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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1answer
130 views

Why is the NIST recommendation for the private key exponent so much smaller than the modulus?

NIST recommends a 256-bit private key exponent for DLP with a 3072-bit modulus. This question answered how the modulus was chosen/calculated, however, why isn't the private key size closer to the ...
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Verify the discrete log of ECDSA is in range

Is it possible to verify the discrete log in elliptic curve is within range without uncovering it? I need to verify that $x$ is within $1$, $2^{64}$ for $xG=P$.
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How can I prove that the plaintext of an elgamal ciphertext is the discrete log of an element?

Is there any (efficient) method to prove that the plaintext of an ElGamal ciphertext is the discrete log of an element? In the scenario I concerned, I have an El Gamal key pair $(pk, sk) = (g^y, y)$....
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Collision resistance from discrete log

Let $G$ be a cyclic group of prime order $q$ generated by $g \in G$. $H$ is a hash function defined over as $H(\alpha_1,\alpha_2,...,\alpha_n)=g_1^{\alpha_1}g_2^{\alpha_2}...g_n^{\alpha_n}$ where $(\...
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How Significant is the New Quasi-Polynomial-Time Attack on Fixed Characteristic Discrete Logarithms?

There is a new paper by Kleinjung and Wesolowski on eprint that claims and proves a new attack on the discrete logarithm problem in finite fixed characteristic fields in quasi-polynomial time. ...
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1answer
77 views

How is the recommended NIST modulus for DLP chosen/calculated?

NIST recommends a 256-bit private key exponent for DLP with a 3072-bit modulus. From this answer it appears that the range of private key numbers is derived by calculating a prime modulus via $2⋅p$ ...
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1answer
48 views

Pollard's Kangaroo— What is the failure probability (assuming random functions)?

I'm reading Pollard's paper on solving the discrete log problem, i.e. to find $x$ given $y = g^x$, where $g$ is a generator of the group. He has a Kangaroo Algorithm (page 4) which allows you, if you ...
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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
68 views

Bit-strength of discrete logarithm for a group of integers modulo a safe prime

Preliminaries Let $p$ be a safe prime number. Let $\mathbb{Z}_p^*$ be the multiplicative group of integers modulo $p$. We have $\mathbb{Z}_p = \{\,a \in \mathbb{Z} \mid 1 \le a \lt p\,\}$ . Let $g \...
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175 views

What information does $g^x$ reveal about $x$?

Let $p$ be a large prime number. Let $G$ be a subgroup of $\mathbb{Z}_p^*$ with order $q$ - again a large prime. Let $g$ be a generator of $G$. Consider the following standard protocol for ...
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Why Smart's attack doesn't work on this ECDLP?

The Problem is as follows: ...
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Why do they use elliptic curve instead of circle or other simpler curves? [duplicate]

I am curious why people use elliptic curve in cryptography. I know the main requirement is DLP, but elliptic curve is not the only curve with such property. Some of curves seem to be even simpler. As ...
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What makes the Discrete Logarithm Problem hard?

I am missing a crucial piece of the maths behind the DLP, and I'm hoping someone can give me a really dumbed down answer.. If $h=g^x \bmod p$ and we're working in the group $Z^*_p$, why can I not ...
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How to determine if $\{n \cdot g^a \mod P\}$ and $\{m \cdot g^a \mod P\}$ generate the same sets? (set size < $P-1$)

given some examples $k_{n_i},k_{m_i}$ out of each value set: $k_{n_i} \in \{n \cdot g^a \mod P\, \forall a \in \mathbb{N}<P\}=G_m$ $k_{m_i} \in \{m \cdot g^a \mod P, \forall a \in \mathbb{N}<P\}...
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Is there a fast way to solve $k = n \cdot g^a \mod P$? (get $a$ for unknown $n$)

Would a factor besides the normal discrete logarithm problem increase or decrease the solving time? $k = n \cdot g^a \mod P$ with given $k,g,P$ and the knowledge $P= 2 \cdot N \cdot f+1$, while $...
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Is there any property of the product you can predict before using $n$-times generator $g$ $\mod P$? Can any $n$'th element have a certain property?

Given a value $v$ which is in same group as the generator $g$ modulo prime $P$. The group size is a prime $s$. $v = g^a \mod P$ Only known values are $v,g,P,s$. Some (possible) computation of other ...
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Which is the fastest way to find a member of a subgroup with known size modulo prime $P$, with know factorization of $P-1$?->$x$ with $x^s \mod P = 1$

Assuming you know the factorization of used prime $P-1$ $P-1 = s \cdot f_2\cdot f_3...f_i$ Now you want to find a member of a subgroup $\mathbb{Z}_s$. This means any $x$ with $x^s \equiv 1 \mod P $ ...
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How safe is a prime with $P=2 \cdot Q \cdot R \cdot S \cdot t+1$ for discrete logarithm? How to enhance/compare?

To get some certain properties for my use case I need a prime $P$ which has the form: $P=2\cdot Q \cdot R \cdot S \cdot t+1$ with $Q,R,S,t$ primes as well. Why that form - Use case Together with ...
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why a group used in cipher based on DLP must be Abelian group?

I can't understand it because $(g^x)^y=(g^y)^x$ in nonabelian group too. thank you very much for read my question
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Is there a way to determine the number of subgroups (with size $s$) while computing $A^b \bmod P$? Constructing a $P$ with $n$ times size $s$?

If you compute $A^b \bmod P$ for all $b$ the set of results $R$ depend at $A$ (and $P$). $R = \{A^b \bmod P, \forall \space b \in \mathbb{N}\}$ In case $R$ contain all numbers from 1 to $P-1$, it ...
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Could Diffie-Hellman protocol serve as a zero-knowledge proof of knowledge of discrete logarithm?

The Schnorr identification protocol is widely recognized as the simplest ZKPoK of the discrete logarithm (more clearly, Honest-Verifier ZKPoK). However, I can't figure out why this simple protocol, ...
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Does a different exponent and base but same key help to resolve discrete logarithm?

E.g.: $k = N^a \mod P$ The attacker knows the prime $P$ and $N$, which is also a prime and (1.) prime root of $P$ or (2.) has a cycle size of $s$, so $1 = N^s \mod P$, (and for $\forall s'<s$, $...
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How to compute a discrete logarithm modulo a power of 2?

This question is related to this one. Specifically, assume that we have $p$ = 2048, $m$ = 13 and $c$ = 357. In this case, $c\ =m^e\ \bmod \ p$. I know that many algorithms rely on the difficulty of ...
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1answer
70 views

Discrete logarithm with exponentiation oracle

Suppose $(n, d)$ is an RSA private keypair. We know the public key only (wlog suppose it's $(n, 2^{16}+1)$ and we are given oracle $E$, decryption oracle for $(n, d)$. Is there any efficient algorithm ...
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Does classifying an integer as a discrete log require it be part of a multiplicative group?

This question is not a question about the discrete log problem, the generalized discrete log problem, or an additive group. The confusion is whether any integer can be considered a discrete log or ...
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Why was the term “discrete” used in discrete logarithm?

Is there anything especially "discrete" about a discrete logarithm? This is not a question of what is a discrete logarithm or why the discrete logarithm problem is an "intractable problem" given ...
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2answers
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Compute discrete logarithm mod $n=p \times q$ knowing factorisation

I read in a document that for a given $n = p\times q$ ($p$, $q$ primes), if you know $p$ and $q$ then you can easily solve the discrete logarithm problem, i.e. for fixed $a,b$, you can find $x$ such ...
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Selection of parameters for Massey-Omura Cryptosystem

I have 4 questions about Massey-Omura Cryptosystem. Are there standards that define these parameters? How to choose a group order? What is better to take the function f? What is the recommended key ...
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1answer
70 views

Understanding Baby-Step Giant-Step Algorithm and discrete logarithm

Studying the Baby-Step/Giant-Step Algorithm, I have some questions: In the algorithm, $p$ is the order of group, $x$ is solution. We rewrite $x = i * m + k $, but why do we make $m =\lfloor\sqrt{p}\...
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Breaking the discrete logarithm problem in subgroups of $G$

I need to find the discrete logarithm of 20 modulo 71 where the generator of the group is 7. I need to break the group $|G|=2 \times 5 \times 7$ in subgroups $|G_1|=2, |G_2|=5, |G_3|=7$. I am new to ...
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2answers
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Difference between when select x from $\mathbb{Z}_{p-1}$ and $\mathbb{Z}_p$ in discrete logarithm Problem?

Reading "Security Arguments for Digital Signatures and Blind Signatures" paper, I confused by some questions. Q1. when it refers to "El Gamal signature scheme", The key generation algorithm: it ...
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1answer
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find $a$ and $k$ for a given el gamal cryptosystem

I am given this question: Suppose Alice is using the ElGamal Signature scheme with parameters $p = 31847$, $\alpha = 5$, and $\beta = 25703$ Assuming that we have received signed messages $(x_1,(\...
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Prove that two commitments are commitments to the same value

Let $x$ be the secret value, $(n,a,b,c)$ a public key, $(n_C, g, h)$ the commitment public key. Furthermore let $r, r_C$ be two random numbers. Define $C = g^x h^{r_C} \bmod n_C$, $C_x = a^x b^r \...
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How should one refer to the smallest remainders generated by a modulus within DH or DLP?

It's my understanding that the integer base and exponents chosen to create the initial public keys in DH are from the remainders of a modulus. For example, if the value of the modulus is $N=11$, a ...
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1answer
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Digital Signature Algorithm (DSA) with medium fields

I understand it when we have to solve Discrete Logarithm Problem $a^x\equiv b\pmod p$, where $a$ and $b$ are given integers and we have to find secret integer $x$ that makes the equation true for some ...
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Can Shamir’s Trick crack the cryptographic strength of ECDSA?

Recently stumbled upon a discussion in the forum What is Shamir’s Trick used for? Are there any such examples?
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In NIST modulus and key size recommendations is `group` equivalent to modulus and `key` equivalent to exponent?

Looking at the following NIST recommendations for a discrete logarithm, for 2016-2030 and beyond they list 3072-bit number for the group and 256-bit for the ...
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1answer
253 views

Diffie-Hellman private key recover with non-prime modulus

Say we have a classic Diffie-Hellman key exchange. We have the following parameters of a public key: p,g,y Where $p$ is the modulus, $g$ is the base, $y$ is the ...
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Elliptical Curve Actual Encryption

Im havirng a had time understanding ECC. For example, I have the equation below: ...
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Distributed key generation (for discrete-log based cryptosystems) with fake shares

Under the definition of Gennaro et al (link), a DKG protocol needs to satisfy “correctness” and “secrecy”. Correctness is divided into three sub-properties: C1. All subsets of $t+1$ shares provided ...
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Using division polynomials to prove that EC discrete log is even

This question is related to the other question I recently asked. I'm trying to figure out if it is possible to use division polynomials to prove that knowing $A = a \cdot G$ we can prove that $a$ is ...
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1answer
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How does Diffie Hellman protocol work in Bitcoin Blockchain Transactions?

Greetings to all! Please explain how the Diffie-Hellman protocol works in Bitcoin? That is, in Blockchain Transactions, there is also a total number of "K" recipient and sender? "K" the recipient and ...
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Are compressed pairings used for Barreto-Naehrig curves in practice?

In 2009 Galbraith and Lin wrote the article "Computing Pairings Using x-Coordinates Only" https://link.springer.com/article/10.1007/s10623-008-9233-3, where they proposed to compute pairings on ...
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2answers
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Random Self-reducibility in Discrete Log

I understand what Random Self-reducibility means and how it is used in the Discrete Log. What is not clear is how does it show that DL is hard in the average case. The probability of success of an ...
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Discrete Logarithm within DH exchange - finding out A and B secret number [duplicate]

How would i find out Alices or Bobs private number from the details exchanged in public. prime - g = 7 primitive root - n = 3 Alice - A = 3 Bob - B = 4 The details visible in public exchange are: ...
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1answer
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Continuous logarithms in cryptography

Cryptographic procedures seem to almost exclusively use discrete logarithms rather than continuous logarithms. Hence, I assume there are good and sound reasons for this. In essence, answers provided ...
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1answer
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Why is Diffie-Hellman Insecure If Order of the Generator Has Only Small Prime Factors?

In this post from security SE, Tom Leek mentioned that, for Diffie-Hellman to be secure order of the group $g$ should have a prime factor at least $2k$ bits long, where $k$ is the security parameter. ...
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2answers
208 views

What prevents the successful use of the Giant-step/Baby-step algorithm solving a discrete log problem implemented with modulo arithmetic?

Does the size of base, exponent, and modulus thwart the Giant-step/Baby-step algorithm in solving DLP using modular arithmetic or is it the use of a property of a particular prime as the modulus, or ...
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54 views

Proving key equivalence across different elliptic curves

We can use the technique described in this answer to prove key equivalence across two elliptic curves of different order. I'm wondering if modifying the technique as described below would compromise ...
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1answer
57 views

Collisions in Diffie-Hellman private keys

Given a generator $g$, a large, safe prime $P$ and a result of the DH key exchange $g^{xy} \mod P$, how would I come up with two different $x', y'$ s.t. $g^{x'y'} = g^{xy} \mod P$