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).

32 questions with no upvoted or accepted answers
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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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125 views

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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142 views

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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171 views

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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82 views

Doubt in computing $g^\frac{1}{\delta+x}$ where $x \in \mathbb{Z}$

I was going through Zero Knowledge Set Membership and came across the following: Given $x \in \mathbb{Z}$ and $g$ is the generator of a multiplicative group $\mathbb{G}$ how do we compute $g^\frac{1}{...
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33 views

EC non-shared cryptosystems - different group for every party

Efficient Identity Based Parameter Selection for Elliptic Curve Cryptosystems by Arjen K. Lenstra contains a proposal for a non-shared elliptic curve cryptosystem. Every party chooses its own field ...
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69 views

Is there a bilinear map on a non abelian group or non cyclic group?

I've recently been studying a pairing map on cryptography. In usual definition, a pairing map is always defined on the cyclic group G. Is it possible to construct a bilinear map on a non-abelian group ...
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94 views

Why does the new encryption scheme proposed by authors stop an adversary from guessing the subspace of the secret key?

In this paper, the authors construct an encryption scheme that is supposed to be resilient to tampering and leaking (as opposed to just leaking). Specifically this scheme: If you look at the ...
2
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142 views

BLS signatures in the G-valued Random Oracle Model

This paper on semi-generic algorithms considers "non-standard properties of the employed hash function". For BLS signatures whose main group is $G$, I'm curious what can be shown when the hash ...
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1answer
155 views

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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113 views

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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110 views

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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266 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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61 views

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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168 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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373 views

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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28 views

Finding the relationship given by the generating elements of a non-abelian group (for a maximum case) where they correspond to finding cycles

Let $G=(\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes_{\phi} \mathbb{Z}_q$, where $p$ and $q$ are odd distinct primes. Let $G$ be generated by the elements $s=(g_1,e_q)$ and $t=(e_p,g_3)$, $g_1,e_p \in \...
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22 views

How do I generate a number from a Z* order q set

I would like to generate a random number based on this set, firstly how do I generate the numbers that belong to the Z* order q set. I have the q value and the prime p value. Also if there's a ...
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41 views

Computing discrete logarithms in the subgroup generated by 1 + N

When I read about DLP, I found that there are groups where the computation is easy. I found that it is known that computing discrete logarithms in the subgroup generated by 1 + N is easy. For example :...
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19 views

Finding the relationship between generating elements represented by a Hamiltonian cycle of a Cayley graph (like in strong RSA condition)

Consider an undirected Cayley graph of a finite group $\mathbb{Z}_p \times \mathbb{Z}_q$, where $p$ and $q$ are distinct primes. Let the generating set for the Cayley graph be $S=\{g_p, g_q\}$, where $...
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15 views

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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39 views

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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34 views

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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36 views

Secret sharing in different groups; Rounding down of elements

I have the following problem: A secret $x \in \mathbb{Z}_q$ is secret shared (additive secret sharing) between $n$ parties, now is it possible to compute the secret shares of $y = \lfloor x \rfloor ...
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104 views

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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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 ...
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40 views

How to encrypt a small number of identities (which are related to each other)? / Which algorithm has the smallest bit-length?

I'm looking for an encryption with as small numbers as possible. Given a small group of identities ($G$) (e.g. numbers from $1$ to $N$). Given one entry (or a small number) $e_i$ allows to compute ...
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48 views

What are some potential applications of trilinear mappings?

In "Applications of Multilinear Forms to Cryptography" Boneh and Silverberg give "one-round n-way Diffie-Hellman key exchange protocol" as a potential application of n-multilinear mappings. If we ...
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178 views

Proof in RSA encryption over multiplicative group

I everyone, I am considering an RSA encryption over the multiplicative group $G = (Z/nZ)$ of the ring $Z/nZ$, where $n = pq$, and $p$ and $q$ are distinct odd primes. First, I want to prove that $H=...
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39 views

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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60 views

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 ...
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344 views

Size of group for Elliptic curves vs RSA for equal security

For my research, I would like to compare the efficiency of a scheme when instantiated with Elliptic curves and RSA. So, I would like to know a "latest" comparison (as of 2018) on what group sizes of ...