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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Finding $x$ such that $g^x\bmod p<p/k$?

In a Schnorr group as used for DSA, of prime modulus $p$, prime order $q$, generator $g$ (with $p/g$ small), how can we efficiently exhibit an $x$ with $0<x<q$ such that $g^x\bmod p<p/k$, for ...
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Variant of Pollard rho using small factors of p - 1

Given an integer $N$ to factor which is divisible by some prime $p$, suppose you know (or guess) that $p - 1$ has a few small factors, e.g. $3, 2^2, 5$. Define $B$ as a product of small prime powers ...
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Are there any cryptographic methods which use multiple cyclic groups?

For some cryptographic methods you can construct them. e.g. elliptic curves (product of two cyclic groups) or Diffie–Hellman (can be product of n-cyclic groups). But they have no usage because at a ...
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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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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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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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Elliptic curve group inverse addition in OpenSSL

I am using group P-256 on OpenSSL with C++. My understanding was that, if you have a point $xP$ and then calculate (xP)^(-1) with EC_POINT_invert(group, xP_inv, ctx), then when I calculate: xP + (xP)^(...
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A related question to “relationship between generating elements given by cycles in Cayley graphs”

I am writing this question with reference to the post at Relationship between generating elements given by cycles in Cayley graph When considering the generating elements $g_qg_p$, does it have the ...
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FLT is partly applying to RSA equation, and also relation between ED mod phi and Phi + 1 mod N

After numerous attempts from myself and all of you guys, I finally came to understand RSA. I can now prove it and understand how I got there. But I still have some very few polishing questions. 1) We ...
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Proof of two pairs with same exponent

Lets assume we have a group $G$ with unknown order. And we have a pair $(A_1,K_1), (A_2,K_2)$ in which all $A_1,K_1,A_2,K_2$ are group elements. The claim is $A_1= K_1 ^ x$ and $A_2 = K_2 ^ x$. or ...
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Attack against factorization of $p-1$ of $\mathbb{Z}_p^*$ group

It is said that for the group $\mathbb{Z}_p^*$, the factorization of $p-1$, is critical. If $p-1$ has some small factors $q_1, q_2, q_3, q_4$, then when we transmit $g^x \bmod p$ where $g$ is a ...
Algorithm to compute DLOG for elliptic curve $E(F_p)$ with order p
I was reading about elliptic curves in this pdf. Page 55 of the pdf states that if number of points on elliptic curve #$E(F_p) = p$, then there exists a p-adic logarithmic map that homomorphically ...