. Schaferexamined the algebras generated by theCayley–Dickson processover afieldand showed that they satisfy the flexible identity.[1]
Besidesassociative algebras, the following classes of nonassociative algebras are flexible:
In the world ofmagmas, there is only a binary multiplication operation with no addition or scaling with from a base ring or field like in algebras. In this setting,alternativeandcommutativemagmas are all flexible - the alternative and commutative laws all imply flexibility. This includes many important classes of magmas: allgroups,semigroupsandmoufang loopsare flexible.
Thesedenionsandtrigintaduonions, and all algebras constructed from these by iterating theCayley–Dickson construction, are also flexible.
|
https://en.wikipedia.org/wiki/Flexible_algebra
|
Inalgebra,n-ary associativityis ageneralizationof theassociative lawton-ary operations.
Aternary operationisternary associativeif one has always
that is, the operation gives the same result when any three adjacent elements are bracketed inside asequenceof five operands.
Similarly, ann-ary operation isn-ary associative if bracketing anynadjacent elements in a sequence ofn+ (n− 1)operands do not change the result.[1]
Thisalgebra-related article is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/N-ary_associativity
|
Inmathematics, aMoufang loopis a special kind ofalgebraic structure. It is similar to agroupin many ways but need not beassociative. Moufang loops were introduced byRuth Moufang(1935). Smooth Moufang loops have an associated algebra, theMalcev algebra, similar in some ways to how aLie grouphas an associatedLie algebra.
AMoufang loopis aloopQ{\displaystyle Q}that satisfies the four following equivalentidentitiesfor allx{\displaystyle x},y{\displaystyle y},z{\displaystyle z}inQ{\displaystyle Q}(the binary operation inQ{\displaystyle Q}is denoted by juxtaposition):
These identities are known asMoufang identities.
Moufang loops differ from groups in that they need not beassociative. A Moufang loop that is associative is a group. The Moufang identities may be viewed as weaker forms of associativity.
By setting various elements to the identity, the Moufang identities imply
Moufang's theorem states that when three elementsx,y, andzin a Moufang loop obey the associative law: (xy)z=x(yz) then they generate an associative subloop; that is, a group. A corollary of this is that all Moufang loops aredi-associative(i.e. the subloop generated by any two elements of a Moufang loop is associative and therefore a group). In particular, Moufang loops arepower associative, so that powersxnare well-defined. When working with Moufang loops, it is common to drop the parenthesis in expressions with only two distinct elements. For example, the Moufang identities may be written unambiguously as
The Moufang identities can be written in terms of the left and right multiplication operators onQ. The first two identities state that
while the third identity says
for allx,y,z{\displaystyle x,y,z}inQ{\displaystyle Q}. HereBz=LzRz=RzLz{\displaystyle B_{z}=L_{z}R_{z}=R_{z}L_{z}}is bimultiplication byz{\displaystyle z}. The third Moufang identity is therefore equivalent to the statement that the triple(Lz,Rz,Bz){\displaystyle (L_{z},R_{z},B_{z})}is anautotopyofQ{\displaystyle Q}for allz{\displaystyle z}inQ{\displaystyle Q}.
All Moufang loops have theinverse property, which means that each elementxhas atwo-sided inversex−1that satisfies the identities:
for allxandy. It follows that(xy)−1=y−1x−1{\displaystyle (xy)^{-1}=y^{-1}x^{-1}}andx(yz)=e{\displaystyle x(yz)=e}if and only if(xy)z=e{\displaystyle (xy)z=e}.
Moufang loops are universal among inverse property loops; that is, a loopQis a Moufang loop if and only if everyloop isotopeofQhas the inverse property. It follows that every loop isotope of a Moufang loop is a Moufang loop.
One can use inverses to rewrite the left and right Moufang identities in a more useful form:
A finite loopQis said to have theLagrange propertyif the order of every subloop ofQdivides the order ofQ.Lagrange's theoremin group theory states that every finite group has the Lagrange property. It was an open question for many years whether or not finite Moufang loops had Lagrange property. The question was finally resolved by Alexander Grishkov and Andrei Zavarnitsine, and independently by Stephen Gagola III and Jonathan Hall, in 2003: Every finite Moufang loop does have the Lagrange property. More results for the theory of finite groups have been generalized to Moufang loops by Stephen Gagola III in recent years.
Anyquasigroupsatisfying one of the Moufang identities must, in fact, have an identity element and therefore be a Moufang loop. We give a proof here for the third identity:
The proofs for the first two identities are somewhat more difficult (Kunen 1996).
Phillips' problemis an open problem in the theory presented by J. D. Phillips at Loops '03 in Prague. It asks whether there exists a finite Moufang loop of odd order with a trivialnucleus.
Recall that the nucleus of aloop(or more generally a quasigroup) is the set ofx{\displaystyle x}such thatx(yz)=(xy)z{\displaystyle x(yz)=(xy)z},y(xz)=(yx)z{\displaystyle y(xz)=(yx)z}andy(zx)=(yz)x{\displaystyle y(zx)=(yz)x}hold for ally,z{\displaystyle y,z}in the loop.
|
https://en.wikipedia.org/wiki/Moufang_loop
|
This is a list of possiblynonassociative algebras. An algebra is amodule, wherein you can alsomultiplytwo module elements. (The multiplication in the module is compatible with multiplication-by-scalars from the basering).
This is a list of fields of algebra.
|
https://en.wikipedia.org/wiki/List_of_algebras
|
Inmathematics, there existmagmasthat arecommutativebut notassociative. A simple example of such a magma may be derived from the children's game ofrock, paper, scissors. Such magmas give rise tonon-associative algebras.
A magma which is both commutative and associative is a commutativesemigroup.
In the game ofrock paper scissors, letM:={r,p,s}{\displaystyle M:=\{r,p,s\}}, standing for the "rock", "paper" and "scissors" gestures respectively, and consider thebinary operation⋅:M×M→M{\displaystyle \cdot :M\times M\to M}derived from the rules of the game as follows:[1]
This results in theCayley table:[1]
By definition, the magma(M,⋅){\displaystyle (M,\cdot )}is commutative, but it is also non-associative,[2]as shown by:
but
i.e.
It is the simplest non-associative magma that isconservative, in the sense that the result of any magma operation is one of the two values given as arguments to the operation.[2]
Thearithmetic mean, andgeneralized meansof numbers or of higher-dimensional quantities, such asFrechet means, are often commutative but non-associative.[3]
Commutative but non-associative magmas may be used to analyzegenetic recombination.[4]
|
https://en.wikipedia.org/wiki/Commutative_non-associative_magmas
|
In thehistorical study of mathematics, anapotomeis a line segment formed from a longer line segment by breaking it into two parts, one of which iscommensurableonly in power to the whole; the other part is the apotome. In this definition, two line segments are said to be "commensurable only in power" when the ratio of their lengths is anirrational numberbut the ratio of their squared lengths is rational.[1]
Translated into modern algebraic language, an apotome can be interpreted as aquadratic irrationalnumber formed by subtracting onesquare rootof a rational number from another.
This concept of the apotome appears inEuclid's Elementsbeginning in book X, whereEucliddefines two special kinds of apotomes. In an apotome of the first kind, the whole is rational, while in an apotome of the second kind, the part subtracted from it is rational; both kinds of apotomes also satisfy an additional condition. Euclid Proposition XIII.6 states that, if a rational line segment is split into two pieces in thegolden ratio, then both pieces may be represented as apotomes.[2]
Thisnumber theory-related article is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/Apotome_(mathematics)
|
Inmathematics, acube rootof a numberxis a numberythat has the given number as itsthird power; that isy3=x.{\displaystyle y^{3}=x.}The number of cube roots of a number depends on thenumber systemthat is considered.
Everyreal numberxhas exactly one real cube root that is denotedx3{\textstyle {\sqrt[{3}]{x}}}and called thereal cube rootofxor simplythe cube rootofxin contexts wherecomplex numbersare not considered. For example, the real cube roots of8and−8are respectively2and−2. The real cube root of anintegeror of arational numberis generally not a rational number, neither aconstructible number.
Every nonzero real orcomplex numberhas exactly three cube roots that are complex numbers. If the number is real, one of the cube roots is real and the two other are nonrealcomplex conjugatenumbers. Otherwise, the three cube roots are all nonreal. For example, the real cube root of8is2and the other cube roots of8are−1+i3{\displaystyle -1+i{\sqrt {3}}}and−1−i3{\displaystyle -1-i{\sqrt {3}}}. The three cube roots of−27iare3i,332−32i,{\displaystyle 3i,{\tfrac {3{\sqrt {3}}}{2}}-{\tfrac {3}{2}}i,}and−332−32i.{\displaystyle -{\tfrac {3{\sqrt {3}}}{2}}-{\tfrac {3}{2}}i.}The number zero has a unique cube root, which is zero itself.
The cube root is amultivalued function. Theprincipal cube rootis itsprincipal value, that is a unique cube root that has been chosen once for all. The principal cube root is the cube root with the largestreal part. In the case of negative real numbers, the largest real part is shared by the two nonreal cube roots, and the principal cube root is the one with positive imaginary part. So, for negative real numbers,the real cube root is not the principal cube root. For positive real numbers, the principal cube root is the real cube root.
Ifyis any cube root of the complex numberx, the other cube roots arey−1+i32{\displaystyle y\,{\tfrac {-1+i{\sqrt {3}}}{2}}}andy−1−i32.{\displaystyle y\,{\tfrac {-1-i{\sqrt {3}}}{2}}.}
In analgebraically closed fieldofcharacteristicdifferent from three, every nonzero element has exactly three cube roots, which can be obtained from any of them by multiplying it by eitherrootof the polynomialx2+x+1.{\displaystyle x^{2}+x+1.}In an algebraically closed field of characteristic three, every element has exactly one cube root.
In other number systems or otheralgebraic structures, a number or element may have more than three cube roots. For example, in thequaternions, a real number has infinitely many cube roots.
The cube roots of a numberxare the numbersywhich satisfy the equationy3=x.{\displaystyle y^{3}=x.\ }
For any real numberx, there is exactly one real numberysuch thaty3=x{\displaystyle y^{3}=x}. Indeed, thecube functionis increasing, so it does not give the same result for two different inputs, and covers all real numbers. In other words, it is abijectionor one-to-one correspondence.
That is, one can definethecube root of a real number as its unique cube root that is also real. With this definition, the cube root of a negative number is a negative number.
However this definition may be confusing when real numbers are considered as specific complex numbers, since, in this casethecube root is generally defined as the principal cube root, and the principal cube root of a negative real number is not real.
Ifxandyare allowed to becomplex, then there are three solutions (ifxis non-zero) and soxhas three cube roots. A real number has one real cube root and two further cube roots which form acomplex conjugatepair. For instance, the cube roots of1are:
The last two of these roots lead to a relationship between all roots of any real or complex number. If a number is one cube root of a particular real or complex number, the other two cube roots can be found by multiplying that cube root by one or the other of the two complex cube roots of 1.
For complex numbers, the principal cube root is usually defined as the cube root that has the greatestreal part, or, equivalently, the cube root whoseargumenthas the leastabsolute value. It is related to the principal value of thenatural logarithmby the formula
If we writexas
whereris a non-negative real number andθ{\displaystyle \theta }lies in the range
then the principal complex cube root is
This means that inpolar coordinates, we are taking the cube root of the radius and dividing the polar angle by three in order to define a cube root. With this definition, the principal cube root of a negative number is a complex number, and for instance−83{\displaystyle {\sqrt[{3}]{-8}}}will not be −2, but rather1+i3{\displaystyle 1+i{\sqrt {3}}}
This difficulty can also be solved by considering the cube root as amultivalued function: if we write the original complex numberxin three equivalent forms, namely
The principal complex cube roots of these three forms are then respectively
Unlessx= 0, these three complex numbers are distinct, even though the three representations ofxwere equivalent. For example,−83{\displaystyle {\sqrt[{3}]{-8}}}may then be calculated to be −2,1+i3{\displaystyle 1+i{\sqrt {3}}}, or1−i3{\displaystyle 1-i{\sqrt {3}}}.
This is related with the concept ofmonodromy: if one follows bycontinuitythe functioncube rootalong a closed path around zero, after a turn the value of the cube root is multiplied (or divided) bye2iπ/3.{\displaystyle e^{2i\pi /3}.}
Cube roots arise in the problem of finding an angle whose measure is one third that of a given angle (angle trisection) and in the problem of finding the edge of a cube whose volume is twice that of a cube with a given edge (doubling the cube). In 1837Pierre Wantzelproved that neither of these can be done with acompass-and-straightedge construction.
Newton's methodis aniterative methodthat can be used to calculate the cube root. For realfloating-pointnumbers this method reduces to the following iterative algorithm to produce successively better approximations of the cube root ofa:
The method is simply averaging three factors chosen such that
at each iteration.
Halley's methodimproves upon this with an algorithm that converges more quickly with each iteration, albeit with more work per iteration:
Thisconverges cubically, so two iterations do as much work as three iterations of Newton's method. Each iteration of Newton's method costs two multiplications, one addition and one division, assuming that1/3ais precomputed, so three iterations plus the precomputation require seven multiplications, three additions, and three divisions.
Each iteration of Halley's method requires three multiplications, three additions, and one division,[1]so two iterations cost six multiplications, six additions, and two divisions. Thus, Halley's method has the potential to be faster if one division is more expensive than three additions.
With either method a poor initial approximation ofx0can give very poor algorithm performance, and coming up with a good initial approximation is somewhat of a black art. Some implementations manipulate the exponent bits of the floating-point number; i.e. they arrive at an initial approximation by dividing the exponent by 3.[1]
Also useful is thisgeneralized continued fraction, based on thenth rootmethod:
Ifxis a good first approximation to the cube root ofaandy=a−x3{\displaystyle y=a-x^{3}}, then:
The second equation combines each pair of fractions from the first into a single fraction, thus doubling the speed of convergence.
Cubic equations, which arepolynomial equationsof the third degree (meaning the highest power of the unknown is 3) can always be solved for their three solutions in terms of cube roots and square roots (although simpler expressions only in terms of square roots exist for all three solutions, if at least one of them is arational number). If two of the solutions are complex numbers, then all three solution expressions involve the real cube root of a real number, while if all three solutions are real numbers then they may be expressed in terms of thecomplex cube root of a complex number.
Quartic equationscan also be solved in terms of cube roots and square roots.
The calculation of cube roots can be traced back toBabylonian mathematiciansfrom as early as 1800 BCE.[2]In the fourth century BCEPlatoposed the problem ofdoubling the cube, which required acompass-and-straightedge constructionof the edge of acubewith twice the volume of a given cube; this required the construction, now known to be impossible, of the length23{\displaystyle {\sqrt[{3}]{2}}}.
A method for extracting cube roots appears inThe Nine Chapters on the Mathematical Art, aChinese mathematicaltext compiled around the second century BCE and commented on byLiu Huiin the third century CE.[3]TheGreek mathematicianHero of Alexandriadevised a method for calculating cube roots in the first century CE. His formula is again mentioned by Eutokios in a commentary onArchimedes.[4]In 499 CEAryabhata, amathematician-astronomerfrom the classical age ofIndian mathematicsandIndian astronomy, gave a method for finding the cube root of numbers having many digits in theAryabhatiya(section 2.5).[5]
|
https://en.wikipedia.org/wiki/Cube_root
|
Inmathematics, afunctional square root(sometimes called ahalf iterate) is asquare rootof afunctionwith respect to the operation offunction composition. In other words, a functional square root of a functiongis a functionfsatisfyingf(f(x)) =g(x)for allx.
Notations expressing thatfis a functional square root ofgaref=g[1/2]andf=g1/2[citation needed][dubious–discuss], or ratherf=g1/2(seeIterated Function), although this leaves the usual ambiguity with taking the function to that power in the multiplicative sense, just asf ² = f ∘ fcan be misinterpreted asx ↦ f(x)².
A systematic procedure to producearbitraryfunctionaln-roots (including arbitrary real, negative, and infinitesimaln) of functionsg:C→C{\displaystyle g:\mathbb {C} \rightarrow \mathbb {C} }relies on the solutions ofSchröder's equation.[3][4][5]Infinitely many trivial solutions exist when thedomainof a root functionfis allowed to be sufficiently larger than that ofg.
Using this extension,sin[1/2](1)can be shown to be approximately equal to 0.90871.[6]
(See.[7]For the notation, see[1]Archived2022-12-05 at theWayback Machine.)
|
https://en.wikipedia.org/wiki/Functional_square_root
|
Innumber theory, theinteger square root(isqrt) of anon-negative integernis the non-negative integermwhich is thegreatest integer less than or equalto thesquare rootofn,isqrt(n)=⌊n⌋.{\displaystyle \operatorname {isqrt} (n)=\lfloor {\sqrt {n}}\rfloor .}
For example,isqrt(27)=⌊27⌋=⌊5.19615242270663...⌋=5.{\displaystyle \operatorname {isqrt} (27)=\lfloor {\sqrt {27}}\rfloor =\lfloor 5.19615242270663...\rfloor =5.}
Lety{\displaystyle y}andk{\displaystyle k}be non-negative integers.
Algorithms that compute (thedecimal representationof)y{\displaystyle {\sqrt {y}}}run foreveron each inputy{\displaystyle y}which is not aperfect square.[note 1]
Algorithms that compute⌊y⌋{\displaystyle \lfloor {\sqrt {y}}\rfloor }do not run forever. They are nevertheless capable of computingy{\displaystyle {\sqrt {y}}}up to any desired accuracyk{\displaystyle k}.
Choose anyk{\displaystyle k}and compute⌊y×100k⌋{\textstyle \lfloor {\sqrt {y\times 100^{k}}}\rfloor }.
For example(settingy=2{\displaystyle y=2}):k=0:⌊2×1000⌋=⌊2⌋=1k=1:⌊2×1001⌋=⌊200⌋=14k=2:⌊2×1002⌋=⌊20000⌋=141k=3:⌊2×1003⌋=⌊2000000⌋=1414⋮k=8:⌊2×1008⌋=⌊20000000000000000⌋=141421356⋮{\displaystyle {\begin{aligned}&k=0:\lfloor {\sqrt {2\times 100^{0}}}\rfloor =\lfloor {\sqrt {2}}\rfloor =1\\&k=1:\lfloor {\sqrt {2\times 100^{1}}}\rfloor =\lfloor {\sqrt {200}}\rfloor =14\\&k=2:\lfloor {\sqrt {2\times 100^{2}}}\rfloor =\lfloor {\sqrt {20000}}\rfloor =141\\&k=3:\lfloor {\sqrt {2\times 100^{3}}}\rfloor =\lfloor {\sqrt {2000000}}\rfloor =1414\\&\vdots \\&k=8:\lfloor {\sqrt {2\times 100^{8}}}\rfloor =\lfloor {\sqrt {20000000000000000}}\rfloor =141421356\\&\vdots \\\end{aligned}}}
Compare the results with2=1.41421356237309504880168872420969807856967187537694...{\displaystyle {\sqrt {2}}=1.41421356237309504880168872420969807856967187537694...}
It appears that the multiplication of the input by100k{\displaystyle 100^{k}}gives an accuracy ofkdecimal digits.[note 2]
To compute the (entire) decimal representation ofy{\displaystyle {\sqrt {y}}}, one can executeisqrt(y){\displaystyle \operatorname {isqrt} (y)}an infinite number of times, increasingy{\displaystyle y}by a factor100{\displaystyle 100}at each pass.
Assume that in the next program (sqrtForever{\displaystyle \operatorname {sqrtForever} }) the procedureisqrt(y){\displaystyle \operatorname {isqrt} (y)}is already defined and — for the sake of the argument — that all variables can hold integers of unlimited magnitude.
ThensqrtForever(y){\displaystyle \operatorname {sqrtForever} (y)}will print the entire decimal representation ofy{\displaystyle {\sqrt {y}}}.[note 3]
The conclusion is that algorithms which computeisqrt()are computationally equivalent toalgorithms which computesqrt().
Theinteger square rootof anon-negative integery{\displaystyle y}can be defined as⌊y⌋=x:x2≤y<(x+1)2,x∈N{\displaystyle \lfloor {\sqrt {y}}\rfloor =x:x^{2}\leq y<(x+1)^{2},x\in \mathbb {N} }
For example,isqrt(27)=⌊27⌋=5{\displaystyle \operatorname {isqrt} (27)=\lfloor {\sqrt {27}}\rfloor =5}because62>27and52≯27{\displaystyle 6^{2}>27{\text{ and }}5^{2}\ngtr 27}.
The following C programs are straightforward implementations.
In the program above (linear search, ascending) one can replace multiplication by addition, using the equivalence(L+1)2=L2+2L+1=L2+1+∑i=1L2.{\displaystyle (L+1)^{2}=L^{2}+2L+1=L^{2}+1+\sum _{i=1}^{L}2.}
Linear searchsequentially checks every value until it hits the smallestx{\displaystyle x}wherex2>y{\displaystyle x^{2}>y}.
A speed-up is achieved by usingbinary searchinstead. The following C-program is an implementation.
Numerical example
For example, if one computesisqrt(2000000){\displaystyle \operatorname {isqrt} (2000000)}using binary search, one obtains the[L,R]{\displaystyle [L,R]}sequence[0,2000001]→[0,1000000]→[0,500000]→[0,250000]→[0,125000]→[0,62500]→[0,31250]→[0,15625]→[0,7812]→[0,3906]→[0,1953]→[976,1953]→[976,1464]→[1220,1464]→[1342,1464]→[1403,1464]→[1403,1433]→[1403,1418]→[1410,1418]→[1414,1418]→[1414,1416]→[1414,1415]{\displaystyle {\begin{aligned}&[0,2000001]\rightarrow [0,1000000]\rightarrow [0,500000]\rightarrow [0,250000]\rightarrow [0,125000]\rightarrow [0,62500]\rightarrow [0,31250]\rightarrow [0,15625]\\&\rightarrow [0,7812]\rightarrow [0,3906]\rightarrow [0,1953]\rightarrow [976,1953]\rightarrow [976,1464]\rightarrow [1220,1464]\rightarrow [1342,1464]\rightarrow [1403,1464]\\&\rightarrow [1403,1433]\rightarrow [1403,1418]\rightarrow [1410,1418]\rightarrow [1414,1418]\rightarrow [1414,1416]\rightarrow [1414,1415]\end{aligned}}}
This computation takes 21 iteration steps, whereas linear search (ascending, starting from0{\displaystyle 0}) needs1414steps.
One way of calculatingn{\displaystyle {\sqrt {n}}}andisqrt(n){\displaystyle \operatorname {isqrt} (n)}is to useHeron's method, which is a special case ofNewton's method, to find a solution for the equationx2−n=0{\displaystyle x^{2}-n=0}, giving the iterative formulaxk+1=12(xk+nxk),k≥0,x0>0.{\displaystyle x_{k+1}={\frac {1}{2}}\!\left(x_{k}+{\frac {n}{x_{k}}}\right),\quad k\geq 0,\quad x_{0}>0.}
Thesequence{xk}{\displaystyle \{x_{k}\}}convergesquadraticallyton{\displaystyle {\sqrt {n}}}ask→∞{\displaystyle k\to \infty }.
One can prove[citation needed]thatc=1{\displaystyle c=1}is the largest possible number for which the stopping criterion|xk+1−xk|<c{\displaystyle |x_{k+1}-x_{k}|<c}ensures⌊xk+1⌋=⌊n⌋{\displaystyle \lfloor x_{k+1}\rfloor =\lfloor {\sqrt {n}}\rfloor }in the algorithm above.
In implementations which use number formats that cannot represent allrational numbersexactly (for example, floating point), a stopping constant less than 1 should be used to protect against round-off errors.
Althoughn{\displaystyle {\sqrt {n}}}isirrationalfor manyn{\displaystyle n}, the sequence{xk}{\displaystyle \{x_{k}\}}contains only rational terms whenx0{\displaystyle x_{0}}is rational. Thus, with this method it is unnecessary to exit thefieldof rational numbers in order to calculateisqrt(n){\displaystyle \operatorname {isqrt} (n)}, a fact which has some theoretical advantages.
For computing⌊n⌋{\displaystyle \lfloor {\sqrt {n}}\rfloor }for very large integersn, one can use the quotient ofEuclidean divisionfor both of the division operations. This has the advantage of only using integers for each intermediate value, thus making the use offloating pointrepresentations of large numbers unnecessary. It is equivalent to using the iterative formulaxk+1=⌊12(xk+⌊nxk⌋)⌋,k≥0,x0>0,x0∈Z.{\displaystyle x_{k+1}=\left\lfloor {\frac {1}{2}}\!\left(x_{k}+\left\lfloor {\frac {n}{x_{k}}}\right\rfloor \right)\right\rfloor ,\quad k\geq 0,\quad x_{0}>0,\quad x_{0}\in \mathbb {Z} .}
By using the fact that⌊12(xk+⌊nxk⌋)⌋=⌊12(xk+nxk)⌋,{\displaystyle \left\lfloor {\frac {1}{2}}\!\left(x_{k}+\left\lfloor {\frac {n}{x_{k}}}\right\rfloor \right)\right\rfloor =\left\lfloor {\frac {1}{2}}\!\left(x_{k}+{\frac {n}{x_{k}}}\right)\right\rfloor ,}
one can show that this will reach⌊n⌋{\displaystyle \lfloor {\sqrt {n}}\rfloor }within a finite number of iterations.
In the original version, one hasxk≥n{\displaystyle x_{k}\geq {\sqrt {n}}}fork≥1{\displaystyle k\geq 1}, andxk>xk+1{\displaystyle x_{k}>x_{k+1}}forxk>n{\displaystyle x_{k}>{\sqrt {n}}}. So in the integer version, one has⌊xk⌋≥⌊n⌋{\displaystyle \lfloor x_{k}\rfloor \geq \lfloor {\sqrt {n}}\rfloor }andxk≥⌊xk⌋>xk+1≥⌊xk+1⌋{\displaystyle x_{k}\geq \lfloor x_{k}\rfloor >x_{k+1}\geq \lfloor x_{k+1}\rfloor }until the final solutionxs{\displaystyle x_{s}}is reached. For the final solutionxs{\displaystyle x_{s}}, one has⌊n⌋≤⌊xs⌋≤n{\displaystyle \lfloor {\sqrt {n}}\rfloor \leq \lfloor x_{s}\rfloor \leq {\sqrt {n}}}and⌊xs+1⌋≥⌊xs⌋{\displaystyle \lfloor x_{s+1}\rfloor \geq \lfloor x_{s}\rfloor }, so the stopping criterion is⌊xk+1⌋≥⌊xk⌋{\displaystyle \lfloor x_{k+1}\rfloor \geq \lfloor x_{k}\rfloor }.
However,⌊n⌋{\displaystyle \lfloor {\sqrt {n}}\rfloor }is not necessarily afixed pointof the above iterative formula. Indeed, it can be shown that⌊n⌋{\displaystyle \lfloor {\sqrt {n}}\rfloor }is a fixed point if and only ifn+1{\displaystyle n+1}is not a perfect square. Ifn+1{\displaystyle n+1}is a perfect square, the sequence ends up in a period-two cycle between⌊n⌋{\displaystyle \lfloor {\sqrt {n}}\rfloor }and⌊n⌋+1{\displaystyle \lfloor {\sqrt {n}}\rfloor +1}instead of converging.
For example, if one computes the integer square root of2000000using the algorithm above, one obtains the sequence1000000→500001→250002→125004→62509→31270→15666→7896→4074→2282→1579→1422→1414→1414{\displaystyle {\begin{aligned}&1000000\rightarrow 500001\rightarrow 250002\rightarrow 125004\rightarrow 62509\rightarrow 31270\rightarrow 15666\rightarrow 7896\\&\rightarrow 4074\rightarrow 2282\rightarrow 1579\rightarrow 1422\rightarrow 1414\rightarrow 1414\end{aligned}}}In total 13 iteration steps are needed. Although Heron's method converges quadratically close to the solution, less than one bit precision per iteration is gained at the beginning. This means that the choice of the initial estimate is critical for the performance of the algorithm.
When a fast computation for the integer part of thebinary logarithmor for thebit-lengthis available (like e.g.std::bit_widthinC++20), one should better start atx0=2⌊(log2n)/2⌋+1,{\displaystyle x_{0}=2^{\lfloor (\log _{2}n)/2\rfloor +1},}which is the leastpower of twobigger thann{\displaystyle {\sqrt {n}}}. In the example of the integer square root of2000000,⌊log2n⌋=20{\displaystyle \lfloor \log _{2}n\rfloor =20},x0=211=2048{\displaystyle x_{0}=2^{11}=2048}, and the resulting sequence is2048→1512→1417→1414→1414.{\displaystyle 2048\rightarrow 1512\rightarrow 1417\rightarrow 1414\rightarrow 1414.}In this case only four iteration steps are needed.
The traditionalpen-and-paper algorithmfor computing the square rootn{\displaystyle {\sqrt {n}}}is based on working from higher digit places to lower, and as each new digit pick the largest that will still yield a square≤n{\displaystyle \leq n}. If stopping after the one's place, the result computed will be the integer square root.
If working inbase 2, the choice of digit is simplified to that between 0 (the "small candidate") and 1 (the "large candidate"), and digit manipulations can be expressed in terms of binary shift operations. With*being multiplication,<<being left shift, and>>being logical right shift, arecursivealgorithm to find the integer square root of anynatural numberis:
Traditional pen-and-paper presentations of the digit-by-digit algorithm include various optimizations not present in the code above, in particular the trick of pre-subtracting the square of the previous digits which makes a general multiplication step unnecessary. SeeMethods of computing square roots § Binary numeral system (base 2)for an example.[1]
The Karatsuba square root algorithm is a combination of two functions: apublicfunction, which returns the integer square root of the input, and a recursiveprivatefunction, which does the majority of the work.
The public function normalizes the actual input, passes the normalized input to the private function, denormalizes the result of the private function, and returns that.
The private function takes a normalized input, divides the input bits in half, passes the most-significant half of the input recursively to the private function, and performs some integer operations on the output of that recursive call and the least-significant half of the input to get the normalized output, which it returns.
Forbig-integersof "50 to 1,000,000 digits",Burnikel-Ziegler Karatsuba divisionandKaratsuba multiplicationare recommended by the algorithm's creator.[2]
An example algorithm for 64-bit unsigned integers is below. The algorithm:
Someprogramming languagesdedicate an explicit operation to the integer square root calculation in addition to the general case or can be extended by libraries to this end.
|
https://en.wikipedia.org/wiki/Integer_square_root
|
Inalgebra, anested radicalis aradical expression(one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression. Examples include
5−25,{\displaystyle {\sqrt {5-2{\sqrt {5}}\ }},}
which arises in discussing theregular pentagon, and more complicated ones such as
2+3+433.{\displaystyle {\sqrt[{3}]{2+{\sqrt {3}}+{\sqrt[{3}]{4}}\ }}.}
Some nested radicals can be rewritten in a form that is not nested. For example,
3+22=1+2,{\displaystyle {\sqrt {3+2{\sqrt {2}}}}=1+{\sqrt {2}}\,,}
23−13=1−23+4393.{\displaystyle {\sqrt[{3}]{{\sqrt[{3}]{2}}-1}}={\frac {1-{\sqrt[{3}]{2}}+{\sqrt[{3}]{4}}}{\sqrt[{3}]{9}}}\,.}
Another simple example,
23=26{\displaystyle {\sqrt[{3}]{\sqrt {2}}}={\sqrt[{6}]{2}}}
Rewriting a nested radical in this way is calleddenesting. This is not always possible, and, even when possible, it is often difficult.
In the case of two nested square roots, the following theorem completely solves the problem of denesting.[2]
Ifaandcarerational numbersandcis not the square of a rational number, there are two rational numbersxandysuch thata+c=x±y{\displaystyle {\sqrt {a+{\sqrt {c}}}}={\sqrt {x}}\pm {\sqrt {y}}}if and only ifa2−c{\displaystyle a^{2}-c~}is the square of a rational numberd.
If the nested radical is real,xandyare the two numbersa+d2{\displaystyle {\frac {a+d}{2}}~}anda−d2,{\displaystyle ~{\frac {a-d}{2}}~,~}whered=a2−c{\displaystyle ~d={\sqrt {a^{2}-c}}~}is a rational number.
In particular, ifaandcare integers, then2xand2yare integers.
This result includes denestings of the forma+c=z±y,{\displaystyle {\sqrt {a+{\sqrt {c}}}}=z\pm {\sqrt {y}}~,}aszmay always be writtenz=±z2,{\displaystyle z=\pm {\sqrt {z^{2}}},}and at least one of the terms must be positive (because the left-hand side of the equation is positive).
A more general denesting formula could have the forma+c=α+βx+γy+δxy.{\displaystyle {\sqrt {a+{\sqrt {c}}}}=\alpha +\beta {\sqrt {x}}+\gamma {\sqrt {y}}+\delta {\sqrt {x}}{\sqrt {y}}~.}However,Galois theoryimplies that either the left-hand side belongs toQ(c),{\displaystyle \mathbb {Q} ({\sqrt {c}}),}or it must be obtained by changing the sign of eitherx,{\displaystyle {\sqrt {x}},}y,{\displaystyle {\sqrt {y}},}or both. In the first case, this means that one can takex=candγ=δ=0.{\displaystyle \gamma =\delta =0.}In the second case,α{\displaystyle \alpha }and another coefficient must be zero. Ifβ=0,{\displaystyle \beta =0,}one may renamexyasxfor gettingδ=0.{\displaystyle \delta =0.}Proceeding similarly ifα=0,{\displaystyle \alpha =0,}it results that one can supposeα=δ=0.{\displaystyle \alpha =\delta =0.}This shows that the apparently more general denesting can always be reduced to the above one.
Proof: By squaring, the equationa+c=x±y{\displaystyle {\sqrt {a+{\sqrt {c}}}}={\sqrt {x}}\pm {\sqrt {y}}}is equivalent witha+c=x+y±2xy,{\displaystyle a+{\sqrt {c}}=x+y\pm 2{\sqrt {xy}},}and, in the case of a minus in the right-hand side,
(square roots are nonnegative by definition of the notation). As the inequality may always be satisfied by possibly exchangingxandy, solving the first equation inxandyis equivalent with solvinga+c=x+y±2xy.{\displaystyle a+{\sqrt {c}}=x+y\pm 2{\sqrt {xy}}.}
This equality implies thatxy{\displaystyle {\sqrt {xy}}}belongs to thequadratic fieldQ(c).{\displaystyle \mathbb {Q} ({\sqrt {c}}).}In this field every element may be uniquely writtenα+βc,{\displaystyle \alpha +\beta {\sqrt {c}},}withα{\displaystyle \alpha }andβ{\displaystyle \beta }being rational numbers. This implies that±2xy{\displaystyle \pm 2{\sqrt {xy}}}is not rational (otherwise the right-hand side of the equation would be rational; but the left-hand side is irrational). Asxandymust be rational, the square of±2xy{\displaystyle \pm 2{\sqrt {xy}}}must be rational. This implies thatα=0{\displaystyle \alpha =0}in the expression of±2xy{\displaystyle \pm 2{\sqrt {xy}}}asα+βc.{\displaystyle \alpha +\beta {\sqrt {c}}.}Thusa+c=x+y+βc{\displaystyle a+{\sqrt {c}}=x+y+\beta {\sqrt {c}}}for some rational numberβ.{\displaystyle \beta .}The uniqueness of the decomposition over1andc{\displaystyle {\sqrt {c}}}implies thus that the considered equation is equivalent witha=x+yand±2xy=c.{\displaystyle a=x+y\quad {\text{and}}\quad \pm 2{\sqrt {xy}}={\sqrt {c}}.}It follows byVieta's formulasthatxandymust be roots of thequadratic equationz2−az+c4=0;{\displaystyle z^{2}-az+{\frac {c}{4}}=0~;}itsΔ=a2−c=d2>0{\displaystyle ~\Delta =a^{2}-c=d^{2}>0~}(≠ 0, otherwisecwould be the square ofa), hencexandymust bea+a2−c2{\displaystyle {\frac {a+{\sqrt {a^{2}-c}}}{2}}~}anda−a2−c2.{\displaystyle ~{\frac {a-{\sqrt {a^{2}-c}}}{2}}~.}Thusxandyare rational if and only ifd=a2−c{\displaystyle d={\sqrt {a^{2}-c}}~}is a rational number.
For explicitly choosing the various signs, one must consider only positive real square roots, and thus assumingc> 0. The equationa2=c+d2{\displaystyle a^{2}=c+d^{2}}shows that|a|>√c. Thus, if the nested radical is real, and if denesting is possible, thena> 0. Then the solution isa+c=a+d2+a−d2,a−c=a+d2−a−d2.{\displaystyle {\begin{aligned}{\sqrt {a+{\sqrt {c}}}}&={\sqrt {\frac {a+d}{2}}}+{\sqrt {\frac {a-d}{2}}},\\[6pt]{\sqrt {a-{\sqrt {c}}}}&={\sqrt {\frac {a+d}{2}}}-{\sqrt {\frac {a-d}{2}}}.\end{aligned}}}
Srinivasa Ramanujandemonstrated a number of curious identities involving nested radicals. Among them are the following:[3]
3+2543−2544=54+154−1=12(3+54+5+1254),{\displaystyle {\sqrt[{4}]{\frac {3+2{\sqrt[{4}]{5}}}{3-2{\sqrt[{4}]{5}}}}}={\frac {{\sqrt[{4}]{5}}+1}{{\sqrt[{4}]{5}}-1}}={\tfrac {1}{2}}\left(3+{\sqrt[{4}]{5}}+{\sqrt {5}}+{\sqrt[{4}]{125}}\right),}
283−273=13(983−283−1),{\displaystyle {\sqrt {{\sqrt[{3}]{28}}-{\sqrt[{3}]{27}}}}={\tfrac {1}{3}}\left({\sqrt[{3}]{98}}-{\sqrt[{3}]{28}}-1\right),}
3255−27553=1255+3255−9255,{\displaystyle {\sqrt[{3}]{{\sqrt[{5}]{\frac {32}{5}}}-{\sqrt[{5}]{\frac {27}{5}}}}}={\sqrt[{5}]{\frac {1}{25}}}+{\sqrt[{5}]{\frac {3}{25}}}-{\sqrt[{5}]{\frac {9}{25}}},}
and
In 1989Susan Landauintroduced the firstalgorithmfor deciding which nested radicals can be denested.[5]Earlier algorithms worked in some cases but not others. Landau's algorithm involves complexroots of unityand runs inexponential timewith respect to the depth of the nested radical.[6]
Intrigonometry, thesines and cosinesof many angles can be expressed in terms of nested radicals. For example,sinπ60=sin3∘=116[2(1−3)5+5+2(5−1)(3+1)]{\displaystyle \sin {\frac {\pi }{60}}=\sin 3^{\circ }={\frac {1}{16}}\left[2(1-{\sqrt {3}}){\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}({\sqrt {5}}-1)({\sqrt {3}}+1)\right]}
andsinπ24=sin7.5∘=122−2+3=122−1+32.{\displaystyle \sin {\frac {\pi }{24}}=\sin 7.5^{\circ }={\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {3}}}}}}={\frac {1}{2}}{\sqrt {2-{\frac {1+{\sqrt {3}}}{\sqrt {2}}}}}.}The last equality results directly from the results of§ Two nested square roots.
Nested radicals appear in thealgebraic solutionof thecubic equation. Any cubic equation can be written in simplified form without a quadratic term, as
x3+px+q=0,{\displaystyle x^{3}+px+q=0,}
whose general solution for one of the roots isx=−q2+q24+p3273+−q2−q24+p3273.{\displaystyle x={\sqrt[{3}]{-{q \over 2}+{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}+{\sqrt[{3}]{-{q \over 2}-{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}.}
In the case in which the cubic has only one real root, the real root is given by this expression with theradicandsof the cube roots being real and with the cube roots being the real cube roots. In the case of three real roots, the square root expression is an imaginary number; here any real root is expressed by defining the first cube root to be any specific complex cube root of the complex radicand, and by defining the second cube root to be thecomplex conjugateof the first one. The nested radicals in this solution cannot in general be simplified unless the cubic equation has at least onerationalsolution. Indeed, if the cubic has three irrational but real solutions, we have thecasus irreducibilis, in which all three real solutions are written in terms of cube roots of complex numbers. On the other hand, consider the equation
x3−7x+6=0,{\displaystyle x^{3}-7x+6=0,}
which has the rational solutions 1, 2, and −3. The general solution formula given above gives the solutionsx=−3+103i93+−3−103i93.{\displaystyle x={\sqrt[{3}]{-3+{\frac {10{\sqrt {3}}i}{9}}}}+{\sqrt[{3}]{-3-{\frac {10{\sqrt {3}}i}{9}}}}.}
For any given choice of cube root and its conjugate, this contains nested radicals involving complex numbers, yet it is reducible (even though not obviously so) to one of the solutions 1, 2, or −3.
Under certain conditions infinitely nested square roots such asx=2+2+2+2+⋯{\displaystyle x={\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots }}}}}}}}}
represent rational numbers. This rational number can be found by realizing thatxalso appears under the radical sign, which gives the equation
x=2+x.{\displaystyle x={\sqrt {2+x}}.}
If we solve this equation, we find thatx= 2(the second solutionx= −1doesn't apply, under the convention that the positive square root is meant). This approach can also be used to show that generally, ifn> 0, thenn+n+n+n+⋯=12(1+1+4n){\displaystyle {\sqrt {n+{\sqrt {n+{\sqrt {n+{\sqrt {n+\cdots }}}}}}}}={\tfrac {1}{2}}\left(1+{\sqrt {1+4n}}\right)}
and is the positive root of the equationx2−x−n= 0. Forn= 1, this root is thegolden ratioφ, approximately equal to 1.618. The same procedure also works to obtain, ifn> 0,n−n−n−n−⋯=12(−1+1+4n),{\displaystyle {\sqrt {n-{\sqrt {n-{\sqrt {n-{\sqrt {n-\cdots }}}}}}}}={\tfrac {1}{2}}\left(-1+{\sqrt {1+4n}}\right),}which is the positive root of the equationx2+x−n= 0.
The nested square roots of 2 are a special case of the wide class of infinitely nested radicals. There are many known results that bind them tosines and cosines. For example, it has been shown that nested square roots of 2 as[7]R(bk,…,b1)=bk22+bk−12+bk−22+⋯+b22+x{\displaystyle R(b_{k},\ldots ,b_{1})={\frac {b_{k}}{2}}{\sqrt {2+b_{k-1}{\sqrt {2+b_{k-2}{\sqrt {2+\cdots +b_{2}{\sqrt {2+x}}}}}}}}}
wherex=2sin(πb1/4){\displaystyle x=2\sin(\pi b_{1}/4)}withb1{\displaystyle b_{1}}in [−2,2] andbi∈{−1,0,1}{\displaystyle b_{i}\in \{-1,0,1\}}fori≠1{\displaystyle i\neq 1}, are such thatR(bk,…,b1)=cosθ{\displaystyle R(b_{k},\ldots ,b_{1})=\cos \theta }forθ=(12−bk4−bkbk−18−bkbk−1bk−216−⋯−bkbk−1⋯b12k+1)π.{\displaystyle \theta =\left({\frac {1}{2}}-{\frac {b_{k}}{4}}-{\frac {b_{k}b_{k-1}}{8}}-{\frac {b_{k}b_{k-1}b_{k-2}}{16}}-\cdots -{\frac {b_{k}b_{k-1}\cdots b_{1}}{2^{k+1}}}\right)\pi .}
This result allows to deduce for anyx∈[−2,2]{\displaystyle x\in [-2,2]}the value of the following infinitely nested radicals consisting of k nested roots asRk(x)=2+2+⋯+2+x.{\displaystyle R_{k}(x)={\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2+x}}}}}}.}
Ifx≥2{\displaystyle x\geq 2}, then[8]Rk(x)=2+2+⋯+2+x=(x+x2−42)1/2k+(x+x2−42)−1/2k{\displaystyle {\begin{aligned}R_{k}(x)&={\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2+x}}}}}}\\&=\left({\frac {x+{\sqrt {x^{2}-4}}}{2}}\right)^{1/2^{k}}+\left({\frac {x+{\sqrt {x^{2}-4}}}{2}}\right)^{-1/2^{k}}\end{aligned}}}
These results can be used to obtain some nested square roots representations ofπ{\displaystyle \pi }. Let us consider the termR(bk,…,b1){\displaystyle R\left(b_{k},\ldots ,b_{1}\right)}defined above. Then[7]π=limk→∞[2k+12−b1R(1,−1,1,1,…,1,1,b1⏟kterms)]{\displaystyle \pi =\lim _{k\rightarrow \infty }\left[{\frac {2^{k+1}}{2-b_{1}}}R(\underbrace {1,-1,1,1,\ldots ,1,1,b_{1}} _{k{\text{ terms }}})\right]}
whereb1≠2{\displaystyle b_{1}\neq 2}.
Ramanujanposed the following problem to theJournal of Indian Mathematical Society:
?=1+21+31+⋯.{\displaystyle ?={\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}.}
This can be solved by noting a more general formulation:?=ax+(n+a)2+xa(x+n)+(n+a)2+(x+n)⋯.{\displaystyle ?={\sqrt {ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\mathrm {\cdots } }}}}}}.}
Setting this toF(x)and squaring both sides gives usF(x)2=ax+(n+a)2+xa(x+n)+(n+a)2+(x+n)⋯,{\displaystyle F(x)^{2}=ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\mathrm {\cdots } }}}},}
which can be simplified toF(x)2=ax+(n+a)2+xF(x+n).{\displaystyle F(x)^{2}=ax+(n+a)^{2}+xF(x+n).}
It can be shown that
F(x)=x+n+a{\displaystyle F(x)={x+n+a}}
satisfies the equation forF(x){\displaystyle F(x)}, so it can be hoped that
it is the true solution. For a complete proof, we would need to show
that this is indeed the solution to the equation forF(x){\displaystyle F(x)}.
So, settinga= 0,n= 1, andx= 2, we have3=1+21+31+⋯.{\displaystyle 3={\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}.}Ramanujan stated the following infinite radical denesting in hislost notebook:5+5+5−5+5+5+5−⋯=2+5+15−652.{\displaystyle {\sqrt {5+{\sqrt {5+{\sqrt {5-{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5-\cdots }}}}}}}}}}}}}}={\frac {2+{\sqrt {5}}+{\sqrt {15-6{\sqrt {5}}}}}{2}}.}The repeating pattern of the signs is(+,+,−,+).{\displaystyle (+,+,-,+).}
Viète's formulaforπ, the ratio of a circle's circumference to its diameter, is2π=22⋅2+22⋅2+2+22⋯.{\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots .}
In certain cases, infinitely nested cube roots such asx=6+6+6+6+⋯3333{\displaystyle x={\sqrt[{3}]{6+{\sqrt[{3}]{6+{\sqrt[{3}]{6+{\sqrt[{3}]{6+\cdots }}}}}}}}}can represent rational numbers as well. Again, by realizing that the whole expression appears inside itself, we are left with the equationx=6+x3.{\displaystyle x={\sqrt[{3}]{6+x}}.}
If we solve this equation, we find thatx= 2. More generally, we find thatn+n+n+n+⋯3333{\displaystyle {\sqrt[{3}]{n+{\sqrt[{3}]{n+{\sqrt[{3}]{n+{\sqrt[{3}]{n+\cdots }}}}}}}}}is the positive real root of the equationx3−x−n= 0for alln> 0. Forn= 1, this root is theplastic ratioρ, approximately equal to 1.3247.
The same procedure also works to get
n−n−n−n−⋯3333{\displaystyle {\sqrt[{3}]{n-{\sqrt[{3}]{n-{\sqrt[{3}]{n-{\sqrt[{3}]{n-\cdots }}}}}}}}}
as the real root of the equationx3+x−n= 0for alln> 1.
An infinitely nested radicala1+a2+⋯{\displaystyle {\sqrt {a_{1}+{\sqrt {a_{2}+\dotsb }}}}}(where allai{\displaystyle a_{i}}arenonnegative) converges if and only if there is someM∈R{\displaystyle M\in \mathbb {R} }such thatM≥an2−n{\displaystyle M\geq a_{n}^{2^{-n}}}for alln{\displaystyle n},[9]or in other wordssupan2−n<+∞.{\textstyle \sup a_{n}^{2^{-n}}<+\infty .}
We observe thata1+a2+⋯≤M21+M22+⋯=M1+1+⋯<2M.{\displaystyle {\sqrt {a_{1}+{\sqrt {a_{2}+\dotsb }}}}\leq {\sqrt {M^{2^{1}}+{\sqrt {M^{2^{2}}+\cdots }}}}=M{\sqrt {1+{\sqrt {1+\dotsb }}}}<2M.}Moreover, the sequence(a1+a2+…an){\displaystyle \left({\sqrt {a_{1}+{\sqrt {a_{2}+\dotsc {\sqrt {a_{n}}}}}}}\right)}is monotonically increasing. Therefore it converges, by themonotone convergence theorem.
If the sequence(a1+a2+⋯an){\displaystyle \left({\sqrt {a_{1}+{\sqrt {a_{2}+\cdots {\sqrt {a_{n}}}}}}}\right)}converges, then it is bounded.
However,an2−n≤a1+a2+⋯an{\displaystyle a_{n}^{2^{-n}}\leq {\sqrt {a_{1}+{\sqrt {a_{2}+\cdots {\sqrt {a_{n}}}}}}}}, hence(an2−n){\displaystyle \left(a_{n}^{2^{-n}}\right)}is also bounded.
|
https://en.wikipedia.org/wiki/Nested_radical
|
Inmathematics, annth rootof anumberxis a numberrwhich, whenraised to the powerofn, yieldsx:rn=r×r×⋯×r⏟nfactors=x.{\displaystyle r^{n}=\underbrace {r\times r\times \dotsb \times r} _{n{\text{ factors}}}=x.}
Thepositive integernis called theindexordegree, and the numberxof which the root is taken is theradicand.A root of degree 2 is called asquare rootand a root of degree 3, acube root. Roots of higher degree are referred by usingordinal numbers, as infourth root,twentieth root, etc. The computation of annth root is aroot extraction.
For example,3is a square root of9, since32= 9, and−3is also a square root of9, since(−3)2= 9.
Thenth root ofxis written asxn{\displaystyle {\sqrt[{n}]{x}}}using theradical symbolx{\displaystyle {\sqrt {\phantom {x}}}}. The square root is usually written asx{\displaystyle {\sqrt {x}}}, with the degree omitted. Taking thenth root of a number, for fixedn{\displaystyle n}, is theinverseof raising a number to thenth power,[1]and can be written as afractionalexponent:
xn=x1/n.{\displaystyle {\sqrt[{n}]{x}}=x^{1/n}.}
For a positive real numberx,x{\displaystyle {\sqrt {x}}}denotes the positive square root ofxandxn{\displaystyle {\sqrt[{n}]{x}}}denotes the positive realnth root. A negative real number−xhas no real-valued square roots, but whenxis treated as a complex number it has twoimaginarysquare roots,+ix{\displaystyle +i{\sqrt {x}}}and−ix{\displaystyle -i{\sqrt {x}}}, whereiis theimaginary unit.
In general, any non-zerocomplex numberhasndistinct complex-valuednth roots, equally distributed around a complex circle of constantabsolute value. (Thenth root of0is zero withmultiplicityn, and this circle degenerates to a point.) Extracting thenth roots of a complex numberxcan thus be taken to be amultivalued function. By convention theprincipal valueof this function, called theprincipal rootand denotedxn{\displaystyle {\sqrt[{n}]{x}}}, is taken to be thenth root with the greatest real part and in the special case whenxis a negative real number, the one with a positiveimaginary part. The principal root of a positive real number is thus also a positive real number. As afunction, the principal root iscontinuousin the wholecomplex plane, except along the negative real axis.
An unresolved root, especially one using the radical symbol, is sometimes referred to as asurd[2]or aradical.[3]Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called aradical expression, and if it contains notranscendental functionsortranscendental numbersit is called analgebraic expression.
Roots are used for determining theradius of convergenceof apower serieswith theroot test. Thenth roots of 1 are calledroots of unityand play a fundamental role in various areas of mathematics, such asnumber theory,theory of equations, andFourier transform.
An archaic term for the operation of takingnth roots isradication.[4][5]
Annth rootof a numberx, wherenis a positive integer, is any of thenreal or complex numbersrwhosenth power isx:
rn=x.{\displaystyle r^{n}=x.}
Every positivereal numberxhas a single positiventh root, called theprincipalnth root, which is writtenxn{\displaystyle {\sqrt[{n}]{x}}}. Fornequal to 2 this is called the principal square root and thenis omitted. Thenth root can also be represented usingexponentiationasx1/n.
For even values ofn, positive numbers also have a negativenth root, while negative numbers do not have a realnth root. For odd values ofn, every negative numberxhas a real negativenth root. For example, −2 has a real 5th root,−25=−1.148698354…{\displaystyle {\sqrt[{5}]{-2}}=-1.148698354\ldots }but −2 does not have any real 6th roots.
Every non-zero numberx, real orcomplex, hasndifferent complex numbernth roots. (In the casexis real, this count includes any realnth roots.) The only complex root of 0 is 0.
Thenth roots of almost all numbers (all integers except thenth powers, and all rationals except the quotients of twonth powers) areirrational. For example,
2=1.414213562…{\displaystyle {\sqrt {2}}=1.414213562\ldots }
Allnth roots of rational numbers arealgebraic numbers, and allnth roots of integers arealgebraic integers.
The term "surd" traces back toAl-Khwarizmi(c.825), who referred to rational and irrational numbers asaudibleandinaudible, respectively. This later led to the Arabic wordأصم(asamm, meaning "deaf" or "dumb") forirrational numberbeing translated into Latin assurdus(meaning "deaf" or "mute").Gerard of Cremona(c.1150),Fibonacci(1202), and thenRobert Recorde(1551) all used the term to refer tounresolved irrational roots, that is, expressions of the formrn{\displaystyle {\sqrt[{n}]{r}}}, in whichn{\displaystyle n}andr{\displaystyle r}are integer numerals and the whole expression denotes an irrational number.[6]Irrational numbers of the form±a,{\displaystyle \pm {\sqrt {a}},}wherea{\displaystyle a}is rational, are calledpure quadratic surds; irrational numbers of the forma±b{\displaystyle a\pm {\sqrt {b}}}, wherea{\displaystyle a}andb{\displaystyle b}are rational, are calledmixed quadratic surds.[7]
Asquare rootof a numberxis a numberrwhich, whensquared, becomesx:
r2=x.{\displaystyle r^{2}=x.}
Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as theprincipal square root, and is denoted with a radical sign:
25=5.{\displaystyle {\sqrt {25}}=5.}
Since the square of every real number is nonnegative, negative numbers do not have real square roots. However, for every negative real number there are twoimaginarysquare roots. For example, the square roots of −25 are 5iand −5i, whereirepresents a number whose square is−1.
Acube rootof a numberxis a numberrwhosecubeisx:
r3=x.{\displaystyle r^{3}=x.}
Every real numberxhas exactly one real cube root, writtenx3{\displaystyle {\sqrt[{3}]{x}}}. For example,
83=2−83=−2.{\displaystyle {\begin{aligned}{\sqrt[{3}]{8}}&=2\\{\sqrt[{3}]{-8}}&=-2.\end{aligned}}}
Every real number has two additionalcomplexcube roots.
Expressing the degree of annth root in its exponent form, as inx1/n{\displaystyle x^{1/n}}, makes it easier to manipulate powers and roots. Ifa{\displaystyle a}is anon-negative real number,
amn=(am)1/n=am/n=(a1/n)m=(an)m.{\displaystyle {\sqrt[{n}]{a^{m}}}=(a^{m})^{1/n}=a^{m/n}=(a^{1/n})^{m}=({\sqrt[{n}]{a}})^{m}.}
Every non-negative number has exactly one non-negative realnth root, and so the rules for operations with surds involving non-negative radicandsa{\displaystyle a}andb{\displaystyle b}are straightforward within the real numbers:
abn=anbnabn=anbn{\displaystyle {\begin{aligned}{\sqrt[{n}]{ab}}&={\sqrt[{n}]{a}}{\sqrt[{n}]{b}}\\{\sqrt[{n}]{\frac {a}{b}}}&={\frac {\sqrt[{n}]{a}}{\sqrt[{n}]{b}}}\end{aligned}}}
Subtleties can occur when taking thenth roots of negative orcomplex numbers. For instance:
−1×−1≠−1×−1=1,{\displaystyle {\sqrt {-1}}\times {\sqrt {-1}}\neq {\sqrt {-1\times -1}}=1,\quad }
but, rather,
−1×−1=i×i=i2=−1.{\displaystyle \quad {\sqrt {-1}}\times {\sqrt {-1}}=i\times i=i^{2}=-1.}
Since the rulean×bn=abn{\displaystyle {\sqrt[{n}]{a}}\times {\sqrt[{n}]{b}}={\sqrt[{n}]{ab}}}strictly holds for non-negative real radicands only, its application leads to the inequality in the first step above.
Anon-nested radical expressionis said to be insimplified formif no factor of the radicand can be written as a power greater than or equal to the index; there are no fractions inside the radical sign; and there are no radicals in the denominator.[8]
For example, to write the radical expression32/5{\displaystyle \textstyle {\sqrt {32/5}}}in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:
325=16⋅25=16⋅25=425{\displaystyle {\sqrt {\frac {32}{5}}}={\sqrt {\frac {16\cdot 2}{5}}}={\sqrt {16}}\cdot {\sqrt {\frac {2}{5}}}=4{\sqrt {\frac {2}{5}}}}
Next, there is a fraction under the radical sign, which we change as follows:
425=425{\displaystyle 4{\sqrt {\frac {2}{5}}}={\frac {4{\sqrt {2}}}{\sqrt {5}}}}
Finally, we remove the radical from the denominator as follows:
425=425⋅55=4105=4510{\displaystyle {\frac {4{\sqrt {2}}}{\sqrt {5}}}={\frac {4{\sqrt {2}}}{\sqrt {5}}}\cdot {\frac {\sqrt {5}}{\sqrt {5}}}={\frac {4{\sqrt {10}}}{5}}={\frac {4}{5}}{\sqrt {10}}}
When there is a denominator involving surds it is always possible to find a factor to multiply both numerator and denominator by to simplify the expression.[9][10]For instance using thefactorization of the sum of two cubes:
1a3+b3=a23−ab3+b23(a3+b3)(a23−ab3+b23)=a23−ab3+b23a+b.{\displaystyle {\frac {1}{{\sqrt[{3}]{a}}+{\sqrt[{3}]{b}}}}={\frac {{\sqrt[{3}]{a^{2}}}-{\sqrt[{3}]{ab}}+{\sqrt[{3}]{b^{2}}}}{\left({\sqrt[{3}]{a}}+{\sqrt[{3}]{b}}\right)\left({\sqrt[{3}]{a^{2}}}-{\sqrt[{3}]{ab}}+{\sqrt[{3}]{b^{2}}}\right)}}={\frac {{\sqrt[{3}]{a^{2}}}-{\sqrt[{3}]{ab}}+{\sqrt[{3}]{b^{2}}}}{a+b}}.}
Simplifying radical expressions involvingnested radicalscan be quite difficult. In particular, denesting is not always possible, and when possible, it may involve advancedGalois theory. Moreover, when complete denesting is impossible, there is no generalcanonical formsuch that the equality of two numbers can be tested by simply looking at their canonical expressions.
For example, it is not obvious that
3+22=1+2.{\displaystyle {\sqrt {3+2{\sqrt {2}}}}=1+{\sqrt {2}}.}
The above can be derived through:
3+22=1+22+2=12+22+22=(1+2)2=1+2{\displaystyle {\sqrt {3+2{\sqrt {2}}}}={\sqrt {1+2{\sqrt {2}}+2}}={\sqrt {1^{2}+2{\sqrt {2}}+{\sqrt {2}}^{2}}}={\sqrt {\left(1+{\sqrt {2}}\right)^{2}}}=1+{\sqrt {2}}}
Letr=p/q{\displaystyle r=p/q}, withpandqcoprime and positive integers. Thenrn=pn/qn{\displaystyle {\sqrt[{n}]{r}}={\sqrt[{n}]{p}}/{\sqrt[{n}]{q}}}is rational if and only if bothpn{\displaystyle {\sqrt[{n}]{p}}}andqn{\displaystyle {\sqrt[{n}]{q}}}are integers, which means that bothpandqarenth powers of some integer.
The radical or root may be represented by theinfinite series:
(1+x)st=∑n=0∞∏k=0n−1(s−kt)n!tnxn{\displaystyle (1+x)^{\frac {s}{t}}=\sum _{n=0}^{\infty }{\frac {\prod _{k=0}^{n-1}(s-kt)}{n!t^{n}}}x^{n}}
with|x|<1{\displaystyle |x|<1}. This expression can be derived from thebinomial series.
Thenth root of a numberAcan be computed withNewton's method, which starts with an initial guessx0and then iterates using therecurrence relation
xk+1=xk−xkn−Anxkn−1{\displaystyle x_{k+1}=x_{k}-{\frac {x_{k}^{n}-A}{nx_{k}^{n-1}}}}
until the desired precision is reached. For computational efficiency, the recurrence relation is commonly rewritten
xk+1=n−1nxk+An1xkn−1.{\displaystyle x_{k+1}={\frac {n-1}{n}}\,x_{k}+{\frac {A}{n}}\,{\frac {1}{x_{k}^{n-1}}}.}
This allows to have only oneexponentiation, and to compute once for all the first factor of each term.
For example, to find the fifth root of 34, we plug inn= 5,A= 34andx0= 2(initial guess). The first 5 iterations are, approximately:
(All correct digits shown.)
The approximationx4is accurate to 25 decimal places andx5is good for 51.
Newton's method can be modified to produce variousgeneralized continued fractionsfor thenth root. For example,
zn=xn+yn=x+ynxn−1+(n−1)y2x+(n+1)y3nxn−1+(2n−1)y2x+(2n+1)y5nxn−1+(3n−1)y2x+⋱.{\displaystyle {\sqrt[{n}]{z}}={\sqrt[{n}]{x^{n}+y}}=x+{\cfrac {y}{nx^{n-1}+{\cfrac {(n-1)y}{2x+{\cfrac {(n+1)y}{3nx^{n-1}+{\cfrac {(2n-1)y}{2x+{\cfrac {(2n+1)y}{5nx^{n-1}+{\cfrac {(3n-1)y}{2x+\ddots }}}}}}}}}}}}.}
Building on thedigit-by-digit calculation of a square root, it can be seen that the formula used there,x(20p+x)≤c{\displaystyle x(20p+x)\leq c}, orx2+20xp≤c{\displaystyle x^{2}+20xp\leq c}, follows a pattern involving Pascal's triangle. For thenth root of a numberP(n,i){\displaystyle P(n,i)}is defined as the value of elementi{\displaystyle i}in rown{\displaystyle n}of Pascal's Triangle such thatP(4,1)=4{\displaystyle P(4,1)=4}, we can rewrite the expression as∑i=0n−110iP(n,i)pixn−i{\displaystyle \sum _{i=0}^{n-1}10^{i}P(n,i)p^{i}x^{n-i}}. For convenience, call the result of this expressiony{\displaystyle y}. Using this more general expression, any positive principal root can be computed, digit-by-digit, as follows.
Write the original number in decimal form. The numbers are written similar to thelong divisionalgorithm, and, as in long division, the root will be written on the line above. Now separate the digits into groups of digits equating to the root being taken, starting from the decimal point and going both left and right. The decimal point of the root will be above the decimal point of the radicand. One digit of the root will appear above each group of digits of the original number.
Beginning with the left-most group of digits, do the following procedure for each group:
Find the square root of 152.2756.
Algorithm terminates: Answer is 12.34
Find the cube root of 4192 truncated to the nearest thousandth.
The desired precision is achieved. The cube root of 4192 is 16.124...
The principalnth root of a positive number can be computed usinglogarithms. Starting from the equation that definesras annth root ofx, namelyrn=x,{\displaystyle r^{n}=x,}withxpositive and therefore its principal rootralso positive, one takes logarithms of both sides (anybase of the logarithmwill do) to obtain
nlogbr=logbxhencelogbr=logbxn.{\displaystyle n\log _{b}r=\log _{b}x\quad \quad {\text{hence}}\quad \quad \log _{b}r={\frac {\log _{b}x}{n}}.}
The rootris recovered from this by taking theantilog:
r=b1nlogbx.{\displaystyle r=b^{{\frac {1}{n}}\log _{b}x}.}
(Note: That formula showsbraised to the power of the result of the division, notbmultiplied by the result of the division.)
For the case in whichxis negative andnis odd, there is one real rootrwhich is also negative. This can be found by first multiplying both sides of the defining equation by −1 to obtain|r|n=|x|,{\displaystyle |r|^{n}=|x|,}then proceeding as before to find |r|, and usingr= −|r|.
Theancient Greek mathematiciansknew how touse compass and straightedgeto construct a length equal to the square root of a given length, when an auxiliary line of unit length is given. In 1837Pierre Wantzelproved that annth root of a given length cannot be constructed ifnis not a power of 2.[11]
Everycomplex numberother than 0 hasndifferentnth roots.
The two square roots of a complex number are always negatives of each other. For example, the square roots of−4are2iand−2i, and the square roots ofiare
12(1+i)and−12(1+i).{\displaystyle {\tfrac {1}{\sqrt {2}}}(1+i)\quad {\text{and}}\quad -{\tfrac {1}{\sqrt {2}}}(1+i).}
If we express a complex number inpolar form, then the square root can be obtained by taking the square root of the radius and halving the angle:
reiθ=±r⋅eiθ/2.{\displaystyle {\sqrt {re^{i\theta }}}=\pm {\sqrt {r}}\cdot e^{i\theta /2}.}
Aprincipalroot of a complex number may be chosen in various ways, for example
reiθ=r⋅eiθ/2{\displaystyle {\sqrt {re^{i\theta }}}={\sqrt {r}}\cdot e^{i\theta /2}}
which introduces abranch cutin thecomplex planealong thepositive real axiswith the condition0 ≤θ< 2π, or along the negative real axis with−π<θ≤π.
Using the first(last) branch cut the principal square rootz{\displaystyle \scriptstyle {\sqrt {z}}}mapsz{\displaystyle \scriptstyle z}to the half plane with non-negative imaginary(real) part. The last branch cut is presupposed in mathematical software likeMatlaborScilab.
The number 1 hasndifferentnth roots in the complex plane, namely
1,ω,ω2,…,ωn−1,{\displaystyle 1,\;\omega ,\;\omega ^{2},\;\ldots ,\;\omega ^{n-1},}
where
ω=e2πin=cos(2πn)+isin(2πn).{\displaystyle \omega =e^{\frac {2\pi i}{n}}=\cos \left({\frac {2\pi }{n}}\right)+i\sin \left({\frac {2\pi }{n}}\right).}
These roots are evenly spaced around theunit circlein the complex plane, at angles which are multiples of2π/n{\displaystyle 2\pi /n}. For example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1,i{\displaystyle i}, −1, and−i{\displaystyle -i}.
Every complex number hasndifferentnth roots in the complex plane. These are
η,ηω,ηω2,…,ηωn−1,{\displaystyle \eta ,\;\eta \omega ,\;\eta \omega ^{2},\;\ldots ,\;\eta \omega ^{n-1},}
whereηis a singlenth root, and 1,ω,ω2, ...ωn−1are thenth roots of unity. For example, the four different fourth roots of 2 are
24,i24,−24,and−i24.{\displaystyle {\sqrt[{4}]{2}},\quad i{\sqrt[{4}]{2}},\quad -{\sqrt[{4}]{2}},\quad {\text{and}}\quad -i{\sqrt[{4}]{2}}.}
Inpolar form, a singlenth root may be found by the formula
reiθn=rn⋅eiθ/n.{\displaystyle {\sqrt[{n}]{re^{i\theta }}}={\sqrt[{n}]{r}}\cdot e^{i\theta /n}.}
Hereris the magnitude (the modulus, also called theabsolute value) of the number whose root is to be taken; if the number can be written asa+bithenr=a2+b2{\displaystyle r={\sqrt {a^{2}+b^{2}}}}. Also,θ{\displaystyle \theta }is the angle formed as one pivots on the origin counterclockwise from the positive horizontal axis to a ray going from the origin to the number; it has the properties thatcosθ=a/r,{\displaystyle \cos \theta =a/r,}sinθ=b/r,{\displaystyle \sin \theta =b/r,}andtanθ=b/a.{\displaystyle \tan \theta =b/a.}
Thus findingnth roots in the complex plane can be segmented into two steps. First, the magnitude of all thenth roots is thenth root of the magnitude of the original number. Second, the angle between the positive horizontal axis and a ray from the origin to one of thenth roots isθ/n{\displaystyle \theta /n}, whereθ{\displaystyle \theta }is the angle defined in the same way for the number whose root is being taken. Furthermore, allnof thenth roots are at equally spaced angles from each other.
Ifnis even, a complex number'snth roots, of which there are an even number, come inadditive inversepairs, so that if a numberr1is one of thenth roots thenr2= −r1is another. This is because raising the latter's coefficient −1 to thenth power for evennyields 1: that is, (−r1)n= (−1)n×r1n=r1n.
As with square roots, the formula above does not define acontinuous functionover the entire complex plane, but instead has abranch cutat points whereθ/nis discontinuous.
It was onceconjecturedthat allpolynomial equationscould besolved algebraically(that is, that all roots of apolynomialcould be expressed in terms of a finite number of radicals andelementary operations). However, while this is true for third degree polynomials (cubics) and fourth degree polynomials (quartics), theAbel–Ruffini theorem(1824) shows that this is not true in general when the degree is 5 or greater. For example, the solutions of the equation
x5=x+1{\displaystyle x^{5}=x+1}
cannot be expressed in terms of radicals. (cf.quintic equation)
Assume thatxn{\displaystyle {\sqrt[{n}]{x}}}is rational. That is, it can be reduced to a fractionab{\displaystyle {\frac {a}{b}}}, whereaandbare integers without a common factor.
This means thatx=anbn{\displaystyle x={\frac {a^{n}}{b^{n}}}}.
Sincexis an integer,an{\displaystyle a^{n}}andbn{\displaystyle b^{n}}must share a common factor ifb≠1{\displaystyle b\neq 1}. This means that ifb≠1{\displaystyle b\neq 1},anbn{\displaystyle {\frac {a^{n}}{b^{n}}}}is not in simplest form. Thusbshould equal 1.
Since1n=1{\displaystyle 1^{n}=1}andn1=n{\displaystyle {\frac {n}{1}}=n},anbn=an{\displaystyle {\frac {a^{n}}{b^{n}}}=a^{n}}.
This means thatx=an{\displaystyle x=a^{n}}and thus,xn=a{\displaystyle {\sqrt[{n}]{x}}=a}. This implies thatxn{\displaystyle {\sqrt[{n}]{x}}}is an integer. Sincexis not a perfectnth power, this is impossible. Thusxn{\displaystyle {\sqrt[{n}]{x}}}is irrational.
|
https://en.wikipedia.org/wiki/Nth_root
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.