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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Get generator from a group in PBC Library [migrated]

I need to find generators from a bilinear group G of order N, using PBC Library. N = P*Q, where P and Q are distinct prime numbers. I have created the group G in the following way - ...
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When using Ristretto or Decaf with Ed25519 and Ed448, do scalars still need pruning/trimming/clamping?

Decaf is a point compression method that builds a prime-order group for (twisted) Edwards curves and Montgomery curves with cofactor $h = 4$ based on the Jacobi quartic [H2015]. The promise is to ...
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All generators for modulo p

so I have to find all generators for modulo p. And I thought: isn't there are rule for it? As far as I remember, every number from {1,...,p-1} is a generator because p is a prime. Is that true? Or am ...
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Why is Multiplicative group used in RSA or Euler's theorem but not additive?

I am on the verge to understanding RSA, but suddenly a question popped into mind. When we are calculating $U(N)$ i.e $U(PQ)$, we are taking invertible elements that are co-prime to $N$. For example, $...
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Research topics related to cryptography and Hamiltonian cycles

I am very interested in pursuing a research where I can show an application of Hamiltonian cycles in Cayley graphs of some group such as reflection groups to the field of cryptography. But currently ...
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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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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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Protocol for proof of knowledge of $l$-th root

Assume we have Group G in which the adaptive root assumption holds. This assumption states that if we choose an element $w$ and after that, if we receive a prime value $l$ it is hard to find the $u$ ...
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78 views

Relation between $N = P \times Q$, and $\Phi(N)$

When studying RSA, and proving simple concepts to myself, I went and understood groups and rings, but I failed to understand Lagrange's theorem. I did understand how from invertible finite groups I ...
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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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(How) Is DDH generally broken in groups of composite order?

In a somewhat recent lecture a claim was made that I couldn't back up myself but it got me curious whether it actually holds: If the order of the group is not prime, then the DDH assumption does ...
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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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Rounding down in the exponent of group element

I have been struggling to find the algorithm $\mathcal{A}$ in the following. Let $(G,g,q)$ be the group parameter, $p << q$, $x\in \mathbb{Z}_q$, can we build the following algorithm: $$\...
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Discrete log problem exponential runtime

I am trying to understand the runtime complexity of the discrete log problem (in the most basic sense). So, if we have $\langle g \rangle = G$ and are trying to find $g^x = a, a \in G, 0 < x < ...
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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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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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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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Variants of Bilinear Diffie-Hellman Assumption

Could someone point me to the paper/reference where the following variant of q-strong Bilinear Diffie-Hellman assumption was used? Given $s \in \mathbb{Z}_p^*$ and $g, g^{\frac{1}{s}}, g^{s}, g^{s^2},...
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How can we evaluate a polynomials in a group instead of a field? (verifiable secret sharing on elliptic curves)

I am trying to understand how we can have cryptographic schemes that builds on both secret sharing, which is build on top of a finite field, and bilinear maps, which are built on top of elliptic curve ...
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How to represent the point-at-infinity(Elliptic Curves) in code? [duplicate]

I am writing code for Elliptic Curve Cryptography. I have a class class EllipticCurvePoint. ...
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Would SHA-256(SHA-256(x)) produce collisions?

Was reviewing some Bitcoin public-key hash literature and the use of RIPEMD-160 and the SHA-256 as below: RIPEMD160(SHA256(ECDSA_publicKey)) The Proof-of-work ...
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Difference in elliptic curve order and finite field size [duplicate]

Must the prime finite field, Fp, an elliptic curve is defined over always have a greater number of elements than the cardinality of an elliptic curve. For example, If I have ...
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Is modulus a prime number important for non-symmetric cryptology?

From this link Generation of a cyclic group of prime order we know how to generate a prime order group. This illustrates why a prime order group is important. But why is modulus a prime number ...
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Calculation of the order of the cosets used in defining the Tate Pairing

I'm working through Pairings for Beginners by Craig Costello, and am trying to understand the preamble to the Tate pairing. (See p. 70 ff., section 5.2 of of the PDF.). I'm having trouble following a ...
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Why are some group representations much easier to compute discrete logarithm for? [duplicate]

The multiplicative group mod $p$ is isometric to the additive group mod $p-1$, yet computing discrete logarithms in the additive group is easy and completing discrete logarithms in the multiplicative ...
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How can we effectively compute the sqrt of some element in the group?

I only know one way, if this group is a cyclic group, and we know the element can be expressed in $g^m$, then $g^{(m/2)}$ is the answer. Another question, if $m$ is an odd number, can we be sure ...
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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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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 ...
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Is that possible to calculate modular inverse of a point on elliptic curves?

Imagine that you are given a point $P$ so that $P=a\times G$. If you have no knowledge of $a$ is that possible to calculate point $I$ so that $I$ is the modular inverse of $P$? We know that over ...
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Is permutation conjugate search problem many answer? Are there many answer equal? [closed]

If conjugate search problem of permutation is difficult , then there are next cryptosystem will appear. A=XYX−1,B=XZX−1,Y and Z are public key.And X is secret permutation. then encryption is C=...
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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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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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Why are the session keys in DHKE for different values of 'a' the same?

In a DHKE scheme with the a prime p = 467. an element g = 4. The element 4 has order 233 and generates thus a subgroup with 233 elements. after Computing the common key for A. a=400, b=134 B. a=167, b=...
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Why does Ed25519 scalar multiplication allow values larger than the subgroup order?

The GeScalarMultBase function is documented like so. From the way it is documented we see that it expects a little-endian value and has a precondition that constrains the range it accepts. ...
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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's the number of unique possible Cayley tables in a 16*16 grid for XOR'ing single hex characters?

A few days ago, I designed and s-box then derived the following Cayley table of all possible XOR outputs of hex digits in the range of ${2^4}$ and was curious how many such "valid" possible ...
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What can be said about the self-power map on groups based on DLP?

Introduction I've been playing with group representation theory some time, concretely representing groups as permutation groups (Cayley's theorem), where the group $G$ has an embedding into the ...
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How to prove element equality in G1 using Groth-Sahai proofs

In the BeleniosRF e-voting scheme, Groth-Sahai proofs with the "Instantiation based on the SXDH assumption" are used (see https://eprint.iacr.org/2007/155, version 20160411:065033, page 24). In the ...
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What does the number 256 in pairing curve BN256 indicate?

There are many pairing based elliptic curves like MNT curves, BN curves, SS curves etc., When we say BN256 curve, what does the number 256 indicate? Is it some group order or number of bits required ...
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Is there any property of the product you can predict before using $n$-times generator $g$ $\mod P$? Can any $n$'th element have a certain property?

Given a value $v$ which is in same group as the generator $g$ modulo prime $P$. The group size is a prime $s$. $v = g^a \mod P$ Only known values are $v,g,P,s$. Some (possible) computation of other ...

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