# 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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### Group Rings on Cryptography

Let $R[G]$ or $RG$ be the group ring where $R=F_q$ and $G$ is any group. Let $Dim(V)=\vert G \vert$. It's clear that $V$ has $\vert R \vert^{\vert G \vert}$ distinct $\vert G \vert$-tuples. This ...
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### Non-commutitive and nonassociative algebraic structures in cryptography

Are there any cryptographic algorithms or primitives that have been developed and studied that make use of non-commutative or non-associative algebraic structures such as quaternion integers or ...
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### number theory question in a group with unknown order

I was reading a paper and I am struggling understanding one part of it. Lets say we have a group $G$ of an unknown order $n$. we know that $B<n<B+C$. both B and C are large values). we choose a ...
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### Is there a quick method of listing certain elements of a cyclic group?

I'm studying applied cryptography and stumbled upon the following question to practice the knowledge about Congruence, Groups etc. "List all Elements $x$, where $x^2 = 2$ in $\mathbb{Z}_{31}$ Okay, ...
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### How to find a small generator with small inverse? Does it have negative impact to security? (for Schnorr subgroup of $\mathbb{Z}/P\mathbb{Z}$)

Given a prime $P$ with $$P= r \cdot q+1$$ with $q$ prime as well. I'm looking for a generator $g$ of the Schnorr subgroup with order $q$ which is small by value and has a inverse (to $\bmod P$) which ...
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### Double discrete logarithm on elliptic curve

Background: I am attempting to implement the paper Publicly Verifiable Secret Sharing. I managed to get it working using modular groups, but when I want to make it more efficient by transferring to ...
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Given the finite cyclic, additive group (G, +), with |G| = n and generator = g, what are the computations and exchanged messages for Diffie-Hellman? What I tried myself: Alice chooses a private $a$ ...
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### Define a Decryption algorithm on a given group-based Cramer Shoup lite scheme

I am currently working on public key encryption schemes and I want some help to figure out how decryption algorithms work. Suppose we have a public key $pk = (G,p,g,e)$ with $e \in Z^*_p$ . (where $G$ ...
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### lcm versus phi in RSA

In textbook RSA, the Euler $\varphi$ function $$\varphi(pq) = (p-1)(q-1)$$ is used to define the private exponent $d$. On the other hand, real-world cryptographic specifications require the Carmichael ...
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### Size of $E$ over $\mathbb{F}_p$ contains $p+1$ points

I am struggling to prove this claim: I proved that the map $x\mapsto x^3+1$ is a bijection from $\mathbb{F}_p$ to itself if we have that $p\equiv 2\bmod{3}$. We have to use this fact to prove that ...
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### Is this an error in the Pinocchio Protocol paper

I am going through the Pinocchio protocol paper and I need 2 clarifications in the section Protocol 1 (Verifiable Computation from strong QAP). The part that explains the Verify process, which ...
I was wondering if a group like the 1536-bit MODP Group from RFC 3526 was a Schnorr group? A Schnorr group must apparently have: $p$ and $q$ being primes $p = q\cdot r+1$ $1 < h < p$ $h^r\not\... 1answer 88 views ### Is it possible to construct a multiplicative group from$\mathbb{Z}_n$if$n$is not a prime number? With$n$being a prime number I know we can generate groups over multiplication. Is it possible the other way around ($n$not being a prime)? 1answer 132 views ### Order of subgroups formed by Elliptic Curves with a Cofactor In this question: Why are the lower 3 bits of curve25519/ed25519 secret keys cleared during creation? The answer indicates that the order of all points on the curve over the finite field ... 1answer 68 views ### Why do we use prime numbers apart from hard factorization? Why are prime numbers important in cryptographic constructs? I am not interested in RSA examples where factorization is the hard problem itself, that makes sense. However wherever I go I encounter ... 0answers 25 views ### What is the current state of the Conjugacy Search Problem in Matrix and Permutation Groups? It results that I have come with a mathematical approach to solve the CSP (Conjugacy Search Problem) where the platform group$G$is either a permutation group or a Matrix Group. From my part I ... 1answer 83 views ### Calculating G for a give cyclic group mod P For DH key agreement, one must begin with a generator of a cyclic group g. However, intuitively to me at least, it seems that g ... 1answer 91 views ### Is there another group structure that is suitable for RSA other than$U_{pq}$? I know that calculating the cardinality of$U_{pq}$is infeasible and therefore it is extremely hard to break a code using Lagrange's theorem. But later on my studies i realized main principle of RSA ... 1answer 35 views ### Multiplying between generators degree I have to elements of multiplicative group of finite field with generator g - $$g^x,g^y$$ Can I get? $$g^{xy}$$ 1answer 778 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 \...
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 ...