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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Pohlig Hellman and small subgroup attacks

While studying Curve25519 I read about the small subgroup attack in chapter 3. So far i know, that you need a point with a small subgroup to do such an attack. Curve25519 has a basepoint with prime ...
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Sage vs. Magma on DLP

Generally speaking, Magma is faster than Sage on several crypto-related computations, however, I have encountered a DLP instance where Sage is significantly faster than Magma. Take the DLP over $GF(p)$...
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What is this problem called and is it hard? given $g^x$ output ($g^y, xy$)

Assume that $G$ is any cyclic group where the discrete log problem is hard, such as the elliptic curve group. Let $g$ be some generator of $G$. The problem is as follows: Given $(g, g^x)$ for unknown $...
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Why discrete logarithm modulo composite moduli not popular and not defined in standards?

The classical discrete logarithm problem is to find $x$ such that $g^x\equiv h\bmod p$ where $p$ is a prime and $g$ is generator of multiplicative group modulo $p$. The demerit of this approach seems ...
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Pohlig-Hellman on ECDLP over extension field $\mathbb{F_p}^6$

Suppose there is an elliptic curve $E$ in form $y^2=x^3+b$ defined over $\mathbb{F_p}$, where $p$ is large prime. #$E(\mathbb{F_p})$ is also a large prime but #$E(\mathbb{F_p})\ne p$. ECDLP on this ...
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Which groups are secure for DL-Problem?

I was wondering why some groups provide more security to cryptosystems relying on DL-Problem. It is not clear to me whether it is just due to the known attacks or if there are some other reasons. So ...
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Is there a multiplicative group of integers modulo p in which the discrete logarithm is easy?

The complexity of computing discrete logarithms in a multiplicative group modulo a prime $p$ is assumed to be sub-exponential time. The complexity is determined by $q$, the biggest factor of the group ...
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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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When will the random bit sequence start to repeat in pseudo random number generator

Let's say we have the Blum-Micali pseudorandom number generator. from wikipedia: Let $p$ be an odd prime, and let $g$ be a primitive root modulo $p$. Let $x_0$ be a seed, and let $x_{i+1} = g^{x_i}\ ...
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Can zksnark prove DLP?

Can one use zksnark to prove the knowledge of a discrete logarithm? In another word, can zksnark (R1CS) encode exponentiation?
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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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implications of shor's algorithm on $F_{2^m}$ elliptic curves and GHASH

The security of elliptic curves depends on the difficulty of the discrete logarithm problem. Should Shor's Algorithm ever prove viable then elliptic curves would cease to offer any useful security ...
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What is the discrete logarithm assumption and why it is not easy with Shank's baby-step/giant-step

When I read about the DLP, it seems that the assumption is that it is not generally possible to solve it in polynomial time. But I also read that there are several algorithms in $\mathcal{O}(\sqrt{n})$...
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Inverse Public Key Proof

Alice has a private key, $x$, and a public key $P = [x] \cdot G$ in a group of order $n$. Alice would like to also publish her inverse public key (inverted modulo the group order) $P_{inv} = [x^{-1} \...
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MOV attack when $E(\mathbb{F}_q)$ is cyclic

Suppose $P\in E(\mathbb{F}_q)$ and $R=dP$. In the MOV attack, we compute $\alpha=e(P,T)$ and $\beta=e(R,T)$ and try to solve the discrete logarithm problem for $\alpha$ and $\beta$ in the finite ...
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Find prime $p$ such that $a^x\equiv b\pmod p$ for many $x\in[1,p)$

Given haphazard large integers $a$ and $b$ (like few thousand bits), can we efficiently find (and how) some integer triplet $(p,x,k)$ with $p$ a large prime (like a thousand bits) $a^x\equiv b\pmod p$...
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Small exponents and the RSA problem

I need some help with the following statement from the book A Graduate Course in Applied Cryptography* - Dan Boneh and Victor Shoup, in 8.10.1 The key derivation problem, page 320 of v0.5: ...
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Excluding specific factors for Pohlig-Hellman

I want to use Pohlig-Hellman and BSGS to solve the discrete log of an Elliptic Curve which has a composite order generator. The ...
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MOV attack on ellipic curves with the correct dlog in the finite field, but wrong dlog in the EC group

I'm following this description of the MOV attack: https://people.cs.nctu.edu.tw/~rjchen/ECC2009/19_MOVattack.pdf (slide 6/8) by implementing it. However, sometimes the computed dlog $k$ (which is ...
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Knowledge of discrete log is needed in the proof of Cramer-Shoup public key scheme?

In the proof of the Cramer-Shoup public key scheme [1], I understand that the adversary's view can be seen as equations such as $\log c = x_1 + w x_2, \log d = y_1 + w y_2$ and so on (equation 1 and 2 ...
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How to construct Undeniable Signature behind and its alternative response and verification

When I am working on Undeniable Signature, I am thinking what is the purpose and how to construct each step. Here is the details of Undeniable Signature For signature $z=m^x$ and $m$ is the message ...
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Variant of Schnorr Protocol (Difference pair of response and verification)

When I am trying to learn deeper to Schnorr Protocol. I found that for deference there is more than one response and verify pair. But I am not sure am I right. We will use Schnorr Protocol to prove ...
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Small complex multiplication field discriminant for solving ECDLP

I've seen from the SafeCurve criteria that one should try to avoid small complex multiplication field discriminant as it can speedup the discret log computation via the Polard Rho method. However, I ...
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How to reduce Computational Diffie–Hellman problem and Decisional Diffie–Hellman problem to Discrete logarithm problem

I'm supposed make some reductions but don't even know where to start. Any help would or explanation on how to do this would be much appreciated.
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The dth root unity in the Pollard Rho Algorithm

In the original paper of Pollard's Monte Carlo Methods for Index Computation (mod p): When the epact is reached, i.e. $$x_i = x_{2i}.$$ then the following equation is formed $$q^m \equiv r^n \pmod p,...
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Standards for finite-field discrete log cryptography

The NIST and TLS standards for Diffie-Hellman key exchange over a finite field all work in a subgroup of ${\mathbb Z}_p^*$ having prime order $q$, where $p = 2q+1$. On the other hand, DSA has a larger ...
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Better understanding what big O is referring to

I thought I understood big-O notation relatively well, but now I'm not sure. In particular, I've seen several posts like this discussing how the discrete logarithm problem is (probably) hard since our ...
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How to Solve Discret Log Computation Problem

Given $g^a$ in $Z_p$, it is hard to get the solution $a$. Everyone says yes, I wonder why it is hard. Could anyone give a specific mathematical way to show it is indeed hard?
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Decisional Discrete Logarithm problem?

Has the decisional version of the discrete logarithm problem been studied somewhere? I mean, for known $G$ in a group, distinguishing $xG$ and $Y$ for unknown integer $x$ and group element $Y$?
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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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Factorization problem

Say, $X= a\cdot b$, where $(a, b) \in Z_q^*$ and $q$ is a large prime. If $X$ is given, then what is the complexity (or hardness) of finding $a$ and $b$? Note that, either $a$ or $b$ can be reused to ...
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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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Lifting point to quadratic twisted curve

How to lift point to it’s quadratic twisted curve? I use secp256k1. Is the diiscrete log still same? Thanks before
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Given generator $g$ with prime order $k$ in $\bmod P$. Does increasing $P = 2 \cdot c \cdot k +1$ decrease security? Increasing $g$ increase security?

An adversary wants to find $a$ in $$m \equiv g^a \bmod P$$ He knows prime $P = 2 \cdot c \cdot k +1$ with it's primes $c,k$, value $m$ and $g$. And he also knows that $g$ only has an order of $k$, ...
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Is the member sum of a subset of $\mathbb{Z}/p\mathbb{Z}$ known (with $g^n \bmod p$)? Is it always $\mod P = 0$?

Let $P$ be a prime and $g$ a value between $2$ and $P$. Let $M$ be the set of numbers which can be generated with $g$: $$M = \{g^n\bmod P, \text{ with } 0 < n <P \}$$ If $g$ is a prime root of $...
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Discrete Logarithm: Given a p, what does it mean to find the discrete logarithm of x to base y?

My understanding is that $a^b \bmod p$ is the discrete logarithm problem. Given the question is worded this way, are we trying to find $ \log_y x \bmod p$. For instance, if we are trying to compute ...
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If they exist a relation between decisional Diffie-Hellman assumption and composite decisional residuosity assumption

From the cryptographic hardness assumptions, we have DDH and CDR assumptions. It is known that the composite decisional residuosity assumption is related to a factoring problem, while the DDH is ...
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Index Calculus for Discrete Logarithm

I'm studying the Index Calculus method for Discrete Logarithm. In the book "Introduction to Cryptography with Coding Theory" by Trappe it's told that, if $$\alpha^k\equiv \prod p_i^{a^i} \mod p$$ ...
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Why does using a prime-order subgroup in DLP improve security?

Let's consider a discrete logarithm $\beta \equiv \alpha ^{x} \bmod \,\, p$ We can solve it using Pohlig-Hellman algorithm. And, if $p-1 = tq$ where $q$ is a large prime factor, we can avoid any ...
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Weakness in Pohlig-Hellman algorithm

Let's try to solve a discrete logarithm: $\beta \equiv \alpha ^{x} \bmod \,\, p$ using the Pohlig-Hellman algorithm. Let's suppose that $p-1=tq$, where $q$ is a large prime number. This means that ...
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Identity Based Encryption: Known Random Value

Let's consider a situation whereby: Alice generates a ciphertext c from a message m using Bob’s ID. An attacker Carol can get c from the open channel. She knows that c is generated by using ...
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Discrete log problem in $N$ and $Z$

Is the discrete log problem hard in $N$ or $Z$?
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For discrete elliptic curves, can you find G, if you are given b and B?

I know you cannot find $b$ if you are given $B$ and $G$, where $B = [b]G$, but can you find $G$ given $b$ and $B$?
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Elliptic curve discrete logarithm problem

I'd like to know what is the maximum bits of the finite field that we can solve the ECDLP in a "regular" computer in trivial time. Is there any recent data about this?
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How does the order of Q affect the time it takes to solve ECDLP?

I use Sagemath's built-in function discrete_log() to solve ECDLP and according to the documentation it uses Pohling-Hellman algorithm to solve an ECDLP. This is ...
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Does the discrete log assumption hold if k > p

In discrete log cryptosystems like ElGamal it is noted that the "private key" $k$ should be chosen as any element of the group $G$ i.e. $k < p-1$. Does the integrity of the cryptosystem rely on $k$ ...
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Retrieving correct ciphertext from additive ElGamal

I have been studying additive ElGamal and I think I have the hang of it except the part where the message $M$ must be retrieved by computing the discrete log of $g^M$. From what I've read, the ...
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How do I interpret the CADO-NFS output for discrete logarithm calculation in GF(p)?

I'm using CADO-NFS to calculate discrete logarithm in a finite field GF(p). The problem is when I type ...

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