# 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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### Does a list of discrete log equations reveal information?

Given public generator $g$ of some cyclic group, a secrets $x\in Z_q$, and public pairs $(a_1,b_1),...,(a_n,b_n)$ (where $a_1,...,a_n$ are selected at random from a big set), and prime p, that ...
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### Distribution of elliptic curves with rank 2?

An elliptic curve defined over a finite field is either cyclic, or a direct sum of two cyclic groups. In cryptography, we use exclusively the former. I was wondering if there is any result on how ...
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### If we can solve discrete log with on $\frac{1}{poly(n)}$ instances, then we can solve, with high probability, for all instances

I am trying to prove the following: Given an ensemble $\{p_n, g_n\}$ ($p_n$ is an $n$-bit prime and $g_n \in \mathbb{Z}^*_{p_n}$ is a generator), if $A$ is a deterministic polynomial time algorithm ...
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### Can there be identical elliptic curve groups of points from different irreducible polynomials in binary extension fields?

Let $E$ be an elliptic curve over a binary extension field $GF(2^m)$, with constructing polynomial $f(z)$ be an irreducible, primitive polynomial over $GF(2)$, and let $G(x_g,y_g)$ be a generator ...
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### How to Run the Public Key Protocol for a Zero-Knowledge Proof of Identity?

In the paper Zero-Knowledge Proofs of Identity (by Feige, Fiat, and Shamir) a ZK protocol is described that leverages quadratic residues. Section 3 describes an "Efficient Identification Scheme,&...
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### Linearization attack on group with automorphism

Recently, I've had an exchange with Lorenz Panny about Xifrat. He says, that the quasigroup that I use can be linearized and then attacked, and he provided a script that linearized the quasigroup. His ...
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### Why is the discrete logarithm problem hard?

Why is the discrete logarithm problem assumed to be hard? Someone else asked the same question but the answers only explain that exponentiation is in $O(\log(n))$ while the fastest known algorithms to ...
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### Division by $2$ or principal root with DH oracle

Assume $g$ is generator of multiplicative group modulo prime $p=2q+1$ where $q$ is prime. Assume we know $g^{2t}\bmod p$ and $g^{2}\bmod p$ and assume we can have access to a Diffie-Hellman oracle. ...
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### How to know if a power is a permutation of an inverse group

Consider the group $$ℤ^*_{55}$$ Is exponentiating to the 3rd power a permutation of: $$ℤ^*_{55}$$ And exponentiation to the 5th power? I'm trying to solve this problem related to groups, but I don't ...
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### Given $N$ with $d$ prime factors. Can the number of unique values $x^d \mod N$ calculated for $d>2$? Does the total amount decrease at some point?

Given a number $N$ with $d$ unique prime factors. Can the number of unique values $v$ with $$v \equiv x^d \mod N$$ $$x\in[0,N-1]$$ $$N = \prod_{i=1}^{d} p_i$$ be calculated for $d>2$? (Q1) Does ...
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### Common exponent problem related to discrete logarithms assuming Diffie Hellman oracle

Let $g$ be a generator of multiplicative group mod $p$ a prime. Suppose we know $$g^{a+km_1}\bmod p$$ $$g^{b-km_2}\bmod p$$ $$g^{a+k'm_3}\bmod p$$ $$g^{b-k'm_4}\bmod p$$ where $m_2m_3-m_4m_1=\phi(p)$ ...
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### Given a cycle $x \mapsto x^a$ with his starting point $x_1$. Can another starting point $x_2$ be transformed to generate the same cycle?

A cyclic sequence can be produced with $$s_{i+1} = s_i^a \mod N$$ with $N = P \cdot Q$ and $P = 2\cdot p+1$ and $Q = 2\cdot q+1$ with $P,Q,p,q$ primes. and $a$ a primitive root of $p$ and $q$. The ...
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### Finding collisions of polynomial rolling hashes

A polynomial hash defines a hash as $H = c_1a^{k-1} + c_2a^{k-2} ... + c_ka^0$, all modulo $2^n$ (that is, in $GF(2^n)$). For brevity, let $c$ be a $k$ dimensional vector (encapsulating all the ...
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### Secret sharing scheme combined with probability theory results?

As a sequel of my previous post I am writing a new one with respect to the secret sharing scheme. I will only cite here the answer because I want to make a question on it. $\textbf{Answer:}$ To be ...
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### How to design such a secure multiparty computation scheme with the players using a majority rule

Suppose that $y$ is a uniform random variable that is defined over the field (or group or abelian group) $Y$. Let us suppose that there are $N=\{1,2,\cdots,i\cdots,N\}$ agents and only one of them, ...
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1 vote
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### Could this be a secure multiparty secret sharing scheme?

Suppose that $y$ is a uniform random variable that is defined over the field (or group or abelian group) $Y$. Let us suppose that there are $N=\{1,2,\cdots,i\cdots,N\}$ agents and only one of them, ...
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### Finding an element of $\mathbb{Z}_p$ if the order of that element is known [duplicate]

I have two prime numbers $p$ (1024 bits) and $q$ (160 bits) such that $q$ divides $p-1$. Now I want to find an element $b$ in $\mathbb{Z}_p$ with the order of $q$. That means that $b^q \equiv 1 \mod p$...
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### How to have a hash function that maps from a group element to a binary string of a certain size in charm-crypto?

I am facing a problem in programming with the charm-crypto library. The hash functions for pairing group elements in charm-crypto can only map from a string to a specific field: $\mathbb Z_r$, $G_1$ ...
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### zkSnark Intro by Maksym Petkus: Is the polynomial defined over $Z$ or is it defined over $Z_n$?

I am reading this explanation of zkSnark written by Maksym Petkus - http://www.petkus.info/papers/WhyAndHowZkSnarkWorks.pdf Here he has a polynomial $p(x) = x^3 − 3x^2 + 2x$ and the homomorphic ...
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### Structure of composition of permutations

If $P_1, P_2$ are finite permutations, what can we say about $P_3 = P_1 \cdot P_2$? That is, what properties of the composition of permutations can be inferred from the properties of the permutations ...
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1 vote
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### Generation of the order $\lambda$ (which is lcm((p-1),(q-1))) element g in modified paillier, why $-a^{2n}$?

As the question states, in variants of paillier cryptosystem, such as CS01 and DT-PKC, when they want an element $g$ of order $\lambda$, they choose a random number $a$ from group $Z^*_{n^2}$ and ...
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### How to find integer point of a ec curve in a given range?

I was looking inside the basics of ecc and found the examples from Internet either uses continuous domain curve or use a very small prime number p like 17 in a ...
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### Setting up the discrete logarithm framework

The discrete logarithm problem over prime cyclic groups consist of finding $x$ satisfying $g^x\equiv h\bmod p$ where $g$ is generator of multiplicative group $\mathbb Z/p\mathbb Z$ at a large prime $p$...
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### Does Poly1305 have weak keys like GCM/GHASH?

Some block cipher keys are weak when used with GCM; see this question. This happens when the multiplier $H$ decided by the key ends up in a small-order subgroup of $\mathbb{F}_{2^{128}}$. Poly1305 ...
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### Why does Index Calculus work?

I understand how the Index Calculus algorithm works - I know & understand the steps. I understand how the steps are derived. However, I am not able to figure out why it works. I can understand why ...
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