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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What are possible caveats when generating a group for use as parameters for Diffie-Hellman key exchange?

As reusing a widely used group for Diffie-Hellman key exchanges might lead to far easier third-party key discovery through precomputation for that specific group, I would like to know what can ...
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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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Solving the discrete logarithm problem for a weak group

I was reading an answer about an attack on a weak group for the discrete logarithm problem and wanted to formalize and verify that the attack was correct. That is, that it was guaranteed to always ...
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Using multiple permutations to strengthen the security of a cipher

In one book it says a set of permutations with the composition operation is a group. This implies that using two permutations one after another cannot strengthen the security of a cipher, ...
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How can I find the generator of a composite group and $Z_p*$?

I was doing some research on elliptic curves. I know how to find the generator of $Z_p$ (this is a prime group). But I came across the term $Z_p*$ (group containing elements that relatively prime to $...
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Finding a solution to a (sort of) discrete logarithm by asking questions

Let $p = kq + 1$ and $q$ be primes such that $log$ $q = n$, $log$ $k = n$ and such that the bit size of every prime factor of $k$ is bounded by $log$ $n$. Let $g$ be a generator of the unique ...
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1answer
76 views

Relationship between generating elements given by cycles in Cayley graph

The strong RSA assumption is that the following problem is hard to solve. "Given a randomly chosen RSA modulus $n$ and a random $z \in \mathbb{Z}_n^*$, find $r>1$ and $y \in \mathbb{Z}_n^*$ such ...
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Given $g,g^t$ in an RSA group modulo $N=pq$, is it hard to compute $g^{t^{-1}}$?

Suppose we have an RSA groug $G=\mathbb{Z}^{*}_{N}$, where $N=pq$ , where $p,q$ are primes. Let $g$ be a random element of $G$ and $t\in \mathbb{Z}^{*}_{N}$. Having $g$ and $g^t$, it seems to be very ...
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229 views

Hardness of CDH in different groups

What is the difference of the CDH problem in different groups? In particular, given a group $\mathbb{G}_1$ of order $q$ that is a subgroup of $\mathbb{Z}_q^*$, $q$ prime, and another group $\mathbb{G}...
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427 views

Do Gap-CDH groups exist?

A Gap-CDH group is such that, given group elements $g, a = g^x, b = g^y$, it is hard to compute $g^{xy}$, but, given a group element $c$, easy to verify if $c = g^{xy}$. While such groups have been ...
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$L^3$ Grover search of NTRU variants

I was reading a text on cryptology by Wayne Patterson and came across the $L^3$ algorithm which reduces integer lattices with respect to their base. I've also read on the NIST CFP A8 that attacks ...
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1answer
167 views

Does Linear Cramer-Shoup have pseudo-random ciphertexts?

"Linear Cramer-Shoup" is defined on pages 4 and 5 of $\:$ eprint.iacr.org/2007/074.pdf . Are the ciphertexts in Linear Cramer-Shoup computationally indistinguishable from uniform under a chosen-...
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How is it decided if $G_1$ and $G_2$ are two “additive” or “multiplicative” cyclic groups?

According to wiki's definition of Bilinear pairing… Let $G_1$ and $G_2$ be two additive cyclic groups of prime order $q$, and $G_T$ another cyclic group of order $q$ written multiplicatively. A ...
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Subgroups generators with respect to group generators of composite order

If I have a group $\mathcal{G}$ of order $N=npq$ and subgroups $\mathcal{G_n,G_p,G_q}$ of order $n$, $p$, $g$ respectively and if $g$ is a generator of $\mathcal{G}$ why then $g^{nq}$ is a generator ...
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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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Why do elliptic curves require fewer bits for the same security level?

I'm studying the basics of cryptography and I didn't understand why elliptic curves use fewer bits. For example, finite-field Diffie-Hellman needs at least 1024 bit and it's a DLP, but elliptic ...
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2answers
979 views

What does the linear assumption over bilinear groups mean?

In the abstract of "Cryptography with Tamperable and Leaky Memory", at the end of the 3rd paragraph, the authors say: In both schemes we rely on the linear assumption over bilinear groups. What ...
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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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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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690 views

Discrete logarithm weak group

I'm looking for weak groups in discrete logarithm, that $x$ can be extracted from $Y$ in polynomial time where $Y \equiv g^x \pmod{p}$ . I thought one way is to produce a prime $p$ that $p-1$ is an ...
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719 views

How to construct a hash function into a cyclic group such that its discrete log is intractable?

From the Linkable Ring Signatures paper: Let $G = \langle g\rangle$ be a cyclic group of prime order $q$ such that the underlying discrete logarithm problem (DLP) is hard. Let $H_1 : {0, 1}^∗ \to \...
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1answer
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What is a cyclic group of prime order q such that the DLP is hard?

On the original paper on Linked Ring Signatures, in order to construct its scheme, the author relies on this: Let $G = \langle g\rangle$ be a cyclic group of prime order $q$ such that the ...
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Finite fields in elliptic curve

I have an elliptic curve defined over finite field where $S_1=aP$ . Is it valid to say that $S_1P$ can also be computed. $P$ is the generator of the group. What my real question is that. Should '$a$' ...
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1answer
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On getting beyond LSB in discrete log

In discrete log we employ sophie germain primes $p=2q+1$ where $q$ is a prime. Then we know least significant bit $x_0$ in $$g^{2x+x_0}=h\bmod p$$ where $2x+x_0$ is discrete logarithm of known $h\...
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Why is only one generator stated in literature for elliptic curve group P-256?

I refer to elliptic curve groups over prime fields and their application in cryptography. If the order of a group is prime, it follows (am I wrong?) that: the group is cyclic and every element ...
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Simple example to describe Bilinear mapping

Notation : $\mathbb{G}$ is an additive group and $\mathbb{G}_T$ is multiplicative group of prime order $q$. Bilinear mapping $e: \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$ has to satisfy ...
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Can we reduce Diffie-Hellman problem to “Discrete-log inversion” problem?

Let $G$ be a cyclic multiplicative group of order $n$. Let $g$ be a (public) generator of $G$. The Diffie-Hellman (DH) problem asks: Given $g^x, g^y\in G$ for $x, y\in \mathbb{Z}^*_n$, to compute $g^{...
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How to determine proportion of quadratic residues in elliptic curve group?

I'm using a 'try and increment' method to hash to an Elliptic Curve point, explained below. With security parameter $k$, EC equation $y^2 = x^3 + ax + b \mbox{ mod } q$, we have: $ u = sha256(\mbox{...
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1answer
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Number generation for Fujisaki-Okamoto commitment scheme parameters

I need to implement the Fujisaki-Okamoto commitment scheme for a project such that I can demonstrate performance of various zero-knowledge proofs in relation to one another, for example Boudot's "...
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1answer
497 views

What's dh-composite test on badssl.com?

The site badssl.com provides examples of bad (red icon) and good (green icon) uses of TLS for the purpose of testing TLS implementations. I'm a bit confused by the test called dh-composite. This ...
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1answer
694 views

Diffie-Hellman key exchange using addition instead of multiplication?

I know that the additive finite group $(Zp,+)$ of prime order $p$ when I perform the Diffie-Hellman key exchange (DHKE) protocol is insecure. I didn't however find many sources online explain why it ...
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Why does Diffie-Hellman need be a cyclic group?

Why is Diffie-Hellman defined on a cyclic group? Doesn't it work for any commutative operation which the inverse is hard to find? Say Alice and Bob agree in a public prime $c$ and both choose a ...
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407 views

Is Triple-DES a group?

I know for a fact that DES is not a group, but are any of the Triple-Des versions a group? Why, or why not?
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1answer
51 views

Mapping a value $g^x \bmod p$ to a small interval $[1…H]$

My question is in $\mathbb{Z}_p^{*}$ context, where $p=q\cdot k+1$ for two primes $p,q$ and $k \in \mathbb{Z}$; $g$ is the generator of the subgroup $G_q$ of $\mathbb{Z}_p^{*}$, of order $q$. Let's ...
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Understanding the groups used in bilinear Ate-pairing

The bilinear ate pairing $e:G_1\times G_2 \rightarrow G_T$ is defined over the following groups: \begin{equation} \begin{aligned} & G_1 = E(\mathbb{F}_p)[r] \cap Ker(\pi_p-[1]), \\ & G_2 = E(...
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1answer
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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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582 views

Modular Arithmetic in RSA

Consider the following the following RSA public key $pk = (N, e) = (1457, 1307)$. (a) Knowing that $187^2 \equiv 1 \pmod {1457}$ find the factorization of $N$. (b) Given the factorization of $N$ ...
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659 views

Generator for Group $QR_{N}$

Let $N=PQ$, where $P=2p+1$ and $Q=2q+1$. $P,Q,p,q$ are prime numbers. $QR_{N}$ is the set of quadratic residues modulo $N$. Please help me to prove $QR_{N}$ is a cyclic group. Note: $QR_{P}$ and $...
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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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1answer
152 views

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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1answer
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Question about the location of the r-torsion in the quotient group used in the Tate pairing

I'm working through Pairings for Beginners by Craig Costello, and am trying to understand the Tate pairing. He defines $rE = \{r*P | P \in E(\mathbb{F}_{q^k})\}$ and then forms the quotient group $E(\...
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Why work in a subgroup for Naor and Pinkas oblivious transfer?

In section 4 (protocol 4.1) of the paper by Naor and Pinkas [1], why did the authors decide to operate in a subgroup? When they say "the messages are in the subgroup" does that mean $x, y, z_0, z_1$, $...
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1answer
153 views

How is a group element converted into a key?

I've only just started research on cryptography so I apologize if this is a basic question or I'm getting terms confused. I'm researching braid group cryptography and currently looking at the Anshel-...
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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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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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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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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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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 ...
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why are non singular curves used in elliptic curve cryptography?

It is not possible to draw tangent at all the points of a singular curve. What is the specialty of this and how it is related to cryptography and elliptic discriminant?