Questions tagged [group-theory]

Groups are an abstract algebraic concept based on a set and a group law (a binary function which closes the set).

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Steps to determine the single element generators for a multiplicative group

As student I've been asked the following question: Consider the specific prime $p=17$. Determine the single element generators (by hand or by Java program) of $F^*_{17}$. Recall $F^*_{17}$ is the ...
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How do pairings behave on G2/twist points off the prime order subgroup?

$\newcommand{\F}{\mathbb{F}}$ Consider the ate pairing defined on a curve $G_1 = E(\F_q)$ and $G_2 = E'(\F_{q^r})$ where $E'$ is a twist of $E$ with the twisting isomorphism defined over $\F_{q^r}$. ...
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65 views

How is the generator used in Feldman's Verifiable Secret Sharing scheme determined? [duplicate]

According to the Wikipedia description of Feldman's VSS scheme First, a cyclic group G of prime order p, along with a generator g of G, is chosen publicly as a system parameter. (Typically, one ...
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Drawbacks of Schnorr Authentication that require Fiat-Shamir and Random Oracles?

I've been going through G. Maxwell's paper on the Borromean Ring Signature, and I don't fully understand this part on Schnorr Signature. If some could explain it more intuitively thank you. "...
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discrete logarithm vs normal logarithm [duplicate]

Crypto schemes normally use discrete logarithm instead of normal logarithm. I think this has to do with the fact that discrete logarithm is hard to solve while normal logarithm isn't. Can someone ...
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261 views

Can Bitcoin mining solve Graph Isomorphism-related problems?

Given a cryptographic hash $H:\{0,1\}^*\mapsto\{0,1\}^N$ and data $D\in\{0,1\}^*$, the Hashcash/Bitcoin Proof-of-Work entails finding a nonce $x$ such that $H(x\Vert D)$ begins with $d$ leading zeros, ...
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Is there a flaw in this ring signature scheme? [closed]

Having read some papers about RSA accumulators applied to ring signatures schemes, I ended up thinking why would we need to accumulate all the members public keys for our specific use case. So I came ...
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Hash function notations and their corresponding existing cryptographic hash algorithm

I am implementing Role-based access control by referencing the paper Enforcing Role-Based Access Control for Secure Data Storage in the Cloud. In page 6, they have mentioned to choose hash functions $...
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167 views

What is the hardness in Decisional Linear Assumption (DLIN)?

I had understood what does the DLIN assumption means and here is a related question. But I fail to understand the 'real hardness' in this problem. I would be grateful if someone can help me to ...
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141 views

Converting a number to a member of a multiplicative cyclic group

I am currently trying to make an implementation of the ElGamal encryption for educational purposes. As I understand it, when using the encryption with multiplicative cyclic groups, one generates a ...
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65 views

How to understand the Bilinear mapping with an example [duplicate]

An efficient bilinear map is given by $ e $: $G_{1}$ Γ— $G_{1}$ β†’ $G_{T} $. How can i prove this equation with the help of an example.
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Best group if one wants the discrete log problem to be hard?

Suppose one is implementing a cryptographic scheme over a group where one needs the discrete logarithm to be hard - what is the recommended group to use? I'm looking for a group where calculations are ...
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Generation of a cyclic group of prime order

I am trying to implement a cyclic group generator in Java, but I am running into some issues. In many cryptosystems, the following phrase is expressed during the key generation stage. Let G be ...
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890 views

Why is ElGamalEngine in bouncy castle restricting input data to be less than the length of the group parameter p?

I am currently using the ElGamalEngine of bouncy castle to implement exponential ElGamal for the purpose of making ElGamal additively homomorphic. To do this i raise the message to be encrypted to the ...
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102 views

If I know an element and it's inverse, can I learn the modulus?

If I know an element $x$ in a group, and it's inverse $x^{-1}$, can I guess the modulo, or with a probability?
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263 views

Show How to Efficiently Solve the Computational Diffie-Hellman Assumption given an Algorithm that Solves the Square-DH Problem

Let $q$ prime number, $G$ a cyclic group with order $q$ and let $g \in G$ be a generator of $G$. Suppose that you have an algorithm $A$ who takes input the element $g^a$ of $G$ and gives as output the ...
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Why are some group representations much easier to compute discrete logarithm for? [duplicate]

The multiplicative group mod $p$ is isometric to the additive group mod $p-1$, yet computing discrete logarithms in the additive group is easy and completing discrete logarithms in the multiplicative ...
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A Question about Notations and Groups

Please consider the following question: Determine the order of all the elements of the following multiplicative groups. You can write a C or Java program to do this. a. $Z_{21}^*$ b. $Z_{23}^*$ Now ...
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Pairing - Is it possible to map two $r$-torsion points to a $r^2$-torsion point?

Let $E(\mathbb F_{q^k})$ be an elliptic curve on finite field $\mathbb F_{q^k}$, where $\mathbb F_{q^k}$ is an extension of $\mathbb F_q$ with $k>1$. Let $e: G_1 \times G_2 \rightarrow G_t$ be a ...
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769 views

What is Group in Diffie-Hellman?

I understand how Diffie-Hellman key-exchange works. Mainly, two parties agrees in a prime $p$ and a generator $g$. Then one party selects its private exponenet $x$, computes its public value $g^x \...
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44 views

Are there any zero knowledge protocols which do not rely on a Group?

To me (new), it seems that a lot of cryptography relies on group theory. Are there any zero knowledge protocols which do not rely on a group?
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Key exchange protocols non-reducible to groups

Some questions I have ended up wondering about while reading through many of key-exchange protocols are: Is there an intrinsic reason, why most key exchange protocols use group-based approaches apart ...
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What does the “description of group $G$” includes?

I was reading here:second discrete log meaning in the solution and also here:key generation, first point where the say given $G$ (or its description). My question is what does this description ...
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271 views

Is solving a modular linear equation a hard problem when the coefficient is not an invertible element?

Assume that we have a linear equation like this: $$ax=b \pmod n$$ when $x$ is the unknown, and $a$ is not an invertible element in $n$. is finding $x$ a hard problem? (by solving I mean finding an ...
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Why is Multiplicative group used in RSA or Euler's theorem but not additive?

I am on the verge to understanding RSA, but suddenly a question popped into mind. When we are calculating $U(N)$ i.e $U(PQ)$, we are taking invertible elements that are co-prime to $N$. For example, $...
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What's the number of unique possible Cayley tables in a 16*16 grid for XOR'ing single hex characters?

A few days ago, I designed and s-box then derived the following Cayley table of all possible XOR outputs of hex digits in the range of ${2^4}$ and was curious how many such "valid" possible ...
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Elliptic curves - operations in larger groups - performance

According to my measurements and to this work, it seems that operations, for example scalar multiplication, are more expensive in larger groups. If I have, for example, an 80-bit elliptic curve and an ...
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232 views

Can someone explain the definition of four square roots as it pertains to groups in Z*p?

So I'm given the following as a problem: When $p$ and $q$ are distinct odd primes and $N = pq$, the points in $Z^βˆ—_N$ have either zero or four square roots. A quarter of the points have four square ...
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534 views

Pollard's Rho - Constructing the random function

Suppose we are aiming to solve the discrete logarithm problem $\alpha^x=\beta$ in some cyclic group $G=<\alpha>$. Then we are looking for a (uniformly) random sequence of elements of the form $\...
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Elliptic curve group over a prime finite field $F_p$

If $p$ is a big prime, and the elliptic curve $E$ is defined over $F_p$ by the equation $y^2=x^3+ax+b$ where $a,b\in F_p$. The point on $E/F_p$ together with the infinite point $\mathcal{O}$ form a ...
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Collusion resistance in proxy re-encryption scheme depending on bilinear map rules

A proxy re-encryption scheme is collusion resistant, if the proxy and a delegatee are not able to recover the secret key of the delegator. For example, when we have a message that was originally ...
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Rounding down in the exponent of group element

I have been struggling to find the algorithm $\mathcal{A}$ in the following. Let $(G,g,q)$ be the group parameter, $p << q$, $x\in \mathbb{Z}_q$, can we build the following algorithm: $$\...
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Is modulus a prime number important for non-symmetric cryptology?

From this link Generation of a cyclic group of prime order we know how to generate a prime order group. This illustrates why a prime order group is important. But why is modulus a prime number ...
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223 views

Is that possible to calculate modular inverse of a point on elliptic curves?

Imagine that you are given a point $P$ so that $P=a\times G$. If you have no knowledge of $a$ is that possible to calculate point $I$ so that $I$ is the modular inverse of $P$? We know that over ...
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Operation Table for Permutation Group

I cannot seem to figure out how the operation table for this permutation is formed. Is it multiplying each index and doing modulus? I can't seem to figure out. This is a Table 4.2 found in "...
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Confusion regarding computing Multiplicative Inverse Modulo P?

May be a silly doubt, please rectify my confusion regarding below problem: For concreteness assume $g=2, p=11, a=6$ and $x=9$ $$A = g^a \bmod p = 2^6 \bmod 11 = 9$$ $$X = g^x \bmod p = 2^9 \bmod 11 ...
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Research topics related to cryptography and Hamiltonian cycles

I am very interested in pursuing a research where I can show an application of Hamiltonian cycles in Cayley graphs of some group such as reflection groups to the field of cryptography. But currently ...
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51 views

Discrete log problem exponential runtime

I am trying to understand the runtime complexity of the discrete log problem (in the most basic sense). So, if we have $\langle g \rangle = G$ and are trying to find $g^x = a, a \in G, 0 < x < ...
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Order of subgroups formed by Elliptic Curves with a Cofactor

In this question: Why are the lower 3 bits of curve25519/ed25519 secret keys cleared during creation? The answer indicates that the order of all points on the curve over the finite field ...
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Multiplying between generators degree

I have to elements of multiplicative group of finite field with generator g - $$g^x,g^y $$ Can I get? $$g^{xy} $$
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Orthogonal generators of a group in Lelantus protocol

In the Lelantus Paper, the authors mentionned this: In our case, the commitment key ck specifies a prime-order group G and three orthogonal group generators $g, h_1$ and $h_2$. G is mentioned in ...
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How can we evaluate a polynomials in a group instead of a field? (verifiable secret sharing on elliptic curves)

I am trying to understand how we can have cryptographic schemes that builds on both secret sharing, which is build on top of a finite field, and bilinear maps, which are built on top of elliptic curve ...
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98 views

How to represent the point-at-infinity(Elliptic Curves) in code? [duplicate]

I am writing code for Elliptic Curve Cryptography. I have a class class EllipticCurvePoint. ...
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133 views

Why are the session keys in DHKE for different values of 'a' the same?

In a DHKE scheme with the a prime p = 467. an element g = 4. The element 4 has order 233 and generates thus a subgroup with 233 elements. after Computing the common key for A. a=400, b=134 B. a=167, b=...
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Which is the fastest way to find a member of a subgroup with known size modulo prime $P$, with know factorization of $P-1$?->$x$ with $x^s \mod P = 1$

Assuming you know the factorization of used prime $P-1$ $P-1 = s \cdot f_2\cdot f_3...f_i$ Now you want to find a member of a subgroup $\mathbb{Z}_s$. This means any $x$ with $x^s \equiv 1 \mod P $ ...
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Can we prove $(a,b,c,d)$ is of form $(g,g^x,g^y,g^{xy})$ without knowing $\phi (N)$?

assume $N=pq$ where $p$ and $q$ are two large prime numbers and all computations are modulo $N$. given $\phi(N)$, one can prove that the tuple $(a,b,c,d)$ is of the form $(g, g^x,g^y, g^{xy})$, i.e....
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How to find g in Factoring-Based Trapdoor Hash Function

Please explain how to find a value of $g$ if $p,q$ are safe primes having $p'=(p-1)/2$ and $q'=(q-1)/2$ are also primes $n=p*q$ $\lambda(𝑛) = \operatorname{lcm}(𝑝 βˆ’ 1, π‘ž βˆ’ 1) = 2𝑝'π‘ž'$. How to ...
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177 views

Working on subgroup of $\mathbb{Z}^*_p$ in practice

It is said that, given a group $\mathbb{Z}^*_p$, we can always have a subgroup whose order is prime. To this end, for a safe prime $p=2q+1$, compute $x_i^2 \bmod p$ for all $x_i \in \mathbb{Z}^*_p$. ...
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How to perform homomorphic multiplication in ElGamal?

How can I compute homomorphic multiplication in ElGamal? That is: Given two ciphertexts $(R_1,c_1)$ and $(R_2,c_2)$ corresponding to plaintexts $m_1$ and $m_2$ under some public key; how can I compute ...
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266 views

Create a field in PBC

Edited (I removed the emphasize on Integers): My question is partly cryptography and partly programming, I would appreciate any help on any aspect of it :) I want to use PBC library to do the ...