Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [discrete-logarithms]

For questions related to the discrete logarithm problem; modulo $p$, in finite fields, over elliptic curves, or in an abstract group.

For a cyclic group $G$ with generator $g$, a discrete logarithm of $b \in G$ to the base $g$ is an integer $k$ such that $g^k = b$.

Finding a discrete logarithm is (believed to be) a computationally hard problem, and various cryptographic protocols are built around it.

The underlying cyclic group is often the multiplicative group of $\mathbb{Z}/p\mathbb{Z}$ for some prime $p$, more generally a multiplicative subgroup of a finite field, or a cyclic group of points on an elliptic curve.

The intended usage of this tag is for questions on or related to the discrete logarithm problem (theoretic as well as algorithmic) and its applications.

Related tags include:

155 questions
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Geometric interpretation of the Logarithm (in $\mathbb{R}$)

(Note: limited to $\mathbb{R}$) (Note: Geometric here means with straightedge and compass) Standard approaches to introducing the concept of Logarithm rely on a previous exposition of the exponential or simply on that of a power. It then receives…
MASL
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Is it possible to find a closed-form expression for $f(n)$?

QUESTİON UPDATED: Here is my problem: $$2^x \equiv a \pmod{3^n}.$$ where, $a\not\equiv 0 \pmod{3}$ and $n\in \mathbb{Z^{+}}$ I want to learn that, If, $x=\left\{ {{3^n-\binom{n}{2}}-1}\right\}-f(n)$ $a=\sum_{j=0}^{n-1} 3^{n-j-1} 2^{3^j -…
lone student
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7
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2 answers

Can irreversibility of trapdoor functions generally not be proved?

The German Wikipedia article on asymmetric cryptography states that asymmetric cryptography is always based on assumptions which can not be proven: Die Sicherheit aller asymmetrischen Kryptosysteme beruht also immer auf unbewiesenen…
radix
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Frey-Rück Attack - Tate-Lichtenbaum Pairing

I am trying to understand the Frey-Rück attack and found different ways of a possible implementation. Since I am not yet very familiar with the Tate-Lichtenbaum pairing and the theory of divisors I wanted to ask which one of the different…
Luca
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5
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Groups where discrete logarithm is hard

What are examples of groups, where DLP (discrete logarithm problem) is hard? Two obvious ones are: integers modulo $p$ ($p$ being prime) and elliptic curves over finite fields. What are the others?
ted.k
  • 181
5
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Can you explain this relation between finite fields and circles?

Let $p$ be a prime such that $p \bmod 4 = 1$, so there exists some $i=\sqrt{-1}$ in $\mathbb{F}_p$. Furthermore, let $r \in \mathbb{N}$ be the radius of a circle such that there are $p-1$ lattice points on it. (The sums of squares function allows to…
Robin
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1 answer

Discrete logarithm - strange polynomials

If $p$ is a prime number and $\omega$ is a fixed primitve root for $\mathbb{Z}/p\mathbb{Z}$, then we can define the discrete logarithm of $x \in (\mathbb{Z}/p\mathbb{Z})^{\times}$ as the unique number $\log_{\omega} x$ between $1$ and $p-1$ such…
user71811
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2 answers

How can I solve this problem : $2^{x} \equiv{2070442609 \cdots 226509} \pmod {6561}$

I want to solve this discrete logarithm problem with Pohlig–Hellman algorithm: $$2^{x} \equiv{…
user548054
4
votes
1 answer

Cubic root of a polynomial to modulo of another polynomial

Is there any algorithm to solve problems like the problem below, any ideas? Find polynomial $f$ such that: $f^3 \equiv x^4 + x^2 \ (mod\ x^{10} + x^3 + 1)$ All numeric coefficients are from $\mathbf{Z_2}$.
Blendamed
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3
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1 answer

Solving DLP using the method of Pohlig-Hellman

I want to solve the DLP for $p=29$, $a=2$ and $b=5$ using the method of Pohlig-Hellman. $$$$ I have done the following: We have that $p-1=28=2^2\cdot 7$. We get \begin{align*}&x_2= x\pmod {2^2} \\ &x_7=x\pmod 7\end{align*} $x_2$ is a number…
Mary Star
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3
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2 answers

Discrete Log solve using Index-Calculus producing incorrect 'r' value.

I have a discrete log that I need to solve to aid in a Cryptography problem, that deals with both programming and mathematics, so I was unsure where to post this problem, feel free to move me if needed. I must use Index-Calculus to solve this…
justanotheruser
  • 97
3
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0 answers

Is Elliptic Curve Discrete Logarithm Problem NP-Hard or NP-Complete

I have trouble classifying Elliptic Curve Discrete Logarithm Problem as NP-Hard or NP-Complete. Where does ECDLP belong? Any brief comprehensive answer is encouraged. Thanks.
alyssaeliyah
  • 223
3
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2 answers

Do there exist any integer solutions for $y=\log_2(1+3^x)$?

I was working on this problem and came to a standstill. I'm not exactly sure how to go about this problem, to find if any integer pairs of $(x,y)$ satisfy this equation. Any guidance would be appreciated! $$ y=\log_2(1+3^x) $$
wjmccann
  • 3,383
3
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1 answer

Pohlig-Hellman Algorithm for solving a DLP - can $x_0$ have > 1 solution?

I have taken generator $g=3$, $h=5$, and prime $p=101$ So; $$3^x=5 \mod 101$$ Following steps of Polig-Helman I get; $p-1=100=5^{2}.2^{2}$ Hence largest prime power divisor $q^e=5^2$. Calculate $g^{\frac{p-1}{q^e}}=81 \mod 101$ Calculate…
unseen_rider
  • 812
3
votes
1 answer

Index Calculus algorithm wrong solution

We want to be able to compute discrete logarithms with basis $a = 89$ in $\mathbb{Z}^*_p$ for $p = 1235789.$ We choose the factor base $B = {−1, 2, 3, 5, 7, 11, 13, 17, 19, 23}$. With the first step of Index Calculus algorithm I get this system of…
Tilen Kavčič
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