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Chris Sullo	handled approved new vulnerabilities added database well manages web checks.addition chris cofounder treasurer open security foundation.nikto sullo author nikto leading open source web security assessment tool.open source web server scanner.nikto known perform comprehensive tests web servers multiple items including thousands potentially dangerous files cgi scripts etc.
Chris Sullo	nikto voted top security tool fyodors top security tools survey.rvasec cofounder rvasec security conference held richmond virginia usa.security advisories sullo long involved world electronic crime security.also published security advisories.one noteworthy advisories advisory verity ultraseek search engine used dozens government agencies universities.
Logjam (computer security)	references external links interview chris sullo cgisecurity.comdiscovered team computer scientists publicly reported may .discoverers able demonstrate attack bit us exportgrade dh systems.estimated state level attacker could bit systems widely used thereby allowing decryption significant fraction internet traffic.recommended upgrading least bits shared prime systems.details diffie–hellman key exchange depends security presumed difficulty solving discrete logarithm problem.
Logjam (computer security)	authors took advantage fact number field sieve algorithm generally effective method finding discrete logarithms consists four large computational steps first three depend order group g specific number whose finite log desired.results first three steps precomputed saved used solve discrete log problem prime group relatively short time.vulnerability known early .turns much internet traffic uses one handful groups order bits less.one approach enabled vulnerability authors demonstrated using maninthemiddle network attacker downgrade transport layer security tls connection use bit dh exportgrade cryptography allowing read exchanged data inject data connection.affects https smtps imaps protocols among others.
Logjam (computer security)	authors needed several thousand cpu cores week precompute data single bit prime.done however individual logarithms could solved minute using two core intel xeon cpus.cve id cve.the authors also estimated feasibility attack bit diffie–hellman primes.design many diffie–hellman implementations use pregenerated prime field.considered secure since discrete log problem still considered hard bigenough primes even group known reused.
Logjam (computer security)	researchers calculated cost creating logjam precomputation one bit prime hundreds millions usd noted well within range fy .billion u.s. consolidated cryptologic program includes nsa.reuse primes generating precomputation one prime would break twothirds vpns quarter ssh servers globally.researchers noted attack fits claims leaked nsa papers nsa able break much current cryptography.recommend using primes bits defense switching ellipticcurve diffie–hellman ecdh.
Security awareness	claims practical implications attack however disputed security researchers eyal ronen adi shamir paper critical review imperfect forward secrecy.test tools tlsbased services web servers offering https checked vulnerability using scanners sslyze qualys ssl server test immuniweb ssl security test ssltrust cryptcheck cyphercraft testssl.sh keycdn.com scanner.ssh servers tested using sshweakdh tool.responses may microsoft released patch internet explorer.on june tor project provided patch logjam tor browser.on june apple released patch os x yosemite ios operating system.on june mozilla project released fix firefox browser.on september google released fix chrome browser.on december ietf published rfc called increase secure shell minimum recommended diffiehellman modulus size bits.see also beast computer securitybreach security exploitcrimepoodleservergated cryptography references external links logjam attacknsa ppoly power precomputationshtetl optimizedlmany organizations require formal security awareness training workers join organization periodically thereafter usually annually.coverage topics covered security awareness training includethe nature sensitive material physical assets may come contact trade secrets privacy concerns government classified informationemployee contractor responsibilities handling sensitive information including review employee nondisclosure agreementsrequirements proper handling sensitive material physical form including marking transmission storage destructionproper methods protecting sensitive information computer systems including password policy use twofactor authenticationother computer security concerns including malware phishing social engineering etc.workplace security including building access wearing security badges reporting incidents forbidden articles etc.consequences failure properly protect information including potential loss employment economic consequences firm damage individuals whose private records divulged possible civil criminal penaltiessecurity awareness means understanding potential people deliberately accidentally steal damage misuse data stored within companys computer systems throughout organization.therefore would prudent support assets institution information physical personal trying stop happening.according european network information security agency awareness risks available safeguards first line defence security information systems networks.the focus security awareness consultancy achieve long term shift attitude employees towards security whilst promoting cultural behavioural change within organisation.security policies viewed key enablers organisation series rules restricting efficient working business.measuring security awareness study researchers developed method measuring security awareness.
ESOFT Metro Campus	specifically measured understanding circumventing security protocols disrupting intended functions systems collecting valuable information getting caught p. .researchers created method could distinguish experts novices people organize different security scenarios groups.experts organize scenarios based centralized security themes novices organize scenarios based superficial themes.see also access controlphysical securitysecuritysecurity controlssecurity managementisoiec internet security awareness training references external links armys security education training awareness sitethe security awareness cyclecomparison security awareness training companiesframeworks interface conversion permissive application designer doesnt mark specific fields immutable possible overwrite fields never intended modified outside e.g.admin permissions flag.these vulnerabilities found applications written ruby rails asp.net mvc java play framework.in mass assignment ruby rails allowed bypassing mapping restrictions resulted proof concept injection unauthorized ssh public keys user accounts github.vulnerabilities ruby rails allowed creation internal objects specially crafted json structure.in asp.net core mapping restriction declared using bindnever attribute.see also data transfer object dto referencesoffers academic professional qualifications computing business management engineering hospitality english.established year company today branches nationwide.headed dr. dayan rajapakse.early years esoft established kirilapone suburban area colombo sri lanka initially offered training services students preparing bcs uk professional examinations.subsequently moved present location bambalapitiya colombo hub training sri lanka.
ESOFT Metro Campus	programmes programmes taught esoft metro campus organised five schools divisions.school computing school business school engineering technology school hospitality management language academy.school computing degree programmes available esoft school computing include bachelors information technology bit awarded university colombo school computing ucsc bsc hons computing beng hons software engineering awarded london metropolitan university.students also opt follow professional qualifications certificate diploma post graduate diploma pgd awarded british computer society bcs chartered institute btec higher national diploma hnd computing systems development awarded edexcel.in addition esoft also conducts several vendor certification programmes including microsoft cisco oracle.school business esoft school business conducts classes bachelors masters level business administration qualifications awarded london metropolitan university.students also opt follow professional qualifications btec higher national diploma hnd business management awarded edexcel.school engineering technology esoft school engineering technology conducts classes bachelors masters level qualifications mechanical engineering electrical electronic engineering civil engineering construction management.
ESOFT Metro Campus	awarded kingston university london.school hospitality management esoft school hospitality management offers professional qualifications btec hnds travel tourism management hospitality management awarded edexcel.language academy language academy offers pearson assured courses academic professional english.recognition although esoft started teaching bcs exams esoft designated accredited course provider bcs year recognition standards maintained course delivery.esoft organisation world accredited three levels bcs higher education qualifications november .
ESOFT Metro Campus	esoft also produced several prize winners high achievers past years bcs bit programmes university moratuwa signed memorandum understanding esoft appointing collaborative partner offering bachelor information technology bit external degree.appointment made extensive evaluation process esoft one three partners appointed.esoft currently conducts bit programme colombo kandy kurunegala jaffna.university moratuwa stopped conducting programme online working purely via partner mode.edexcel uk accredited esoft conduct higher national diploma hnd programmes computing since business management since .in september esoft metro campus colombo recognised ministry higher education university grants commission ugc non state degree awarding institution.means esoft officially award degrees.
ESOFT Metro Campus	first degree introduced bachelor information technology honours degree.bachelor business management honours degree approved first intake scheduled february.partnerships affiliations esoft appointed accredited course provider bcs heq professional examinations uk.especially noteworthy since esoft organisation accredited three levels higher education qualifications heq world.in esoft appointed approved centre edexcel conducting level programmes sri lanka.in esoft entered agreement university moratuwa appointed collaborative partner university offering bachelor external degree.in esoft partnered kingston university uk offer undergraduate postgraduate qualifications various disciplines engineering.later partnership expanded cover undergraduate postgraduate qualifications it.in esoft entered strategic exclusive partnership london metropolitan university uk offer undergraduate postgraduate qualifications sri lanka.in esoft virtusa signed mou campus reach initiative provide internship training opportunities undergraduate students esoft.
ESOFT Metro Campus	esoft partnered thesmartbridge singapore offer projectbased training undergraduate students it.‘smart internz’ global platform academia corporate giving complete training transforming knowledge work experience via guided projects externship programmes.csr activities esoft involved several corporate social responsibility activities one recent platinum level sponsorship edex education careers expo held january .esoft also mobile unit bus fitted workstations sent rural areas schools provide positive experiences students may even seen used computer before.as esoft cosponsor royal college blue gold hockey tournament organised participation school teams around nation esoft started ethilina project partnership sirasa tv.aim project donate complete computing labs deserving rural schools provide children opportunity learn itict early age.
ESOFT Metro Campus	project continued year well.lab consists minimum modern networked pcs along laser printer.tables chairs carpeting also provided complete cost borne esoft.sirasa tv provides coverage via media network.two computer labs opened president sri lanka hon mohomed munshid.
ESOFT Metro Campus	ethilina project concluded successful run years computer labs donated public schools around country.esoft also partnered ranaviru seva authority state ministry defence ministry defence offer scholarships savings course fees members armed forces families.special loyalty card known virusara privilege launched president maithripala sirisena.in esoft launched islandwide scholarship programme english partnership ministry youth sports state ministry digital technology enterprise development.total full scholarships half scholarships awarded part programme.awards recognitions awarded pearson previously edexcel fastest developing centre sri lanka.awarded gold partner pearson remarkable growth student numbers revenue.
ESOFT Metro Campus	awarded gold bcs best centre large scale sector bcs academy awards.national business excellence award education services merit award.national business excellence award education services runnersup.awarded gold partner pearson remarkable growth student numbers revenue.awarded gold bcs best centre large scale sector bcs academy awards held .. bmich colombo awarded gold partner pearson remarkable growth student numbers revenue.
ESOFT Metro Campus	awarded platinum partner pearson high growth student numbers revenue market leader.awarded year .awarded platinum partner pearson second year running exceptional performance market leader.awarded year .awarded platinum partner pearson third year running.
List of data breaches	awarded degree awarding powers recognition nonstate higher education institution ministry education sri lanka.awarded platinum partner pearson fourth consecutive year chairman dr dayan rajapakse received national provincial gold awards entrepreneur year organised federation chambers commerce sri lanka admitted member association commonwealth universities acu esoft wins ‘best education institute’ award esoft digital campus mobile app south asian business excellence awards branching esoft started expanding regional areas launched accelerated expansion project aep increase total number branches end year branches opened thereafter referenceslist includes involving theft compromise records although many smaller breaches occur continually.breaches large organizations number records still unknown also listed.addition various methods used breaches listed hacking common.most breaches occur north america.estimated average cost data breach million global annual cost forecast .trillion.
List of data breaches	result data breaches estimated first half alone .billion records exposed.collection .billion identity records consisting million unique email addresses million unique passwords posted web sale.references sources
Web literacy	described content activity – i.e.web users learn web also make website.history concept latter part literacy researchers started explore differences printed text networkenabled devices screens.research largely focused two areas credibility information found world wide web difference hypertext makes reading writing.skills included definitions information literacy included sconul position paper .
Web literacy	paper became pillars information literacy last updated .web literacy map mozilla foundation nonprofit organization promotes openness innovation participation internet.created web literacy map consultation community stakeholders formal informal education well industry.web literacy described skills competencies needed reading writing participating web.work originally entitled web literacy standard began early .
Web literacy	version .launched mozilla festival later year.going forward standard seen problematic ethos mozilla community trying achieve.literacy version .web literacy map released early underpins mozilla foundations webmaker resources section learners mentors find activities help teach related areas.although web literacy map list strands skills competencies commonly represented competency grid.the mozilla community finalised version .
Noodle	web literacy map end march .involves small changes competencies layer comprehensive review skills contain.exploring navigating webnavigation using software tools browse webweb mechanics understanding web ecosystemsearch locating information people resources via webcredibility critically evaluating information found websecurity keeping systems identities content safe building creating webcomposing web creating curating contentremixing modifying existing web resources create something newdesign accessibility creating universally effective communications web resourcescodingscripting creating interactive experiences webinfrastructure understanding internet stack connecting participating websharing creating web resources otherscollaborating providing access web resourcescommunity participation getting involved web communities understanding practicesprivacy examining consequences sharing data onlineopen practices helping keep web democratic universally accessible see also cyber selfdefensecomputer literacydigital literacyinformation literacy referencesnoodles refrigerated shortterm storage dried stored future use.noodles usually cooked boiling water sometimes cooking oil salt added.also often panfried deepfried.noodle dishes include sauce noodles put soup.material composition geocultural origin specific type wide variety noodles.noodles staple food many cultures see chinese noodles japanese noodles korean noodles filipino noodles vietnamese noodles italian pasta.
Noodle	etymology word derived th century german word nudel.history origin earliest written record noodles found book dated eastern han period – ce.noodles made wheat dough became prominent food people han dynasty.oldest evidence noodles years ago china.team archaeologists reported finding earthenware bowl contained yearold noodles lajia archaeological site.
Noodle	noodles said resemble lamian type chinese noodle.analyzing husk phytoliths starch grains present sediment associated noodles identified millet belonging panicum miliaceum setaria italica.however researchers cast doubt lajias noodles made specifically millet difficult make pure millet noodles unclear whether analyzed residue directly derived lajias noodles starch morphology cooking shows distinctive alterations fit lajias noodles uncertain whether starchlike grains laijias noodles starch show nonstarch characteristics.food historians generally estimate pastas origin among mediterranean countries homogenous mixture flour water called itrion described nd century greek physician galen among rd th century palestinians itrium described jerusalem talmud itriyya arabic cognate greek word stringlike shapes made semolina dried cooking defined th century aramean physician lexicographer isho bar ali.historical variations east asia wheat noodles japan udon adapted chinese recipe early th century.innovations continued noodles made buckwheat naengmyeon developed joseon dynasty korea –.
Noodle	ramen noodles based southern chinese noodle dishes guangzhou named northern chinese lamian became common japan .central asia kesme erishte noodles eaten turkic peoples th century.west asia ash reshteh noodles thick soup herbs one popular dishes middle eastern countries iran brought turcomongol.europe st century bce horace wrote fried sheets dough called lagana.however cooking method doesnt correspond current definition either fresh dry pasta product.
Noodle	italy first concrete information pasta products italy dates th th centuries.pasta taken variety shapes often based regional specializations.germany area would become germany documents dating mention spätzle.medieval illustrations believed place noodle even earlier date.ancient israel diaspora latinized word itrium referred kind boiled dough.
Noodle	arabs adapted noodles long journeys fifth century first written record dry pasta.muhammad alidrisi wrote itriyya manufactured exported norman sicily.itriya also known persian jews early persian rule spoke aramaic islamic rule.referred small soup noodle greek origin prepared twisting bits kneaded dough shape resembling italian orzo.polish jews zacierki type noodle found polish jewish cuisine.it part rations distributed jewish victims łódź ghetto nazis.out major ghettos łódź affected hunger starvation malnutritionrelated deaths.the diary young jewish girl łódź recounts fight father spoonful zacierki taken familys meager supply grams week.
Noodle	types primary ingredient types dishes baked noodles boiled drained noodles combined ingredients baked.common examples include many casseroles.basic noodles cooked water broth drained.foods added noodles added foods see fried noodles noodles served plain dipping sauce oil added table.general noodles soft absorb flavors.chilled noodles noodles served cold sometimes salad.examples include thai glass noodle salad cold udon.fried noodles dishes made noodles stir fried various meats seafood vegetables dairy products.
HTTPS	typical examples include chow mein lo mein mie goreng hokkien mee varieties pancit yakisoba curry noodles pad thai.noodle soup noodles served broth.examples phở beef noodle soup chicken noodle soup ramen laksa saimin batchoy.preservation instant noodlesfrozen noodles see also references bibliography external links dictionary definition noodle wiktionaryused secure communication computer network widely used internet.https communication protocol encrypted using transport layer security tls formerly secure sockets layer ssl.protocol therefore also referred http tls http ssl.the principal motivations https authentication accessed website protection privacy integrity exchanged data transit.protects maninthemiddle attacks bidirectional encryption communications client server protects communications eavesdropping tampering.authentication aspect https requires trusted third party sign serverside digital certificates.
HTTPS	historically expensive operation meant fully authenticated https connections usually found secured payment transaction services secured corporate information systems world wide web.campaign electronic frontier foundation support web browser developers led protocol becoming prevalent.https used often web users original nonsecure http primarily protect page authenticity types websites secure accounts keep user communications identity web browsing private.overview uniform resource identifier uri scheme https identical usage syntax http scheme.however https signals browser use added encryption layer ssltls protect traffic.
HTTPS	ssltls especially suited http since provide protection even one side communication authenticated.case http transactions internet typically server authenticated client examining servers certificate.https creates secure channel insecure network.ensures reasonable protection eavesdroppers maninthemiddle attacks provided adequate cipher suites used server certificate verified trusted.because https piggybacks http entirely top tls entirety underlying http protocol encrypted.includes request url particular web page requested query parameters headers cookies often contain identifying information user.however website addresses port numbers necessarily part underlying tcpip protocols https cannot protect disclosure.
HTTPS	practice means even correctly configured web server eavesdroppers infer ip address port number web server sometimes even domain name e.g.rest url user communicating along amount data transferred duration communication though content communication.web browsers know trust https websites based certificate authorities come preinstalled software.certificate authorities way trusted web browser creators provide valid certificates.therefore user trust https connection website following truethe user trusts browser software correctly implements https correctly preinstalled certificate authorities.the user trusts certificate authority vouch legitimate websites.the website provides valid certificate means signed trusted authority.the certificate correctly identifies website e.g.browser visits httpsexample.com received certificate properly example.com entity.the user trusts protocols encryption layer ssltls sufficiently secure eavesdroppers.https especially important insecure networks networks may subject tampering.
HTTPS	insecure networks public wifi access points allow anyone local network packetsniff discover sensitive information protected https.additionally freetouse paid wlan networks observed tampering webpages engaging packet injection order serve ads websites.practice exploited maliciously many ways injecting malware onto webpages stealing users private information.https also important connections tor network malicious tor nodes could otherwise damage alter contents passing insecure fashion inject malware connection.one reason electronic frontier foundation tor project started development https everywhere included tor browser.as information revealed global mass surveillance criminals stealing personal information use https security websites becoming increasingly important regardless type internet connection used.even though metadata individual pages user visits might considered sensitive aggregated reveal lot user compromise users privacy.deploying https also allows use http predecessor nowdeprecated protocol spdy new generation http designed reduce page load times size latency.it recommended use http strict transport security hsts https protect users maninthemiddle attacks especially ssl stripping.https confused seldomused secure http shttp specified rfc .
HTTPS	usage websites april .alexa top websites use https default .internets popular websites secure implementation https page loads measured firefox telemetry use https.browser integration browsers display warning receive invalid certificate.older browsers connecting site invalid certificate would present user dialog box asking whether wanted continue.
HTTPS	newer browsers display warning across entire window.newer browsers also prominently display sites security information address bar.extended validation certificates show legal entity certificate information.browsers also display warning user visiting site contains mixture encrypted unencrypted content.additionally many web filters return security warning visiting prohibited websites.the electronic frontier foundation opining ideal world every web request could defaulted https provided addon called https everywhere mozilla firefox google chrome chromium android enables https default hundreds frequently used websites.forcing web browser load https content supported firefox starting version .
HTTPS	security security https underlying tls typically uses longterm public private keys generate shortterm session key used encrypt data flow client server.x. certificates used authenticate server sometimes client well.consequence certificate authorities public key certificates necessary verify relation certificate owner well generate sign administer validity certificates.beneficial verifying identities via web trust mass surveillance disclosures drew attention certificate authorities potential weak point allowing maninthemiddle attacks.important property context forward secrecy ensures encrypted communications recorded past cannot retrieved decrypted longterm secret keys passwords compromised future.
HTTPS	web servers provide forward secrecy.for https effective site must completely hosted https.sites contents loaded http scripts images example certain page contains sensitive information login page loaded https rest site loaded plain http user vulnerable attacks surveillance.additionally cookies site served https must secure attribute enabled.site sensitive information user session get exposed every time site accessed http instead https.technical difference http https urls begin https use port default whereas http urls begin http use port default.http encrypted thus vulnerable maninthemiddle eavesdropping attacks let attackers gain access website accounts sensitive information modify webpages inject malware advertisements.
HTTPS	https designed withstand attacks considered secure exception https implementations use deprecated versions ssl.network layers http operates highest layer tcpip model—the application layer tls security protocol operating lower sublayer layer encrypts http message prior transmission decrypts message upon arrival.strictly speaking https separate protocol refers use ordinary http encrypted ssltls connection.https encrypts message contents including http headers requestresponse data.exception possible cca cryptographic attack described limitations section attacker able discover connection taking place two parties along domain names ip addresses.server setup prepare web server accept https connections administrator must create public key certificate web server.
HTTPS	certificate must signed trusted certificate authority web browser accept without warning.authority certifies certificate holder operator web server presents it.web browsers generally distributed list signing certificates major certificate authorities verify certificates signed them.acquiring certificates number commercial certificate authorities exist offering paidfor ssltls certificates number types including extended validation certificates.lets encrypt launched april provides free automated service delivers basic ssltls certificates websites.according electronic frontier foundation lets encrypt make switching http https easy issuing one command clicking one button.
HTTPS	majority web hosts cloud providers leverage lets encrypt providing free certificates customers.use access control system also used client authentication order limit access web server authorized users.site administrator typically creates certificate user user loads browser.normally certificate contains name email address authorized user automatically checked server connection verify users identity potentially without even requiring password.case compromised secret private key important property context perfect forward secrecy pfs.
HTTPS	possessing one longterm asymmetric secret keys used establish https session make easier derive shortterm session key decrypt conversation even later time.diffie–hellman key exchange dhe elliptic curve diffie–hellman key exchange ecdhe schemes known property.firefox opera chromium browser sessions used nearly apples safari microsoft internet explorer sessions.tls .published august dropped support ciphers without forward secrecy.
HTTPS	february .web servers surveyed support form forward secrecy .use forward secrecy browsers.certificate revocation certificate may revoked expires example secrecy private key compromised.newer versions popular browsers firefox opera internet explorer windows vista implement online certificate status protocol ocsp verify case.
HTTPS	browser sends certificates serial number certificate authority delegate via ocsp online certificate status protocol authority responds telling browser whether certificate still valid not.ca may also issue crl tell people certificates revoked.crls longer required cabrowser forum nevertheless still widely used cas.revocation statuses internet disappear soon expiration certificates.limitations ssl secure sockets layer tls transport layer security encryption configured two modes simple mutual.
HTTPS	simple mode authentication performed server.mutual version requires user install personal client certificate web browser user authentication.either case level protection depends correctness implementation software cryptographic algorithms use.ssltls prevent indexing site web crawler cases uri encrypted resource inferred knowing intercepted requestresponse size.allows attacker access plaintext publicly available static content encrypted text encrypted version static content permitting cryptographic attack.because tls operates protocol level http knowledge higherlevel protocols tls servers strictly present one certificate particular address port combination.past meant feasible use namebased virtual hosting https.
HTTPS	solution called server name indication sni exists sends hostname server encrypting connection although many old browsers support extension.support sni available since firefox opera apple safari .google chrome internet explorer windows vista.from architectural point viewan ssltls connection managed first front machine initiates tls connection.reasons routing traffic optimization etc.front machine application server decipher data solutions found propagate user authentication information certificate application server needs know going connected.for ssltls mutual authentication ssltls session managed first server initiates connection.
HTTPS	situations encryption propagated along chained servers session timeout management becomes extremely tricky implement.security maximal mutual ssltls clientside way properly end ssltls connection disconnect user except waiting server session expire closing related client applications.a sophisticated type maninthemiddle attack called ssl stripping presented blackhat conference.type attack defeats security provided https changing https link http link taking advantage fact internet users actually type https browser interface get secure site clicking link thus fooled thinking using https fact using http.attacker communicates clear client.prompted development countermeasure http called http strict transport security.https shown vulnerable range traffic analysis attacks.traffic analysis attacks type sidechannel attack relies variations timing size traffic order infer properties encrypted traffic itself.
HTTPS	traffic analysis possible ssltls encryption changes contents traffic minimal impact size timing traffic.may research paper researchers microsoft research indiana university discovered detailed sensitive user data inferred side channels packet sizes.researchers found despite https protection several highprofile topoftheline web applications healthcare taxation investment web search eavesdropper could infer illnessesmedicationssurgeries user hisher family income investment secrets.although work demonstrated vulnerability https traffic analysis approach presented authors required manual analysis focused specifically web applications protected https.the fact modern websites including google yahoo amazon use https causes problems many users trying access public wifi hot spots wifi hot spot login page fails load user tries open https resource.several websites neverssl.com nonhttps.com guarantee always remain accessible http.
HTTPS	history netscape communications created https netscape navigator web browser.originally https used ssl protocol.ssl evolved transport layer security tls https formally specified rfc may .google announced february chrome browser would mark http sites secure july .move encourage website owners implement https effort make world wide web secure.
Prime number	see also bullrun decryption program – secret antiencryption program run us national security agencycomputer securityhstsopportunistic encryptionstunnel references external links rfc http tlsrfc transport layer security protocol .rfc secure sockets layer ssl protocol version .how https works ...in comicis tls fast yetnatural number greater prime called composite number.example prime ways writing product × × involve itself.however composite product × numbers smaller .primes central number theory fundamental theorem arithmetic every natural number greater either prime factorized product primes unique order.the property prime called primality.simple slow method checking primality given number n displaystyle n called trial division tests whether n displaystyle n multiple integer n displaystyle sqrt n .faster algorithms include miller–rabin primality test fast small chance error aks primality test always produces correct answer polynomial time slow practical.
Prime number	particularly fast methods available numbers special forms mersenne numbers.december largest known prime number mersenne prime decimal digits.there infinitely many primes demonstrated euclid around bc.known simple formula separates prime numbers composite numbers.however distribution primes within natural numbers large statistically modelled.first result direction prime number theorem proven end th century says probability randomly chosen large number prime inversely proportional number digits logarithm.several historical questions regarding prime numbers still unsolved.
Prime number	include goldbachs conjecture every even integer greater expressed sum two primes twin prime conjecture infinitely many pairs primes one even number them.questions spurred development various branches number theory focusing analytic algebraic aspects numbers.primes used several routines information technology publickey cryptography relies difficulty factoring large numbers prime factors.abstract algebra objects behave generalized way like prime numbers include prime elements prime ideals.definition examples natural number etc.
Prime number	called prime number prime greater cannot written product two smaller natural numbers.numbers greater prime called composite numbers.words n displaystyle n prime n displaystyle n items cannot divided smaller equalsize groups one item possible arrange n displaystyle n dots rectangular grid one dot wide one dot high.for example among numbers numbers prime numbers numbers divide evenly without remainder.prime specifically excluded definition.× × composite.the divisors natural number n displaystyle n natural numbers divide n displaystyle n evenly.every natural number divisor.
Prime number	divisor cannot prime.idea leads different equivalent definition primes numbers exactly two positive divisors number itself.yet another way express thing number n displaystyle n prime greater one none numbers … n − displaystyle dots n divides n displaystyle n evenly.the first prime numbers prime numbers less sequence oeis.no even number n displaystyle n greater prime number expressed product × n displaystyle times n .therefore every prime number odd number called odd prime.similarly written usual decimal system prime numbers larger end .numbers end digits compositedecimal numbers end even decimal numbers end divisible .the set primes sometimes denoted p displaystyle mathbf p boldface capital p p displaystyle mathbb p blackboard bold capital p. history rhind mathematical papyrus around bc egyptian fraction expansions different forms prime composite numbers.
Prime number	however earliest surviving records explicit study prime numbers come ancient greek mathematics.euclids elements c. bc proves infinitude primes fundamental theorem arithmetic shows construct perfect number mersenne prime.another greek invention sieve eratosthenes still used construct lists primes.around ad islamic mathematician ibn alhaytham alhazen found wilsons theorem characterizing prime numbers numbers n displaystyle n evenly divide n − displaystyle n .also conjectured even perfect numbers come euclids construction using mersenne primes unable prove it.another islamic mathematician ibn albanna almarrakushi observed sieve eratosthenes sped testing divisors square root largest number tested.
Prime number	fibonacci brought innovations islamic mathematics back europe.book liber abaci first describe trial division testing primality using divisors square root.in pierre de fermat stated without proof fermats little theorem later proved leibniz euler.fermat also investigated primality fermat numbers n displaystyle n marin mersenne studied mersenne primes prime numbers form p − displaystyle p p displaystyle p prime.christian goldbach formulated goldbachs conjecture every even number sum two primes letter euler.euler proved alhazens conjecture euclid–euler theorem even perfect numbers constructed mersenne primes.
Prime number	introduced methods mathematical analysis area proofs infinitude primes divergence sum reciprocals primes ⋯ displaystyle tfrac tfrac tfrac tfrac tfrac cdots .at start th century legendre gauss conjectured x displaystyle x tends infinity number primes x displaystyle x asymptotic x log ⁡ x displaystyle xlog x log ⁡ x displaystyle log x natural logarithm x displaystyle x .weaker consequence high density primes bertrands postulate every n displaystyle n prime n displaystyle n n displaystyle n proved pafnuty chebyshev.ideas bernhard riemann paper zetafunction sketched outline proving conjecture legendre gauss.although closely related riemann hypothesis remains unproven riemanns outline completed hadamard de la vallée poussin result known prime number theorem.another important th century result dirichlets theorem arithmetic progressions certain arithmetic progressions contain infinitely many primes.many mathematicians worked primality tests numbers larger trial division practicably applicable.
Prime number	methods restricted specific number forms include pépins test fermat numbers proths theorem c. lucas–lehmer primality test originated generalized lucas primality test.since largest known primes found using tests computers.search ever larger primes generated interest outside mathematical circles great internet mersenne prime search distributed computing projects.idea prime numbers applications outside pure mathematics shattered publickey cryptography rsa cryptosystem invented using prime numbers basis.the increased practical importance computerized primality testing factorization led development improved methods capable handling large numbers unrestricted form.mathematical theory prime numbers also moved forward green–tao theorem arbitrarily long arithmetic progressions prime numbers yitang zhangs proof exist infinitely many prime gaps bounded size.primality one early greeks even consider number could consider primality.
Prime number	mathematicians time also considered prime numbers subdivision odd numbers also consider prime.however euclid majority greek mathematicians considered prime.medieval islamic mathematicians largely followed greeks viewing number.by middle ages renaissance mathematicians began treating number included first prime number.midth century christian goldbach listed prime correspondence leonhard euler however euler consider prime.th century many mathematicians still considered prime lists primes included continued published recently .if definition prime number changed call prime many statements involving prime numbers would need reworded awkward way.
Prime number	example fundamental theorem arithmetic would need rephrased terms factorizations primes greater every number would multiple factorizations different numbers copies .similarly sieve eratosthenes would work correctly handled prime would eliminate multiples numbers output single number .technical properties prime numbers also hold number instance formulas eulers totient function sum divisors function different prime numbers .early th century mathematicians began agree listed prime rather special category unit.elementary properties unique factorization writing number product prime numbers called prime factorization number.
Prime number	example ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ .displaystyle beginalignedcdot cdot cdot cdot cdot cdot cdot .endaligned terms product called prime factors.prime factor may occur example two copies prime factor .displaystyle .prime occurs multiple times exponentiation used group together multiple copies prime number example second way writing product displaystyle denotes square second power .
Prime number	displaystyle .central importance prime numbers number theory mathematics general stems fundamental theorem arithmetic.theorem states every integer larger written product one primes.stronglythis product unique sense two prime factorizations number numbers copies primesalthough ordering may differ.although many different ways finding factorization using integer factorization algorithm must produce result.
Prime number	primes thus considered basic building blocks natural numbers.some proofs uniqueness prime factorizations based euclids lemma p displaystyle p prime number p displaystyle p divides product b displaystyle ab integers displaystyle b displaystyle b p displaystyle p divides displaystyle p displaystyle p divides b displaystyle b both.conversely number p displaystyle p property divides product always divides least one factor product p displaystyle p must prime.infinitude infinitely many prime numbers.another way saying sequence ...of prime numbers never ends.statement referred euclids theorem honor ancient greek mathematician euclid since first known proof statement attributed him.
Prime number	many proofs infinitude primes known including analytical proof euler goldbachs proof based fermat numbers furstenbergs proof using general topology kummers elegant proof.euclids proof shows every finite list primes incomplete.key idea multiply together primes given list add .displaystyle .list consists primes p p … p n displaystyle ppldots pn gives number n p ⋅ p ⋯ p n .displaystyle npcdot pcdots pn.
Prime number	fundamental theorem n displaystyle n prime factorization n p ′ ⋅ p ′ ⋯ p ′ displaystyle npcdot pcdots pm one prime factors.n displaystyle n evenly divisible factors n displaystyle n remainder one divided prime numbers given list none prime factors n displaystyle n given list.finite list primes must infinitely many primes.the numbers formed adding one products smallest primes called euclid numbers.first five prime sixth ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ displaystyle big cdot cdot cdot cdot cdot big cdot composite number.formulas primes known efficient formula primes.
Prime number	example nonconstant polynomial even several variables takes prime values.however numerous expressions encode primes primes.one possible formula based wilsons theorem generates number many times primes exactly once.also set diophantine equations nine variables one parameter following property parameter prime resulting system equations solution natural numbers.used obtain single formula property positive values prime.other examples primegenerating formulas come mills theorem theorem wright.
Prime number	assert real constants displaystyle μ displaystyle mu ⌊ n ⌋ ⌊ ⋯ μ ⌋ displaystyle leftlfloor anrightrfloor text leftlfloor cdots mu rightrfloor prime natural number n displaystyle n first formula number exponents second formula.⌊ ⋅ ⌋ displaystyle lfloor cdot rfloor represents floor function largest integer less equal number question.however useful generating primes primes must generated first order compute values displaystyle μ .displaystyle mu .open questions many conjectures revolving primes posed.
Prime number	often elementary formulation many conjectures withstood proof decades four landaus problems still unsolved.one goldbachs conjecture asserts every even integer n displaystyle n greater written sum two primes.conjecture verified numbers n ⋅ .displaystyle ncdot .weaker statements proven example vinogradovs theorem says every sufficiently large odd integer written sum three primes.
Prime number	chens theorem says every sufficiently large even number expressed sum prime semiprime product two primes.also even integer greater written sum six primes.branch number theory studying questions called additive number theory.another type problem concerns prime gaps differences consecutive primes.the existence arbitrarily large prime gaps seen noting sequence n n … n n displaystyle nndots nn consists n − displaystyle n composite numbers natural number n .displaystyle n. however large prime gaps occur much earlier argument shows.example first prime gap length primes much smaller .
Prime number	displaystyle .conjectured infinitely many twin primes pairs primes difference twin prime conjecture.polignacs conjecture states generally every positive integer k displaystyle k infinitely many pairs consecutive primes differ k .displaystyle k. andricas conjecture brocards conjecture legendres conjecture oppermanns conjecture suggest largest gaps primes displaystyle n displaystyle n approximately n displaystyle sqrt n result known follow riemann hypothesis much stronger cramér conjecture sets largest gap size log ⁡ n .displaystyle olog n. prime gaps generalized prime k displaystyle k tuples patterns differences two prime numbers.
Prime number	infinitude density subject first hardy–littlewood conjecture motivated heuristic prime numbers behave similarly random sequence numbers density given prime number theorem.analytic properties analytic number theory studies number theory lens continuous functions limits infinite series related mathematics infinite infinitesimal.this area study began leonhard euler first major result solution basel problem.the problem asked value infinite sum … displaystyle tfrac tfrac tfrac dots today recognized value ζ displaystyle zeta riemann zeta function.function closely connected prime numbers one significant unsolved problems mathematics riemann hypothesis.euler showed ζ π displaystyle zeta pi .the reciprocal number π displaystyle pi limiting probability two random numbers selected uniformly large range relatively prime factors common.the distribution primes large question many primes smaller given large threshold described prime number theorem efficient formula n displaystyle n th prime known.dirichlets theorem arithmetic progressions basic form asserts linear polynomials p n b n displaystyle pnabn relatively prime integers displaystyle b displaystyle b take infinitely many prime values.stronger forms theorem state sum reciprocals prime values diverges different linear polynomials b displaystyle b approximately proportions primes.although conjectures formulated proportions primes higherdegree polynomials remain unproven unknown whether exists quadratic polynomial integer arguments prime infinitely often.
Prime number	analytical proof euclids theorem eulers proof infinitely many primes considers sums reciprocals primes ⋯ p .displaystyle frac frac frac frac cdots frac p. euler showed arbitrary real number x displaystyle x exists prime p displaystyle p sum bigger x displaystyle x .shows infinitely many primes finitely many primes sum would reach maximum value biggest prime rather growing past every x displaystyle x .the growth rate sum described precisely mertens second theorem.comparison sum ⋯ n displaystyle frac frac frac cdots frac n grow infinity n displaystyle n goes infinity see basel problem.sense prime numbers occur often squares natural numbersalthough sets infinite.
Prime number	bruns theorem states sum reciprocals twin primes ⋯ displaystyle leftfrac frac rightleftfrac frac rightleftfrac frac rightcdots finite.bruns theorem possible use eulers method solve twin prime conjecture exist infinitely many twin primes.number primes given bound primecounting function π n displaystyle pi n defined number primes greater n displaystyle n .example π displaystyle pi since five primes less equal .methods meissel–lehmer algorithm compute exact values π n displaystyle pi n faster would possible list prime n displaystyle n .
Prime number	prime number theorem states π n displaystyle pi n asymptotic n log ⁡ n displaystyle nlog n denoted π n ∼ n log ⁡ n displaystyle pi nsim frac nlog n means ratio π n displaystyle pi n righthand fraction approaches n displaystyle n grows infinity.implies likelihood randomly chosen number less n displaystyle n prime approximately inversely proportional number digits n displaystyle n .it also implies n displaystyle n th prime number proportional n log ⁡ n displaystyle nlog n therefore average size prime gap proportional log ⁡ n displaystyle log n .a accurate estimate π n displaystyle pi n given offset logarithmic integral π n ∼ li ⁡ n ∫ n log ⁡ .displaystyle pi nsim operatorname li nint nfrac dtlog t. arithmetic progressions arithmetic progression finite infinite sequence numbers consecutive numbers sequence difference.difference called modulus progression.example ...is infinite arithmetic progression modulus .
Prime number	arithmetic progression numbers remainder divided modulus example remainder .modulus remainder multiples every element sequence.therefore progression contains one prime number itself.general infinite progression q q q … displaystyle aaqaqaqdots one prime remainder displaystyle modulus q displaystyle q relatively prime.relatively prime dirichlets theorem arithmetic progressions asserts progression contains infinitely many primes.the green–tao theorem shows arbitrarily long finite arithmetic progressions consisting primes.
Prime number	prime values quadratic polynomials euler noted function n − n displaystyle nn yields prime numbers ≤ n ≤ displaystyle leq nleq although composite numbers appear among later values.search explanation phenomenon led deep algebraic number theory heegner numbers class number problem.hardylittlewood conjecture f predicts density primes among values quadratic polynomials integer coefficientsin terms logarithmic integral polynomial coefficients.quadratic polynomial proven take infinitely many prime values.the ulam spiral arranges natural numbers twodimensional grid spiraling concentric squares surrounding origin prime numbers highlighted.visually primes appear cluster certain diagonals others suggesting quadratic polynomials take prime values often others.
Prime number	zeta function riemann hypothesis one famous unsolved questions mathematics dating one millennium prize problems riemann hypothesis asks zeros riemann zeta function ζ displaystyle zeta located.this function analytic function complex numbers.complex numbers displaystyle real part greater one equals infinite sum integers infinite product prime numbers ζ ∑ n ∞ n ∏ p prime − p − .displaystyle zeta ssum ninfty frac nsprod ptext primefrac ps.equality sum product discovered euler called euler product.euler product derived fundamental theorem arithmetic shows close connection zeta function prime numbers.it leads another proof infinitely many primes finitely manythen sumproduct equality would also valid displaystyle sum would diverge harmonic series … displaystyle tfrac tfrac dots product would finite contradiction.the riemann hypothesis states zeros zetafunction either negative even numbers complex numbers real part equal .
Prime number	original proof prime number theorem based weak form hypothesis zeros real part equal although elementary proofs found.the primecounting function expressed riemanns explicit formula sum term comes one zeros zeta function main term sum logarithmic integral remaining terms cause sum fluctuate main term.in sense zeros control regularly prime numbers distributed.riemann hypothesis true fluctuations small theasymptotic distribution primes given prime number theorem also hold much shorter intervals length square root x displaystyle x intervals near number x displaystyle x .abstract algebra modular arithmetic finite fields modular arithmetic modifies usual arithmetic using numbers … n − displaystyle dots n natural number n displaystyle n called modulus.any natural number mapped system replacing remainder division n displaystyle n .modular sums differences products calculated performing replacement remainderon result usual sum difference product integers.equality integers corresponds congruence modular arithmetic x displaystyle x displaystyle congruent written x ≡ displaystyle xequiv mod n displaystyle n remainder division n displaystyle n .however system numbers division nonzero numbers possible modulus prime.
Prime number	instance prime number displaystyle modulus division displaystyle possible ≡ mod displaystyle equiv bmod clearing denominators multiplying sides displaystyle gives valid formula ≡ mod displaystyle equiv bmod .however composite modulus displaystyle division displaystyle impossible.valid solution ≡ x mod displaystyle equiv xbmod clearing denominators multiplying displaystyle causes lefthand side become displaystyle righthand side becomes either displaystyle displaystyle .in terminology abstract algebra ability perform division means modular arithmetic modulo prime number forms field specifically finite field moduli give ring field.several theorems primes formulated using modular arithmetic.instance fermats little theorem states ≢ displaystyle anot equiv mod p displaystyle p p − ≡ displaystyle apequiv mod p displaystyle p .summing choices displaystyle gives equation ∑ p − p − ≡ p − ⋅ ≡ − mod p displaystyle sum apapequiv pcdot equiv pmod p valid whenever p displaystyle p prime.giugas conjecture says equation also sufficient condition p displaystyle p prime.wilsons theorem says integer p displaystyle p prime factorial p − displaystyle p congruent − displaystyle mod p displaystyle p .composite number n r ⋅ displaystyle nrcdot cannot hold since one factors divides n n − displaystyle n n − ≡ − mod n displaystyle nequiv pmod n impossible.
Prime number	padic numbers p displaystyle p adic order ν p n displaystyle nu pn integer n displaystyle n number copies p displaystyle p prime factorization n displaystyle n .concept extended integers rational numbers defining p displaystyle p adic order fraction n displaystyle mn ν p − ν p n displaystyle nu pmnu pn .p displaystyle p adic absolute value q p displaystyle qp rational number q displaystyle q defined q p p − ν p q displaystyle qppnu pq .multiplying integer p displaystyle p adic absolute value cancels factors p displaystyle p factorization leaving primes.distance two real numbers measured absolute value distance distance two rational numbers measured p displaystyle p adic distance p displaystyle p adic absolute value difference.
Prime number	definition distance two numbers close together small distance difference divisible high power p displaystyle p .way real numbers formed rational numbers distances adding extra limiting values form complete field rational numbers p displaystyle p adic distance extended different complete field p displaystyle p adic numbers.this picture order absolute value complete field derived generalized algebraic number fields valuations certain mappings multiplicative group field totally ordered additive group also called orders absolute values certain multiplicative mappings field real numbers also called norms places extensions complete fields given field dense set also called completions.extension rational numbers real numbers instance place distance numbers usual absolute value difference.corresponding mapping additive group would logarithm absolute value although meet requirements valuation.according ostrowskis theorem natural notion equivalence real numbers p displaystyle p adic numbers orders absolute values valuations absolute values places rational numbers.
Prime number	localglobal principle allows certain problems rational numbers solved piecing together solutions places underlining importance primes number theory.prime elements rings commutative ring algebraic structure addition subtraction multiplication defined.integers ring prime numbers integers generalized rings two different ways prime elements irreducible elements.element p displaystyle p ring r displaystyle r called prime nonzero multiplicative inverse unit satisfies following requirement whenever p displaystyle p divides product x displaystyle xy two elements r displaystyle r also divides least one x displaystyle x displaystyle .element irreducible neither unit product two nonunit elements.
Prime number	ring integers prime irreducible elements form set … − − − − − … .displaystyle dots dots .arbitrary ring prime elements irreducible.converse hold general hold unique factorization domains.the fundamental theorem arithmetic continues hold definition unique factorization domains.example domain gaussian integers z displaystyle mathbb z ring complex numbers form b displaystyle abi displaystyle denotes imaginary unit displaystyle b displaystyle b arbitrary integers.
Prime number	prime elements known gaussian primes.every number prime among integers remains prime gaussian integers instance number written product two gaussian primes displaystyle − displaystyle .rational primes prime elements integers congruent mod gaussian primes rational primes congruent mod not.consequence fermats theorem sums two squareswhich states odd prime p displaystyle p expressible sum two squares p x displaystyle pxy therefore factorable p x x − displaystyle pxiyxiy exactly p displaystyle p mod .prime ideals every ring unique factorization domain.
Prime number	instance ring numbers b − displaystyle absqrt integers displaystyle b displaystyle b number displaystyle two factorizations ⋅ − − − displaystyle cdot sqrt sqrt neither four factors reduced unique factorization.order extend unique factorization larger class rings notion number replaced ideal subset elements ring contains sums pairs elements products elements ring elements.prime ideals generalize prime elements sense principal ideal generated prime element prime ideal important tool object study commutative algebra algebraic number theory algebraic geometry.prime ideals ring integers ideals ... fundamental theorem arithmetic generalizes lasker–noether theorem expresses every ideal noetherian commutative ring intersection primary ideals appropriate generalizations prime powers.the spectrum ring geometric space whose points prime ideals ring.arithmetic geometry also benefits notion many concepts exist geometry number theory.example factorization ramification prime ideals lifted extension field basic problem algebraic number theory bears resemblance ramification geometry.
Prime number	concepts even assist numbertheoretic questions solely concerned integers.example prime ideals ring integers quadratic number fields used proving quadratic reciprocity statement concerns existence square roots modulo integer prime numbers.early attempts prove fermats last theorem led kummers introduction regular primes integer prime numbers connected failure unique factorization cyclotomic integers.the question many integer prime numbers factor product multiple prime ideals algebraic number field addressed chebotarevs density theorem applied cyclotomic integers dirichlets theorem primes arithmetic progressions special case.group theory theory finite groups sylow theorems imply power prime number p n displaystyle pn divides order group group subgroup order p n displaystyle pn .lagranges theorem group prime order cyclic groupand burnsides theorem group whose order divisible two primes solvable.computational methods long time number theory general study prime numbers particular seen canonical example pure mathematics applications outside mathematics use prime numbered gear teeth distribute wear evenly.
Prime number	particular number theorists british mathematician g. h. hardy prided work absolutely military significance.this vision purity number theory shattered publicly announced prime numbers could used basis creation publickey cryptography algorithms.these applications led significant study algorithms computing prime numbers particular primality testing methods determining whether given number prime.the basic primality testing routine trial division slow useful large numbers.one group modern primality tests applicable arbitrary numbers efficient tests available numbers special types.primality tests tell whether argument prime not.routines also provide prime factor composite arguments prime factors called factorization algorithms.prime numbers also used computing checksums hash tables pseudorandom number generators.trial division basic method checking primality given integer n displaystyle n called trial division.
Prime number	method divides n displaystyle n integer square root n displaystyle n .integer dividing n displaystyle n evenly establishes n displaystyle n composite otherwise prime.integers larger square root need checked whenever n ⋅ b displaystyle nacdot b one two factors displaystyle b displaystyle b less equal square root n displaystyle n .another optimization check primes factors range.for instance check whether prime method divides primes range displaystyle sqrt .division produces nonzero remainder indeed prime.although method simple describe impractical testing primality large integers number tests performs grows exponentially function number digits integers.however trial division still used smaller limit square root divisor size quickly discover composite numbers small factors using complicated methods numbers pass filter.
Prime number	sieves computers mathematical tables listing primes prime factorizations given limit commonly printed.oldest method generating list primes called sieve eratosthenes.animation shows optimized variant method.another asymptotically efficient sieving method problem sieve atkin.advanced mathematics sieve theory applies similar methods problems.primality testing versus primality proving fastest modern tests whether arbitrary given number n displaystyle n prime probabilistic monte carlo algorithms meaning small random chance producing incorrect answer.for instance solovay–strassen primality test given number p displaystyle p chooses number displaystyle randomly displaystyle p − displaystyle p uses modular exponentiation checkwhether p − ± displaystyle appm divisible p displaystyle p .
Prime number	answers yes otherwise answers no.p displaystyle p really prime always answer yes p displaystyle p composite answers yes probability probability least .if test repeated n displaystyle n times numberthe probability composite number could pass test every time n displaystyle n .decreases exponentially number tests provides high confidence although certainty number passes repeated test prime.hand test ever fails number certainly composite.a composite number passes test called pseudoprime.in contrast algorithms guarantee answer always correct primes always determined prime composites always determined composite.for instance true trial division.the algorithms guaranteedcorrect output include deterministic nonrandom algorithms aks primality testand randomized las vegas algorithms random choices made algorithm affect final answer variations elliptic curve primality proving.when elliptic curve method concludes number prime provides primality certificate verified quickly.the elliptic curve primality test fastest practice guaranteedcorrect primality tests runtime analysis based heuristic arguments rather rigorous proofs.aks primality test mathematically proven time complexity slower elliptic curve primality proving practice.
Prime number	methods used generate large random prime numbers generating testing random numbers finding one primewhen faster probabilistic test quickly eliminate composite numbers guaranteedcorrect algorithm used verify remaining numbers prime.the following table lists tests.running time given terms n displaystyle n number tested probabilistic algorithms number k displaystyle k tests performed.moreover ε displaystyle varepsilon arbitrarily small positive number log logarithm unspecified base.big notation means time bound multiplied constant factor convert dimensionless units units time factor depends implementation details type computer used run algorithm input parameters n displaystyle n k displaystyle k .specialpurpose algorithms largest known prime addition aforementioned tests apply natural number numbers special form tested primality quickly.for example lucas–lehmer primality test determine whether mersenne number one less power two prime deterministicallyin time single iteration miller–rabin test.
Prime number	since december largest known prime always mersenne prime.it conjectured infinitely many mersenne primes.the following table gives largest known primes various types.primes found using distributed computing.great internet mersenne prime search project awarded us prize first discovering prime least million digits.electronic frontier foundation also offers primes least million digits billion digits respectively.integer factorization given composite integer n displaystyle n task providing one prime factors referred factorization n displaystyle n .
Prime number	significantly difficult primality testing although many factorization algorithms known slower fastest primality testing methods.trial division pollards rho algorithm used find small factors n displaystyle n elliptic curve factorization effective n displaystyle n factors moderate size.methods suitable arbitrary large numbers depend size factors include quadratic sieve general number field sieve.primality testing also factorization algorithms require input special form including special number field sieve.december largest number known factored generalpurpose algorithm rsa decimal digits bits product two large primes.shors algorithm factor integer polynomial number steps quantum computer.
Prime number	however current technology run algorithm small numbers.october largest number factored quantum computer running shors algorithm .computational applications several publickey cryptography algorithms rsa diffie–hellman key exchange based large prime numbers bit primes common.rsa relies assumption much easier efficient perform multiplication two large numbers x displaystyle x displaystyle calculate x displaystyle x displaystyle assumed coprime product x displaystyle xy known.diffie–hellman key exchange relies fact efficient algorithms modular exponentiation computing b mod c displaystyle abbmod c reverse operation discrete logarithm thought hard problem.prime numbers frequently used hash tables.
Prime number	instance original method carter wegman universal hashing based computing hash functions choosing random linear functions modulo large prime numbers.carter wegman generalized method k displaystyle k independent hashing using higherdegree polynomials modulo large primes.well hash function prime numbers used hash table size quadratic probing based hash tables ensure probe sequence covers whole table.some checksum methods based mathematics prime numbers.instance checksums used international standard book numbers defined taking rest number modulo prime number.prime method detect singledigit errors transpositions adjacent digits.
Prime number	another checksum method adler uses arithmetic modulo largest prime number less displaystyle .prime numbers also used pseudorandom number generators including linear congruential generators mersenne twister.applications prime numbers central importance number theory also many applications areas within mathematics including abstract algebra elementary geometry.example possible place prime numbers points twodimensional grid three line every triangle formed three points large area.another example eisensteins criterion test whether polynomial irreducible based divisibility coefficients prime number square.the concept prime number important generalized different ways various branches mathematics.generally prime indicates minimality indecomposability appropriate sense.
Prime number	example prime field given field smallest subfield contains .either field rational numbers finite field prime number elements whence name.often second additional meaning intended using word prime namely object essentially uniquely decomposed prime components.example knot theory prime knot knot indecomposable sense cannot written connected sum two nontrivial knots.knot uniquely expressed connected sum prime knots.
Prime number	prime decomposition manifolds another example type.beyond mathematics computing prime numbers potential connections quantum mechanics used metaphorically arts literature.also used evolutionary biology explain life cycles cicadas.constructible polygons polygon partitions fermat primes primes form f k k displaystyle fkk k displaystyle k nonnegative integer.named pierre de fermat conjectured numbers prime.first five numbers – – prime f displaystyle f composite fermat numbers verified .
Prime number	regular n displaystyle n gon constructible using straightedge compass odd prime factors n displaystyle n distinct fermat primes.likewise regular n displaystyle n gon may constructed using straightedge compass angle trisector prime factors n displaystyle n number copies together possibly empty set distinct pierpont primes primes form b displaystyle ab .it possible partition convex polygon n displaystyle n smaller convex polygons equal area equal perimeter n displaystyle n power prime number known values n displaystyle n .quantum mechanics beginning work hugh montgomery freeman dyson mathematicians physicists speculated zeros riemann zeta function connected energy levels quantum systems.prime numbers also significant quantum information science thanks mathematical structures mutually unbiased bases symmetric informationally complete positiveoperatorvalued measures.biology evolutionary strategy used cicadas genus magicicada makes use prime numbers.
Prime number	insects spend lives grubs underground.pupate emerge burrows years point fly breed die weeks most.biologists theorize primenumbered breeding cycle lengths evolved order prevent predators synchronizing cycles.in contrast multiyear periods flowering bamboo plants hypothesized smooth numbers small prime numbers factorizations.arts literature prime numbers influenced many artists writers.the french composer olivier messiaen used prime numbers create ametrical music natural phenomena.works la nativité du seigneur quatre études de rythme – simultaneously employs motifs lengths given different prime numbers create unpredictable rhythms primes appear third étude neumes rythmiques.

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