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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Is it possible to generate backdoored DH parameters?

I know it has been already asked and answered whether it's possible to generate weak DH parameters. But "recentely" we experienced the Logjam attack, which makes use of the pre-computation ...
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30 views

Testing PRNG quality from ECC public keys?

Having a large set of ECC public keys $P_i = n_iB$ on a fixed curve $E$ over a prime field, is there a way to determine if coefficients $n_i$ were generated using a bad PRNG? In other words, can a ...
17
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500 views

Hardness of finding mutual discrete logarithms of small generators in $\mathbb{Z}_p$

Suppose you want to select a prime $p$ such that finding e.g. $log_2(3)$ in $\mathbb{Z}_p$ is expected to be either at least as hard as the general Discrete Logarithm Problem in $\mathbb{Z}_p$, or at ...
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40 views

Reduction to cdp,dl or cdh?

Assuming a randomize encoding scheme that takes as input a secret key $\mathsf{sk} \in \mathbb{Z}_p$ for a large prime number $p$. Then the algorithm outputs $g^{\mathsf{sk}x}, x \in \mathbb{Z}_p^*$. ...
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1answer
197 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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1answer
44 views

Brute forcing the secret key in Elgamal encryption

Crypto noob here, I am attempting to do this programming challenge. I do not have the secret key that is used to decrypt the message. However, the key is small enough for a brute force approach. I am ...
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2answers
472 views

Reuse of a DH / ECDH public key

I was wondering whether it is safe to use the same DH or ECDH key pair in more than one key agreement, particularly if these public keys are in a public registry. These public keys could be used by ...
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1answer
39 views

Degenerate discrete logarithm in binary field

Given a field $\mathbb{F}_{2^n}$, are there any choices of primitive element $g$ that make the discrete logarithm easier for that generator? That is, are there any degenerate cases? For example, if I ...
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33 views

El-Gamal and Lines on Planes

I've been thinking about a geometric picture for El-Gamal. The idea is to understand the set $\{(my^{x},g^x) \mid x \in Z_p\}$ (the set of encryption of $m$ for fixed $g$ and $y$) by taking the ...
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2answers
183 views

In a group, is it hard to calculate the base $g$ given $g^a$ and $a$?

Discrete logarithm, that is: calculate $a$ given $g$ and $g^a$, is assumed to be a hard problem in some groups. Is it also hard to calculate $g$ given $g^a$ and $a$?
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1answer
164 views

Discrete Logarithm (512 bits numbers) - Find exponent parameter

I'm trying to resolve a discrete logarithm equation: $$y = g^x \bmod p$$ Every parameter is a 512-bit number. I know the values for $g$, $y$ and $p$ and I need to find the $x$ value. Finally, I know ...
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1answer
171 views

Finding discrete logarithm with baby-step-giant-step algorithm

I am trying to use the Baby Step Giant Step algorithm to find discrete logarithm in: $$a^x= B \pmod p$$ with using BSGS: $$x = im+j$$ $$a^j = B a^{-im}$$ where $m = \sqrt{p}$ Wikipedia says: ...
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1answer
334 views

Solving a discrete logarithm using GDlog

I am trying to calculate an $x$, such that $t = g^x \pmod p$ in order to crack a weak ElGamal encryption for university. I found GDlog, but I cant figure out how I can use the input to calculate my ...
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1answer
602 views

Difference between Pedersen commitment and commitment based on ElGamal

Does any of you know what is the difference between the Pedersen commitment and the commitment that uses the ElGamal encryption scheme? For the sake of completeness, I recall what both of them look ...
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1answer
90 views

Is $(a,g^{ab})$ computationally indistinguishable from $(a, g^c)$?

From wikipedia, the DDH assumption says,given a cyclic group $G$ of order $q$ with generator $g$, $(g^a, g^b, g^{ab})$ looks like $(g^a, g^b, g^c)$ where $a,b,c$ are randomly and independently chosen ...
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42 views

Solving discrete logarithm when p is not a safe prime

If you have the cyclic group of integers modulo $p$, where $p$ is not a safe prime, as well as a generator $g$ with which for all factors $q$ of $(p-1)$, $g^{(p-1)/q} \ne 1$, This answer says that ...
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0answers
43 views

Discrete log in Galois Extension Field

I was reading 'Pinocchio Coin' paper by Danezis et al. where they have said, "If we use the efficient pairing groups of Pinocchio, computing discrete logarithms in the exponent field $\mathbb{F}_p$ ...
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1answer
167 views

Logjam on Elliptic Curves?

I think we're all aware of the Logjam attack. From now on we know that re-using primes for DH is a bad idea. But we also say that elliptic curves are safe from the attack (relying on the NFS), ...
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2answers
82 views

Discrete logarithm over prime modulo: small input, large exponent, larger prime

I am considering using modular exponentiation as a one-way hash function. More specifically, here is the scenario. 1) Input ($m$): the input messages are small (16-bit) 2) Exponent ($e$): the ...
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2answers
121 views

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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1answer
102 views

What does signed fixed window method mean in ECC?

I am studying (sliding) window method in Elliptic Curve Cryptography (ECC) but I am confused by the term, signed fixed window method. By the way term is used in a research paper and not in the book ...
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69 views

Is the reverse of the “discrete logarithm problem” equally dificult? [duplicate]

It is not easy to understand why this becomes a hard problem. The discrete logarithm problem as defined here: “any integer k that solves $b^k = \{g\mod{n}\}$ is termed a discrete logarithm” i.e.: ...
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2answers
100 views

Zero knowledge-proof for discrete log that is not honest-verifier

Take a cyclic group of prime order. The Schnorr-protocol for proving knowledge of the discrete logarithm of some group element is honest-verifier zero-knowledge, meaning that if the verifier chooses ...
2
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1answer
48 views

Discrete log accumulator without pairings

Here $g$ is some fixed generator of a discrete log group. I don't want the group to be bilinear for efficiency and BDH-skepticism reasons. Is anyone aware of a discrete log accumulator? What I mean ...
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2answers
85 views

What does variable-base point/scalar multiplication mean in ECC?

I am a bit confuse about the term, variable-base point/scalar multiplication, in Elliptic Curve Cryptography. What I have understood so far. It means that the base or point on EC is variable/unknown. ...
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1answer
43 views

Is this discrete log generalization a well known cryptographic assumption?

Assume you have a finite group $\mathbb{G}$ and an integer $n$. Given $g_1,\dots,g_n,t$ chosen uniformly from $\mathbb{G}$, consider the problem of finding a vector $(a_1,\dots,a_n)\in \mathbb{Z}^n$ ...
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26 views

Looking for an adjustable compute bound decryption function

I'm looking for a way to decrypt data, but make it require varying amounts of CPU power to do so. Maybe this could be illustrated as e(m) = m' and ...
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2answers
77 views

Find the prime factorization in the DLP

Suppose we have $g^x \equiv h \pmod N$ Where $N = pq$ and $p$, $q$ are distinct primes. Also $(p-1)/2$ and $(q-1)/2$ are primes. If we know what $x$, $g$ and $h$ are, is it possible for us to know ...
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112 views

Zero Knowledge Non Interactive Proof with random oracle

I am trying to write an assay about Non Interactive Zero-Knowledge proofs and would like to take the simple discrete logarithm problem example fallowing the Feige-Fiat-Shamir heuristics. I understand ...
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1answer
50 views

In a additive group is it hard to calculate $bg$ given $ag, g, abg$

The ECDH problem defined that given $g,ag,bg$ it is difficult to calculate $abg$. But it is also difficult to calculate $bg$ given $ag,g,abg$. where $g$ is generator and a,b are elements of group.
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Parallel Pollard's Rho: Number of distinguished points

When using the parallel version of Pollard's Rho algorithm for discrete logs, each processor performs its own random walk to find distinguished points, and reports the starting point and the ...
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1answer
68 views

Diffie-Hellman on infinite groups

The most common groups to be used as examples for the DH protocol are modular multiplication and elliptic curves. But I've realised that the groups doesn't need to be finite, a suitable infinite group ...
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80 views

Exploration of Blum Micali Security By Seed Size

I'm new to cryptography and am most intrigued by mathematically based pseudo random number generators. With reference to the Blum Micali algorithm: $X_{i+1} = G^{X_i} \bmod P$ can security be ...
3
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57 views

Pollard's Rho - Restricting the random function to the exponents

Pollard's Rho is usually constructed using a function $f:G \rightarrow G$ which behaves 'random enough' in order to detect a collision with Floyd's cycle detection trick. It is easy enough to observe, ...
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1answer
74 views

What is the difficulty of DLP in GF(P^Q) with a subgroup with a prime order of L

Given a finite field GF(P^Q), having a subgroup with a prime order of L (P,Q,L are all primes), how difficult is it to find the discrete log, is it related to P and Q or is it related to L, or to ...
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1answer
128 views

subgroup of quadratic residue

if a number is said to be a subgroup of a quadratic residue of Z∗p, can I affirm that it is a generator of a cyclic group ? ie, say i am looking for a number which is an element of order q in Z*p, ...
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1answer
162 views

Discrete logarithm modulo a smooth number

I am solving the discrete logarithm problem modulo $N$. $N$ is a composite number, I found its factors — lots of small primes and two big primes ($> 2^{50}$). Does the factorization of $N$ somehow ...
3
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100 views

Relations between RSA and DLOG, factoring and DLOG

Definition: (The generalized Diffie-Hellman problem) Let $n=pq$ for two large primes $p,q$. Given $x, x^a, x^b,n$, find $x^{ab}\pmod{n}$. (1) Is there a known reduction from the GDH problem ...
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1answer
97 views

Prime Numbers in Discrete Log

I am implementing a security protocol based on discrete log. I came across the equation $p = kq + 1$. Understand that based on number theories that both $p$ and $q$ should be large enough to be ...
3
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1answer
299 views

Generating a valid signature on El-Gamal without knowing the private key

Suppose we are given $p$, the large prime, $g$ which is the primitive root for $p$, $b$ which is calculated as $b=g^x$ mod $p$ where $x$ is the private key and $0<x<p-1$. Also suppose we know ...
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1answer
272 views

How can we prove that two discrete logarithms are equal?

Suppose there are two elements $a = g^x$ and $b = h^x$, where $g$ and $h$ are generators in $Z^*_p$ and $p$ is a large prime. How can we prove that $a$ and $b$ have the same discrete logarithms with ...
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47 views

DLOG in $\mathbb{F}_{p^n}^*$?

Assume that we are given an element $g\in \mathbb{F}_{p^n}^*$ and $g$ does not belong to any of the smaller subfields contained in $\mathbb{F}_{p^n}$. If the degree of $g$ is some number $q$, how much ...
4
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1answer
66 views

NIZK Proof of knowledge N of M discrete logarithms (threshold)

It is well known how to produce a NIZK that curvepoints $aG$ and $aP$ have the same discrete logarithm $a$ with respect to the curvepoints they are multiplied by. There is also a way to prove that a ...
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1answer
125 views

Pollard's Rho - Constructing the random function

Suppose we are aiming to solve the discrete logarithm problem $\alpha^x=\beta$ in some cyclic group $G=<\alpha>$. Then we are looking for a (uniformly) random sequence of elements of the form ...
3
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3answers
185 views

SRP-6 vulnerabilities when N is small

I'm one of the developers of an application which uses SRP-6 as the authentication mechanism. The authentication part of the code is very old and uses N with only 256 bits (all arithmetic is done in ...
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540 views

How compared encryption algorithm in terms of efficiency

I want to compare two cryptographic algorithms. The first algorithm is RSA, and second algorithm is ElGamal elliptic curve cryptography. Now, I’m looking for a way to compare the speed of the two ...
2
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1answer
256 views

Adding points on Elliptic Curves

How do we add the integer points $P=(-1, 4)$ and $Q=(2, 5)$ on the elliptic curve of the form $y^2=x^3+17$ ?
3
votes
1answer
228 views

Security equivalent to Diffie–Hellman problem?

I've been doing the security proof for one of my Theorem. Basically, given $g^a$, $g^b$, $g^{cb}$, $g$ and $c$ as known values. Is the problem of computing $g^{acb^{-1}}$ equivalent to the Diffie ...
2
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2answers
208 views

Given $g$, $b$, $g^{ab}$, is finding $g^a$ a hard problem?

As in the title, given $g$, $g^{ab}$ are big elements in a prime group $Z_p$ and $b$ in prime group $Z_r$ ($p > r$, $g$ is one generator of $Z_p$). $a$ is unknown and also in $Z_r$, is finding ...