# 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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### 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 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)?
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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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### 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 ...
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### 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 ...
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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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### 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 ...
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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 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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### 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 ...
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### Which properties of a group are used in the steps of Diffie Hellman?

I’m trying to understand which properties of a group are used in DHKE at each step. For example, Alice and Bob’s public keys appear to only use the closure property of a group and maybe identity (e....
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### 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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### 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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### Cyclic Groups Other than $\mathbb{Z}^*_n$ or Elliptic Curves

I see two types of cyclic groups are most commonly used in cryptography: modulo multiplicative group of integers with prime order elliptic curves Are there any other cyclic groups used in ...
In a cyclic group with randomly looking behavior like the one used in secp256k1, is there any known efficient algorithm to compare the order of two randomly given elements $P_1$ and $P_2$ and find out ...