Questions tagged [discrete-logarithm]

In cryptography, a discrete logarithm is the number of times a generator of a group must be multiplied by itself to produce a known number. By choosing certain groups, the task of finding a discrete logarithm can be made intractable.

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if they exist a relation between Decisional Composite Residuosity Assumption and Square roots in elliptic curve groups assumption

We have DCRA and ECSQRT assumptions. ECSQRT: Square roots in elliptic curve groups over Z/nZ Definition: Let E(Z/nZ) be the elliptic curve group over Z/nZ. Given a point Q ∈ E(Z/nZ). Compute all ...
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Zero-Knowledge Proof of Equality between RSA Modulus and Prime Order Group

Assume there is an RSA public key $(e,n)$ such that factarization of $n$ is unknown to both prover and verifier parties. We also have a prime order group $G$ and a generator $g$ for the group. For $m\...
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Why are elliptic curves over binary fields used less than those over prime fields?

In practical applications, elliptic curves over $F_p$ seem to be more popular than those over $F_{2^n}$. Is it because operations over prime fields are faster than those over $F_{2^n}$ for the same ...
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Gap between DLog and CDH

Is there any concrete group in which one CDH is exponentially easier (even it's still hard) than DLog.
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Time complexity of DLP over Elliptic curve group

Consider NIST 192 elliptic curve group https://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.800-186-draft.pdf. What is the time complexity of discrete log problem of it? Is it Pollard $\rho$ ...
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Super basic questions to DLP and DH

This is a super basic cryptography and I don't get it. My professor only explained it to us on a very abstract level, which I just agreed to without questioning it. The DLP says it is easy to ...
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The complexity of Pohlig-Hellman algorithm

The Wikipedia Pohlig Hellman algorithm says that the complexity of Pohlig-Hellman algorithm is $$\displaystyle {\mathcal {O}}\left(\sum _{i}{e_{i}(\log n+{\sqrt {p_{i}}})}\right)$$ I understand that ...
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Pollard Rho pseudorandom function

In Pollard Rho's Algorithm, a function $f$ with pseudorandom properties is required. Through this property, the birthday paradox can be leveraged to find a collision. But not all functions are ...
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Pohlig-Hellman: While solving in a subgrp, why is multiplication done mod the parent group's $p$ while the exponent is expanded as per $p_i$ of subgrp

In the Pohlig-Hellman algorithm, we take a Discrete Log Problem (DLP) in a group & solve it in subgroups $p_1^{n_1}$, $p_2^{n_2}$, $p_3^{n_3}$ etc & then combine it with the Chinese Remainder ...
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Discrete logarithm problem and the chinese remainder theorem

Say that $G_q$ is a group of order $q = \Pi_{i=1}^{s} q_i$, where $log(q) = n$ and $q_i$ is an odd prime $\forall i$ such that $log(q_i) = O(log(n))$. I'm tasked with arguing that $G_q$ is a cyclic ...
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Finding an elliptic curve of specific order

I wish to use elliptic curves for cryptographic operations like commitments etc. I see that most standard elliptic curves like $\operatorname{secp256k1, sect571r1}$ have a certain specific and fixed ...
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Security proof of schnorr identification scheme

I have studied the Schnorr identification scheme, and I came across the security proof. My question is regarding the following: $$\begin{align} \text{Pr}\left[\text{DLog}_{\mathcal{A}',\mathcal{G}}\...
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How to factorize the group order in Pohlig-Hellman algorithm

The Pohlig-Hellman algorithm is for computing discrete logarithms in a group whose order is a smooth integer. This algorithm requires the factorization of the group order. However we know that ...
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Independent parameters basis for torsion-groups in SIDH: Is the Weil-pairing necessary?

In the original SIDH paper by De Feo, Jao and Plût, the basis points $P_A$ and $Q_A$ are supposed to be independent points in $E(\mathbb{F}_{p^2})$ of order $\ell_A^{e_A}$ for some small prime $\ell_A$...
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Use the Index Calculus to solve for $19^x \equiv 205\pmod{337}$, using the factor base $B=\{2,3,5,7\}$

I'm supposed to use the following information to solve the question, but I don't know how. $$\begin{align} 19^2 &\equiv 2^3 \times 3^1 \times 5^0 \times 7^0&\pmod{337}\\ 19^5 &\equiv 2^5 \...
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Solution to Discrete Log as a Commitment

Is the solution to a discrete logarithm a reasonable commitment scheme? By my analysis, the following scheme is a reasonable commitment scheme: Let $p$ and $q$ be large primes such that $q∣(p−1)$, let ...
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Reduction to the discrete log problem

Let $G$ be a group of prime order $p$ and generator $g$. Let $msk_i = (x_i, y_i) \in Z_p^2$ be two master secret keys and $mpk_i = (g^{x_i}, g^{y_i})$ the corresponding master public keys, $i \in [0, ...
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Zero Knowledge Proof: Prove correct ElGamal encryption without leaking message

I know it is possible to prove zero knowledge that a given ElGamal ciphertext $(c,d)$ encrypts a plaintext belonging to some set ($\{0,1\}$ is frequently used for electronic voting applications). Next ...
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ElGamal Encryption over a Subgroup

Say for instance we are encrypting over a subgroup of ($Z_{67}^*, \bmod 67$). To perform a round of El Gamal's encryption scheme we should: Choose a subgroup of ($Z_{67}^*, \bmod 67$), of prime order....
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Problem about complexity of Chinese remainder theorem

I have a question about CRT. Assuming, that we have this system (S): x=a0 mod n0 x=a1 mod n1 with N=n0*n1 and n0,n1 are two distinct prime numbers. Then the complexity in terms of binary operation is ...
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Security of an ECDSA Adaptor Signature Implementation

I'm currently working on an implementation of ECDSA Adaptor Signatures, and part of the signature scheme calls for a NIZK proof to verify knowledge of exponent over two public keys that share a ...
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Is this an instance of a Diffie-Hellman problem?

Let $\mathbb{G}$ be a cyclic group of order $p$ with generator $g$, and let $m\in\mathbb{G}$. Problem: Given $c=m.g^{k.a}$ and $v=g^a$, where $k,a \in \mathbb{Z}^*_p$, output $k$. Is this an instance ...
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Is there any way to encrypt the path in between a small number of 3D positions?

Or more general: given two valid random cipher $c_0, c_1$ a function $D$ with $$D(c_0,c_1) = (a,b,c)$$ should exist but hard to compute (for most cases). The result $(a,b,c)$ represents the path from $...
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Proof of knowledge of discrete log

Suppose that $P = pG = H(rK)G$, where $P$ is a point on an elliptic curve, $H$ is a hash function, $rK = kR$ is a shared secret between the party that knows $r$ and the party that knows $k$, and $G$ ...
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In which case number field sieve/index calculus is faster for solving discrete logarithm?

Given the normal discrete logarithm problem: $$a = b^c \mod{P}$$ with prime $P$ and numbers $a,b,c$ For which kind of $P,b$ the NFS/IC algorithm is faster than Baby-Step/Giant-Step+ Pollard's Rho ($\...
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114 views

How does the security of Elliptic curve compare to normal discrete logarithm?

Intro: EC are often compared with RSA but how about a more safe version of the discrete logarithm? All 3 can be reduced to the problem: $$b = g^a \mod{P}$$ In RSA $P$ is a product of two primes. To ...
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Clarification of Advantage vs Probability/Success of an Adversary

In cryptography, for a polynomial time-bounded adversary $\mathcal{A}$, given a scheme $\Pi$, the success or probability of succeeding $\mathcal{A}$ is the likelihood for $\mathcal{A}$ to break $\Pi$, ...
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How can the Number Field Sieve attack the discrete log in $\mathbb Z_p^*$ of DSA?

The Digital Signature Algorithm (DSA) uses $L$-bit prime $p$ and $N$-bit prime $q$ with $q| p-1$, i.e., $p = r\cdot q +1$ ( Schnorr group if $r>2$ and safe prime if $r=2$). In a way, the security ...
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what is the probability for an adversary to find the new key after adding new entropy in a group where computational diffie hellman is hard?

Let's say I have an Elliptic curve group $E(\mathbb{F}_q)$ with base Point $G$ and large prime order $n$. Computational Diffie-Hellman is assumed to be hard in that group. $H: \{0,1\}^*\rightarrow \{...
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How hard will it be to solve an equation in elliptic curve group/ cyclic group where Discrete Logarithm is hard?

Given an Elliptic curve group $E(\mathbb{F}_q)$ where the Discrete Logarithm Problem (DLP) is hard and a base point $G \in E(\mathbb{F}_q)$ with large prime order $n$, what will be the advantage of a ...
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How to reconstruct the static private key of a Diffie Hellman client, when I can freely choose A, g and p?

I am struggling with a Diffie Hellman crypto challenge based on a client that uses a static private key. My goal is to trick the client into revealing enough information to reconstruct the private key ...
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Does this describe an invalid key share attack?

I recently came across a mention of invalid key share attacks on this site. It is done with DH groups over $\mathbb{ℤ}_p$ :prime $p$, where $p = qh + 1$, where $q$ is the prime order of DH subgroup ...
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Modular equation system

I have $N=p\cdot q$ and the following system where I know $A,B,C,D, k$: $$A = B \cdot q^k \pmod N$$ and $$C = D \cdot p^k \pmod N$$ Is there an easy way to recover $p$ and $q$?
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Discrete logarithm with 2 solutions? A clarification request

I need some clarification on the discrete logarithm problem... When a friend and I were solving for the discrete logarithm problem of 9 = 2 ^ x mod 11, we got two ...
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Does resuing pederson commitment preserve the hiding property?

Assume you have a pedersen commitment scheme where the commitment is: $$\mathcal C_1 =C(m,r)=g^m\cdot h^r$$ with $g,h$ being public generators in a public group $(G,\cdot)$ in which the discrete ...
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Question about hiding commitment scheme for integers

Given a generic group $\mathbb{G}$ of an unknown order (such as a $3000$-bit RSA group) and a randomly generated element $g \in \mathbb{G}$, is the commitment scheme $\mathrm{Com}(x)= g^x$ not ...
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Computing $g^y\bmod p$ from $g^{y^2}\bmod p$ if Diffie-Hellman is compromised?

Given generator $g$ of a multiplicative group mod a prime $p$ the Diffie Hellman problem is to find $$g^{xy}\bmod p$$ from $g^x\bmod p$ and $g^y\bmod p$. The best way to solve this is through discrete ...
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A card game (for mental poker or any other card game)

I thought of a way to produce trustless card game in a flexible way. One feature that I want is it should be flexible (It should work for any type of card game, though I indeed started it as a ...
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Question about the proof of the change of base formula for the discrete logarithm

I was looking at the proof of a change of base formula for the discrete logarithm in this paper (page 6, 4th bullet indent). In the intruduction, the paper states: Let $F_q$ be a finite field of order ...
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By what modulo calculations with discrete logarithms are performed?

For odd prime $p$, I have been given a group $\mathbb{Z}_p^*$ of all invertible elements from $\mathbb{Z}_p$. Basically, $\mathbb{Z}_p^* = \{1,2,\ldots , p-1 \}$. I also have $a$ and $b$, which are ...
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Show how an efficient algorithm for computing discrete logarithms with base a can be used to efficiently compute discrete logarithms with base b

I have an exercise that says the following: Let $p$ be an odd prime, and let $a$ and $b$ be generators $\mathbb{Z}​_p^*$. Suppose that we have an efficient algorithm $A$ for computing discrete ...
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Breaking Elgamal private key

I have been given a task to explain if - given a public key and a portion of a private key (over 300bits) with a remain unknown of 80 bits - the private key can be broken with an algorithm faster than ...
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Is there a concept of Pedersen commitment “in the base”?

This question Can Elgamal be made additively homomorphic and how could it be used for E-voting? says ElGamal can be made homomorphic over multiplication. So you can have $(g^r, h^r g^m)$ (i.e., ...
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Where is factoring if discrete logarithm is broken?

Assume given $g^X\equiv h\bmod p$ where $g$ is of order $\frac{\lambda(p)}2$ where $\lambda(p)$ is Carmichael Lambda function applied to prime $p$ (so $2$ is invertible in exponent) we can compute $X$ ...
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Why do the discriminant and primality of the group order of an elliptic curve affect security?

In a book about cryptography and elliptic curves, there was a mention that not all curves are secure, and a statement than in order to pick a secure curve the curve must satisfy 3 requirements. The ...
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Is it hard to determine an automorphism when the mapped value by several compositions of the automorphism is given

Generalization of the Discrete logarithm problem to non-abelian groups is discussed by many authors. One of the generalizations is shown in MOR cryptosystem as in the below link, by considering the ...
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What is a “constant time” work around when dealing with the point at infinity for prime curves?

I've been working for some time, on designing a constant time solution for dealing with the "point at infinity" for prime curves. So, far I'm using the Standard Projective Coordinates for ...
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Why is this signature independent of the message?

Assume that we have the following signature scheme CL Signature: Choose a cyclic group $G = \langle g \rangle$ of order $q$. Uniformly and randomly choose two elements $x,y \in \mathbb{Z}_q$, and ...
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The Generator point and Mod P in ECDSA

I've been reading about The discrete logarithm problem as of recent and i decided to try it out on a small portion of numbers myself and i actually came to a mental gridlock after watching this Video....
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How does $g^{x^2} \bmod p$ help you find $x$?

I was thinking about the Diffie-Hellman key exchange. One fact that we seem to know is that given a group generator $g$, a prime $p$ and the expression $g^x \bmod p$ its believed to be hard to find $x$...

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