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.

learn more… | top users | synonyms

0
votes
1answer
37 views

What is base of $log$ in Subexponential Algo for DLP?

I am currently going through this paper - "A Subexponential Algorithm for the Discrete Logarithm Problem" by Leonard Adleman. On page 56, author mentions that Dixon's algorithm - Asymptotically Fast ...
2
votes
1answer
101 views

Given $g,g^t$ in a cyclic group of order $pq$, is it hard to compute $g^{t^{-1}}$?

Suppose we have a group $G$ cyclic of order $pq$ , where $p,q$ are primes. Let $g$ be a generator of $G$ and $t\in \mathbb{Z}_{pq}$. Having $g$ and $g^t$, it seems to be very hard to find ...
0
votes
2answers
67 views

Discrete logarithm hash function Exercises

I have a problem with this exercise: Let $G$ be a group of order a prime $q$ and let $g, h$, be two randomly selected elements of $G$, with $g,h\ne 1$. Consider the following hash function on ...
4
votes
2answers
98 views

The difficulty of computing discrete logs

I understand that in Diffie-Hellman it should be hard to compute $a$ given $g$ and $g^a$. In computational Diffie-Hellman, it appears to be hard to compute $(g^{ab})$ from $g^a$ and $g^b$. As for ...
2
votes
0answers
38 views

Recover secret $x$ when $c\equiv m^x \pmod p$ with public $p$ (modified)

Given an encryption system where $c\equiv m^x \pmod p$, $p$ is a known prime, 1. Is it possible to recover $x$ with a known plaintext attack? Given $(p,\text{factorization of }\varphi(p),m,c)$ 2. Is ...
2
votes
0answers
56 views

Convert messages to elliptic curve points [duplicate]

Let $E$ be an elliptic curve; $\alpha,\beta$ two points of $E$; and $a$ a private key such that $\beta=a\cdot\alpha$. We choose random integer $k$ and plain text $x\in E$. Encryption and decryption ...
1
vote
1answer
85 views

How is El Gamal different from Diffie Hellman Key Exchange [duplicate]

I am Reading RSA and Public-Key Cryptography by Mollin and I can't make out how El Gamal is different from Diffie Hellman Key Exchange. Any thoughts?
10
votes
4answers
1k views

RSA & DH at risk due to math advances, will this eventually affect elliptic curves too?

I was looking into the predictions by some researchers that RSA and Diffie-Hellman may not be secure in the next few years due to advances in math and being able to calculate the discrete logarithm ...
7
votes
3answers
173 views

Do I understand (below) why Q = dP is easy while finding d is hard

As we all know for discussion of Dual_EC_DBRG, the point on an elliptic curve Q can be calculated from P and some (large) integer d $Q = dP$ And we know that knowledge of Q and P is not sufficient ...
4
votes
1answer
76 views

Diffie-Hellman: *assumed* largest prime factor of order of $g$?

I've read in several places: some of my confusions about DDH assumption - comment by Thomas Pornin on his own post Is there a key length definition for DH or DHE? - the answer by Tom Leek that ...
2
votes
1answer
62 views

Elliptic Curve ElGamal and DSA - smooth group order and element of large prime order

In regular ElGamal and DSA, we choose large primes $p$ and $q$ such that $p\equiv 1\pmod{q}$, and a group element $g$ of order $q$ by computing $a^{(p-1)/q}$ for some random $a$. This is to prevent ...
2
votes
1answer
35 views

Generalization of the DL-assumption in bilinear group pair

When thinking about a pairing-based cryptographic scheme, I encountered the following problem. Let $e \colon G_1, G_2 \to G_T$ be a Type 3 pairing. Then: Given $P, zP \in G_1$ and $Q, zQ \in G_2$, ...
3
votes
0answers
39 views

Cost of attack on DSA with attack on DLP

Are there any (recent) estimates of cost of attack on DSA by solving the discrete logarithm? I'm especially interested in attacks that use Pollard's rho algorithm. Are there any optimized ...
1
vote
1answer
37 views

Bilinear map assumpion

Is there an assumption that says from a tag $k\cdot e(g,g_1)^{rx}$ ($k,r$ are secret) it is difficult to forge it with some x': $k\cdot e(g,g_1)^{rx'}$, as long as you cannot solve DL in ...
1
vote
0answers
36 views

How to find the integer multiplicand from given two points?

I have two points on an elliptic curve $P(x_1,y_1)$ and $Q(x_2,y_2)$ and a scalar value $x$, where $P=x \cdot Q$. What is the best way that I could figure out the value of $x$? Given that I know all ...
3
votes
1answer
38 views

How many bits of an exponent are leaked when doing a powmod?

How many bits of the exponent $x$ are leaked when you calculate and reveal $g^x \pmod p$ for some generator $g$ of $\mathbb Z^∗_p$? The low bit of $x$ is obviously leaked: the low bit equals ...
0
votes
1answer
77 views

How to prove knowledge of discrete logarithm in a product?

Definitions Suppose I have two large safe primes $p$ and $q$, and a composite number $N=pq$. I have $G$, a large cyclic subgroup of $\mathbb{Z}^{*}_{N}$; $g$ and $h$ are generators of $G$. I commit ...
3
votes
2answers
63 views

Discrete Logarithm problem with inverse

Let $\mathbb G$ be a cyclic group of order $q$. The Discrete Logarithm Problem (DLP) is, given $g, g^x \in \mathbb G$, to compute $x \in \mathbb Z_q $. I'm interested to know if there is a known ...
4
votes
1answer
50 views

State level “Weak Diffie-Hellman” working for SRP too?

I've read about the "Weak Diffie-Hellman" attack (paper, website), where a resourceful entity like a state can pre-compute values for known primes to aid solving the discrete logarithm problem for ...
1
vote
2answers
591 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 ...
8
votes
3answers
1k views

Perfect zero knowledge for the Schnorr protocol?

Can somebody explain (or point to a reference) why the Schnorr protocol cannot be proved zero knowledge?
2
votes
0answers
39 views

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 ...
2
votes
0answers
42 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 ...
19
votes
0answers
628 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 ...
0
votes
1answer
45 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 $g^{\mathsf{sk}x}, x \in \mathbb{Z}_p^*$. ...
4
votes
1answer
217 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 ...
0
votes
1answer
51 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 ...
1
vote
1answer
51 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 ...
3
votes
0answers
35 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 ...
4
votes
2answers
190 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 $a$?
1
vote
1answer
263 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 ...
1
vote
1answer
465 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: ...
1
vote
1answer
355 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 ...
3
votes
1answer
887 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 ...
1
vote
1answer
96 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 ...
0
votes
0answers
58 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 ...
0
votes
0answers
47 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 $\mathbb{F}_p$ ...
10
votes
1answer
212 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), ...
4
votes
2answers
104 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 ...
0
votes
2answers
177 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 ...
3
votes
1answer
124 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 ...
4
votes
0answers
74 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.: ...
2
votes
2answers
174 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
votes
1answer
53 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 ...
1
vote
2answers
126 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. ...
1
vote
1answer
50 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 $(a_1,\dots,a_n)\in \mathbb{Z}^n$ ...
1
vote
0answers
29 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 ...
0
votes
2answers
82 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 ...
1
vote
0answers
151 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 ...