instruction stringlengths 15 30k |
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Given some group in which discrete logarithms are hard. Furthermore, two random, unrelated group generators $G_a, G_b$, and a third generator defined by $G_c = G_a + G_b$. Can you compute $xG_a$ if you know only $xG_c$, $G_a$, and $G_b$? |
Given two unrelated generators $G_a$ and $G_b$, and a third with $G_c = G_a + G_b$. Is it hard to compute $xG_a$ from $xG_c$? |
Given some group in which discrete logarithms are hard. Furthermore, two random, unrelated group generators $G_1, G_2$, and a third generator defined by $H = G_1 + G_2$. Can you compute $xG_1$ if you know only $xH$, $G_1$, and $G_1$?
**My guess**: I would assume so, because otherwise it would be easy to compute ... |
Given two unrelated generators $G_1$ and $G_2$, and a third with $H = G_1 + G_2$. Is it hard to compute $xG_1$ from $xH$? |
Given some group in which discrete logarithms are hard. Furthermore, two random, unrelated group generators $G_1, G_2$, and a third generator defined by $H = G_1 + G_2$. Can you compute $xG_1$ if you know only $xH$, $G_1$, and $G_2$?
**My guess**: I would assume so, because otherwise it would be easy to compute ... |
Given some group in which discrete logarithms are hard. Furthermore, two random, unrelated group generators $G_1, G_2$, and a third generator defined by $H = G_1 + G_2$. Can you compute $xG_1$ if you know only $G_1$, $G_2$, and $xH$?
**My guess**: I would assume it's hard, because otherwise it would be easy to c... |
Given some group in which both discrete logarithms and the computational Diffie-Hellman problem are hard. Furthermore, two random, unrelated group generators $G_1, G_2$, and a third generator defined by $H = G_1 + G_2$. Can you compute $xG_1$ if you know only $G_1$, $G_2$, and $xH$?
**My guess**: I would assume ... |
How do we know that hash functions cannot be reversed? An example is often given of two primes and their product, but any composite number that is the product of two primes has, by definition, exactly 2 natural factors. In the case of hash functions, things are different.
Have there been attempts to reverse hash fun... |
> an attacker would be able to compute the private key by using $\log_g(g^x)$ to find $x$.
No, because we are supposed to assume/understand that the public key really is $(p,g,g^x\bmod p)$, and that $p$ is such that the Discrete Logarithm Problem is hard. A meta reason is that if it was possible to compute the priva... |
> an attacker would be able to compute the private key by using $\log_g(g^x)$ to find $x$.
No, because we are supposed to assume/understand that the public key really is $(p,g,g^x\bmod p)$, and that $p$ is such that the Discrete Logarithm Problem is hard. A meta reason is that if it was possible to compute the priva... |
The CEO of the organization XYZ decides to hold a vote to decide whether employees should be allowed to work from home (WFH) either one, two or three days a week.
All 4 employees of XYZ (excluding the CEO), need to vote either “WFH one”, “WFH two” or “WFH three”. To ensure privacy, the CEO asks employees to send th... |
I am constructing a two-party OR gate and trying to do this with oblivious transfer. Yet I am very new to oblivious transfer, wishing to know whether the following construction makes sense.
Goal: Alice inputs a random bit $a$, Bob inputs a random bit $b$, and Bob outputs the logical OR bit $a\oplus b$.
Constructi... |
Suppose that Peggy(prover) and Victor(verifier) are running some zero knowledge proof protocol that does not rely on hidden verifier secrets. The verifier generates randomly chosen challenge values only. Such protocols can be Fiat-Shamir transformed into NI(non-interactive)ZKPs.
There is significant work on NIZKPs. ... |
The shape of the edwards curve is equal to $a x^2 + b y^2 = d x^2 y^2 + 1$ in $\mathbb{Z}_p$. Is there a way to know $p$ when we know the five points on the edwards curve without knowing $a, b, d, p,$ and generator $G$?
I tried to find p by using the five points given to me by concatenating the relationship $a x... |
How to recover $p$ from edwards curve? |
Do you know a blockchain that does not use at all cryptographic primitives standardized by USA or other countries? It is strange to me that the security of many cryptocurrencies is based on ciphers, hash functions, elliptic curves, etc. from American standards.
It is normal when a cryptographic product contains sta... |
It really depends on what you need it for. Protecting top secret confidential files? Better use 384. Computing an example for your class? Maybe 5-bit. There is a lot in between and it is your choice. |
Here is a CTF crypto challenge likes(its write up is public on https://ctftime.org/writeup/15438):
$$N = p*q\\
c1 = (2*p + 3*q)^{e_{1}} mod N\\
c2 = (5*p + 7*q)^{e_{2}} mod N$$
After i transform these:
$$(c^{e_2}_1)\equiv (2p)^{e_1e_2}+(3q)^{e_1e_2}\pmod{N}\\
(c^{e_1}_2)\equiv (5p)^{e_1e_2}+(7q)^{e_1e_2}\pmod{N}$... |
When attacking RSA with Square&Multiply, one can figure out the secret key by looking at the exponentiation algorithm itself. To prevent this in software, we could use dummy multiplications after each square.
Yet, there are attacks like correlation power analysis on AES, which is vulnerable to it by definition. How... |
How to prevent power analysis on software level? |
The cycle structure of the squaring map modulo $N$ was studied in [BBS,Sections 7 and 8].
A summary follows.
Let's assume that $N=pq$ for $p=2p'+1$ and $q=2q'+1$.
The cardinality of $QR_N:=\{x^2:x\in\mathbb{Z}_N^*\}$, [the subgroup of quadratic residues modulo $N$](https://en.wikipedia.org/wiki/Quadratic_residue)... |
I have some questions about the setting of the *one-way authenticated key exchange* of *Kyber.AKE* as defined in section 5 of the [paper][1].
1. How does $P_2$ authenticate itself to $P_1$? It is not obvious to me in the paper how it is ensured that $P_1$ really communicates with $P_2$. So how does $P_1$ know that i... |
I have some questions about the setting of the *one-way authenticated key exchange* of *Kyber.AKE* as defined in section 5 of the [paper][1].
1. How does $P_2$ authenticate itself to $P_1$? It is not obvious to me in the paper how it is ensured that $P_1$ really communicates with $P_2$. So how does $P_1$ know that i... |
Could anyone explain me the difference between "key_share" extension and "server_key_exchange" message in TLS (1.3) ?
If I understood right, these "packets" of data are used to send a key material to generate premaster keys. But what's the difference ?
In which case a server must use one or another |
> Could anyone explain me the difference between "key_share" extension and "server_key_exchange" message in TLS (1.3) ?
server_key_exchange isn't used in TLS 1.3. It was used in TLS 1.2; however they modified the protocol to go to TLS 1.3, they opted not use it. However, they wanted to keep the same handshake type... |
> Could anyone explain me the difference between "key_share" extension and "server_key_exchange" message in TLS (1.3) ?
server_key_exchange isn't used in TLS 1.3. It was used in TLS 1.2; however when they modified the protocol to go to TLS 1.3, they opted not use it. However, they wanted to keep the same handshake... |
1. I ask because some vendors of HSM try to avoid the export of wrapped secret key from HSM to insecure storage – storage that does not belong to these vendor’s HSM infrastructure.
For example, Thales prefer to backup keys to another Thales HSM – most of their documentation is about cloning between their device... |
From the [PLONK paper](https://eprint.iacr.org/2019/953.pdf).
On pages 19 & 20, the paper describes the prescribed permutation check in PLONK.
[![enter image description here][1]][1]
My question is about the Step 3 in the protocol which I have marked in red
I am interpreting $1 \le j < i$ as $j =1$ to $j = ... |
Is the permuation check range in the PLONK Paper incorrect? |
From the [PLONK paper](https://eprint.iacr.org/2019/953.pdf).
On pages 19 & 20, the paper describes the prescribed permutation check in PLONK.
[![enter image description here][1]][1]
**------------------------------------------------------------------------------------------------**
My question is about the... |
When I want to encrypt a 16 byte long message with AES-128-CBC using int TLS 1.2 do I need to add a 16 byte block of padding or can I just encrypt the message without padding ?
|
Its Tls 1.2 Padding required when message ist multiple of block length? |
> When I want to encrypt a 16 byte long message with AES-128-CBC using int TLS 1.2 do I need to add a 16 byte block of padding or can I just encrypt the message without padding ?
You always need to add padding. If you didn't, then the decryptor would be unable to distinguish between "this message was padded" and "t... |
For example is it harder to break one 256 bit encryption than two 255 bit encryption for RSA and AES?
|
How do different encryption schemes scale as a relation between their security parameter and computational requirement? |
For example is it harder to break one 256 bit encryption than two 255 bit encryption for RSA and AES?
For example, I understand that 256 bit RSA can be cracked in one minute according to this article [1] So would two independent 255 bit RSA message take thirty seconds each, and four 254 bit messages take 15 seconds ... |
For example is it harder to break one 256 bit encryption than two 255 bit encryption for RSA and AES?
For example, I understand that 256 bit RSA can be cracked in one minute according to this article [1] So would two independent 255 bit RSA message take thirty seconds each, and four 254 bit messages take 15 seconds ... |
From the [PLONK paper](https://eprint.iacr.org/2019/953.pdf).
On pages 19 & 20, the paper describes the prescribed permutation check in PLONK.
[![enter image description here][1]][1]
**------------------------------------------------------------------------------------------------**
My question is about the... |
When using AES-256 (cipher mode CBC and padding mode PKCS7) in combination with HMAC-SHA for authenticated encryption (assuming alternatives like TLS and AES-GCM cannot be used),
should we use SHA-256 or SHA-512?
[This answer][1] seems to indicate SHA512.
Is this interpretation correct?
I've seen an implementation... |
When using AES-256 in combination with HMAC-SHA, should we use SHA-256 or SHA-512? |
I need to implement a Merkle tree using a 128 bit hash function. In general, any hash function that guarantees pre-image, second pre-image and collission resistance should be fine to implement a Markle tree. Is it correct? Probably, it is not even necessary to have pre-image resistance as long as the other two properti... |
Implementing a Merkle tree using a 128 bit hash function? |
[fgrieu][1] has shown that $\lim\limits_{h\to\infty} H(h,k) = k$ for any fixed $k$, so I will try to address the second question, that is, whether or not $$\lim\limits_{h\to\infty} h - H(2h,h) = 0$$
In particular, if we let $n = 2^h$, then we are asking whether
$$\lim\limits_{n\to\infty} \lg(n) - n\sum_{j=0}^{n^2}\bi... |
[fgrieu][1] has shown that $\lim\limits_{h\to\infty} H(h,k) = k$ for any fixed $k$, so I will try to address the second question, that is, whether or not $$\lim\limits_{h\to\infty} h - H(2h,h) = 0$$
In particular, if we let $n = 2^h$, then we are asking whether
$$\lim\limits_{n\to\infty} \lg(n) - n\sum_{j=0}^{n^2}\bi... |
Alice has master EC key pair: $a$ - private key, $A$ - corresponding public key
Bob generates 2 random integers $r_1$ and $r_2$ and wants Alice to derive 2 new key pairs:
$a_1$ = $a$ + $r_1$ and $a_2$ = $a$ + $r_2$ and corresponding public keys $A_1$ and $A_2$
questions:
1. Is it possible for Alice to prove ... |
Summary: The operation of the Enigma Bombe is well documented. I manage to use it and a candidate checking machine to recover the plugboard pairs and the ring setting for the fast rotor. I struggle to recover ring settings for the slow and the middle rotors.
Question: Where can I find out more technical details on t... |
How to recover ring settings for the slow and the middle rotors on Enigma Checking Machine? |
I am working on a chat feature to use both post quantum cryptography along with RSA, and want to confirm my thoughts.
As these algorithms haven't been fully battle tested, I have decided to use a combination of both PQE and usual public, secret key cryptography
I decided upon the best algorithms to use for securi... |
Alice has master EC key pair: $a$ - private key, $A$ - corresponding public key
Bob generates 2 random integers $r_1$ and $r_2$ and wants Alice to derive 2 new key pairs:
$a_1$ = $a$ + $r_1$ and $a_2$ = $a$ + $r_2$ and corresponding public keys $A_1$ and $A_2$
Questions:
1. Is it possible for Alice to prove ... |
Summary: The operation of the Enigma Bombe is well documented. I manage to use it and a candidate checking machine to recover the plugboard pairs and the ring setting for the fast rotor. I struggle to recover ring settings for the slow and the middle rotors.
Question: Where can I find out more technical details on t... |
I'd like for two machines on a network to be able to prove to each other that they both have knowledge of a pre-shared secret, without revealing the secret to each other. Let's assume that all traffic over the connection between the parties, A and B, is encrypted.
Here are the steps I'm currently imagining:
#### ... |
Summary: The operation of the Enigma Bombe is well documented. I manage to use it and a candidate checking machine to recover the plugboard pairs and the ring setting for the fast rotor. I struggle to recover ring settings for the slow and the middle rotors.
Question: Where can I find out more technical details on t... |
I have been researching the best encryption to use in a .NET application for managing a sensitive database field (column). This encryption is on top of e.g. AWS at-rest encryption applied to the whole of the database and is aimed at frustrating use of the sensitive data by anyone other than the application (which knows... |
AES-GCM for sensitive database field - good solution? |
Summary: The operation of the Enigma Bombe is well documented. I manage to use it and a candidate checking machine to recover the plugboard pairs and the ring setting for the fast rotor. I struggle to recover ring settings for the slow and the middle rotors.
Question: Where can I find out more technical details on t... |
I'm drafing an RFC for a low computation crypto algorithm, intended for low power bluetooth communication. Likely without a connection using advertisements only.
It's going to include raw C code examples so it should be quite portable.
Is there a good site to post my RFC to, where it will be more visible than say... |
When using AES-256 (cipher mode CBC and padding mode PKCS7) in combination with HMAC-SHA for authenticated encryption (assuming alternatives like TLS and AES-GCM cannot be used),
should we use SHA-256 or SHA-512?
[This answer][1] seems to indicate SHA512.
Is this interpretation correct?
I've seen an implementation... |
assuming that I have an RSA key of length 4k bit which I'm interested to wrap using AES-GCM, and I have a (limited) AES-GCM cipher which can only encrypt limited input in size (say 256-bit/512-bit input) per invocation.
Is there a way/conditions to fulfill when splitting this 4k bit key into smaller keys to fit my (li... |
How does JOSE/JWE make use of ECDH when encrypting/decrypting messages? |
When we take the coin flip in Blum's algorithm in "Coin flipping by telephone a protocol for solving impossible problems", where Alice and Bob both want ownership over the same car, then one party (the adversary / Alice) can abort the protocol. Provided there does exist a powerful third party (e.g. the law), the protoc... |
When we take the coin flip in Blum's algorithm in "Coin flipping by telephone a protocol for solving impossible problems", where Alice and Bob both want ownership over the same car, then one party (the adversary / Alice) can abort the protocol. Provided there does exist a powerful third party (e.g. the law), the protoc... |
When we take the coin flip in Blum's algorithm in "Coin flipping by telephone a protocol for solving impossible problems", where Alice and Bob both want ownership over the same car, then one party (the adversary / Alice) can abort the protocol. Provided there does exist a powerful third party (e.g. the law), the protoc... |
What does preprocessed polynomial mean in the context of PLONK? |
I'm trying to understand some cryptographic properties of the s-box so I can have my own code. Example of balanced properties I read in [this document][1] that they say
>A boolean function S : $ GF(2^n) $ $ \to $ $ GF(2) $ is called balanced if the output
set contains equal number of ones and zeros in the correspo... |
I'm trying to understand some cryptographic properties of the s-box so I can have my own code. Example of balanced properties I read in [this document][1] that they say
>A boolean function S : $ GF(2^n) $ $ \to $ $ GF(2) $ is called balanced if the output
set contains equal number of ones and zeros in the correspo... |
I'm trying to understand some cryptographic properties of the s-box so I can have my own code. Example of balanced properties I read in [this document][1] that they say
>A boolean function S : $ GF(2^n) $ $ \to $ $ GF(2) $ is called balanced if the output
set contains equal number of ones and zeros in the correspo... |
> how is the S-box in AES represented ?
[FIPS 197 Figure 7][1] is the truth table of the AES S-box, stated in hexadecimal for compactness. It's it's most usual and arguably most convenient representation. For example, for input $\mathtt{53_h}$, then the output would be determined by the intersection
of the row with... |
> how is the S-box in AES represented ?
[FIPS 197 Figure 7][1] is the truth table of the AES S-box, stated in hexadecimal for compactness. It's it's most usual and arguably most convenient representation. For example, for input $\mathtt{53_h}$, then the output would be determined by the intersection
of the row... |
> how is the S-box in AES represented ?
[FIPS 197 Figure 7][1] is the truth table of the AES S-box, stated in hexadecimal for compactness. It's it's most usual and arguably most convenient representation. For example, for input $\mathtt{53_h}$, then the output would be determined by the intersection
of the row... |
I am working with lossy ID-schemes and their security in the QROM. Following the article of [Kiltz et al.][1] , I am at a loss of the number 8 appearing in most reductions throughout the article. I know it comes from the *Generic Search Problem for Bounded Probabilites*, however how?
The Lemma from the article as we... |
Where does the 8 come from? Generic Search Problem with Bounded Probabilities |
> how is the S-box in AES represented ?
[FIPS 197 Figure 7][1] is the truth table of the AES S-box, stated in hexadecimal for compactness. That's the most usual and arguably most convenient representation. For example, for input $\mathtt{53_h}$, then the output would be determined by the intersection of the row... |
> how is the S-box in AES represented ?
[FIPS 197 Figure 7][1] is the truth table of the AES S-box, stated in hexadecimal for compactness. That's the most usual and arguably most convenient representation. For example, for input $\mathtt{53_h}$, the output is determined by the intersection of the row with index... |
> how is the S-box in AES represented ?
[FIPS 197 Figure 7][1] is the truth table of the AES S-box, stated in hexadecimal for compactness. That's the most usual and arguably most convenient representation. For example, for input $\mathtt{53_h}$, the output is determined by the intersection of the row with index... |
I am working on MPCs (multi party computation) in crypto, and now I am developing a implementation of [GG 18][1].
In sign phase, algorithm needs MtA (multiply to addiction) and uses Paillier key pair for this.
Paillier uses modulus N (= p * q, prime numbers used in key generation ). But we need modulus q (order o... |
> how is the S-box in AES represented ?
[FIPS 197 Figure 7][1] is the truth table of the AES S-box, stated in hexadecimal for compactness. That's the most usual and arguably most convenient representation. For example, for input $\mathtt{53_h}$, the output is determined by the intersection of the row with index... |
I am working with lossy ID-schemes and their security in the QROM. Following the article of [Kiltz et al.][1] , I am at a loss of the number 8 appearing in most reductions throughout the article. I know it comes from the *Generic Search Problem for Bounded Probabilites*, however how?
The Lemma from the article as we... |
I am working on MPCs (multi party computation) in crypto, and now I am developing a implementation of [GG 18][1].
In sign phase, algorithm needs MtA (Multiplicative to Additive) and uses a [Paillier][2] key pair for this.
Paillier uses modulus $N$ ($N=p_1 * p_2$†, prime numbers drawn at key generation). But we ne... |
From this blog post: https://medium.com/@VitalikButerin/exploring-elliptic-curve-pairings-c73c1864e627
> if P = G * p, Q = G * q and R = G * r, you can check whether or not p * q = r, having just P, Q and R as inputs.
How?
It's easy to see `p * Q = q * P = R`. But I don't see the leap to proving `p * q = r`. |
In a [previous question](https://crypto.stackexchange.com/q/81988/36960), I asked how to implement arbitrary s-box in side-channel-free fashion. The code I posted in the question loops over all 256 values of **a byte** to avoid timing channel, and I want to change it to loop over the values of a **word**.
Before get... |
Is it fatal if a blockcipher uses a sbox with a fixed point at 0? |
> This question wants to ask, whether and how fatal it is, to have a fixed point at 0 in the sbox of a block cipher?
Whether a fixed point at 0 in an sbox is fatal would depend on the cipher. For AES, it is fatal, because the AES sbox is fixed and expects a value of 0x63 at location 0.
On the other hand, it is a... |
The cycle structure of the squaring map modulo $N$ was studied in [BBS,Sections 7 and 8].
A summary follows.
Let's assume that $N=pq$ for $p=2p'+1$ and $q=2q'+1$.
The cardinality of $QR_N:=\{x^2:x\in\mathbb{Z}_N^*\}$, [the subgroup of quadratic residues modulo $N$](https://en.wikipedia.org/wiki/Quadratic_residue)... |
I'm trying to understand some cryptographic properties of the s-box so I can have my own code. Example of balanced properties I read in [this document][1] that they say
>A boolean function S : $ GF(2^n) $ $ \to $ $ GF(2) $ is called balanced if the output
set contains equal number of ones and zeros in the correspo... |
Passkeys have this complicated QR code + Bluetooth dance to enroll a new device which *seems* like it ought to be securely creating a key on the device in a TPM. However the messaging around "sync" seems to suggest that while that is happening, the device key is only used to encrypt a software key which is encrypted an... |
Are passkeys actually hardware keys or are they just reversibly encrypted keys secured by a hardware key? |
> how is the S-box in AES represented ?
[FIPS 197 Figure 7][1] is the truth table of the AES S-box, stated in hexadecimal for compactness. That's the most usual and arguably most convenient representation. For example, for input $\mathtt{53_h}$, the output is determined by the intersection of the row with index... |
Why is IV in RFC8448 examples only 12 octets instead of 16? |
I am new at cryptography and start to learn some terms. I just saw an question in the book and didn't got a clue about this question. Could someone can explain me the answer of this question? Because all of the questions seems to be fine for me.
Assuming the block length is 8 bytes (L = 8) Which of the following is/... |
I am new at cryptography and start to learn some terms. I just saw an question in the book and didn't got a clue about this question. Could someone can explain me the answer of this question? Because all of the questions seems to be fine for me.
Assuming the block length is 8 bytes (L = 8) Which of the following is/... |
Suppose that Peggy(prover) and Victor(verifier) are running some public-coin zero knowledge proof protocol. The verifier generates randomly chosen challenge values only. Such protocols can be Fiat-Shamir transformed into NI(non-interactive)ZKPs.
There is significant work on NIZKPs. Using them as-is in contexts where... |
The Weil Pairing is a bilinear pairing, which is a function $e$ that maps two points in an elliptic curve $E(\mathbb{F}_q)$ to a element in the finite field $\mathbb{F}_{q^k}$, where $k$ is the embedding degree associated with the curve.
The bilinearity means that $e(rP,sQ) = e(P,Q)^{rs}$ for points $P, Q$.
We k... |
The Weil Pairing is a bilinear pairing, which is a function $e$ that maps two points in an elliptic curve $E(\mathbb{F}_q)$ to a element in the finite field $\mathbb{F}_{q^k}$, where $k$ is the embedding degree associated with the curve.
If the elliptic curve's generator is $G$ & $a$ & $b$ are scalars, then the bil... |
The Weil Pairing is a bilinear pairing, which is a function $e$ that maps two points in a pairing friendly elliptic curve $E(\mathbb{F}_q)$ to a element in the finite field $\mathbb{F}_{q^k}$, where $k$ is the embedding degree associated with the curve.
If the elliptic curve's generator is $G$ & $a$ & $b$ are scala... |
The Weil Pairing is a bilinear pairing, which is a function $e$ that maps two points in a pairing friendly elliptic curve $E(\mathbb{F}_q)$ to an element in the finite field $\mathbb{F}_{q^k}$, where $k$ is the embedding degree associated with the curve.
If the elliptic curve's generator is $G$ & $a$ & $b$ are scal... |
Suppose that Peggy(prover) and Victor(verifier) are running some zero knowledge proof protocol that does not rely on hidden verifier secrets. The verifier generates randomly chosen challenge values only. Such protocols can be Fiat-Shamir transformed into NI(non-interactive)ZKPs.
There is significant work on NIZKPs. ... |
The Weil Pairing is a bilinear pairing, which is a function $e$ that maps two points in a pairing friendly elliptic curve $E(\mathbb{F}_q)$ to an element in the finite field $\mathbb{F}_{q^k}$, where $k$ is the embedding degree associated with the curve.
If the elliptic curve's generator is $G$ & $a$ & $b$ are scal... |
I am using the NIST SP 800-22 to evaluate the performance of the Random Number Generator.
NIST SP 800-22 has a total of 15 tests. In each test, it has a recommendation for the size of the tested sequence.
For example, in the Discrete Fourier Transform (Spectral) Test :
"It is recommended that each sequence to be... |
I am currently working through the exercises in A Graduate Course in Applied Cryptography by Dan Boneh. I am stuck on exercise 4.16 (page 188 in [this PDF][1])
In the GGM tree construction for constructing a PRF from a PRG, the secret key is used at the root of the tree and the input is used to trace a path through ... |
Exchanging key and input in GGM tree? |
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