Finding discrete logarithm in $F_p^* $

TL;DR

This paper introduces a practical alternative to the Pohlig-Hellman algorithm for computing discrete logarithms in finite fields, offering faster runtimes by eliminating the need for Chinese Remainder Theorem computations.

Contribution

The paper proposes a new method for discrete logarithm calculation that improves efficiency over Pohlig-Hellman by simplifying the combination process.

Findings

Faster computation times compared to Pohlig-Hellman.

No need for Chinese Remainder Theorem in the new method.

Effective even when p=2q+1, q prime.

Abstract

Difficulty of calculation of discrete logarithm for any arbitrary Field is the basis for security of several popular cryptographic solutions. Pohlig-Hellman method is a popular choice to calculate discrete logarithm in finite field $F_p^*$. Pohlig-Hellman method does yield good results if p is smooth ( i.e. p-1 has small prime factors). We propose a practical alternative to Pohlig-Hellman algorithm for finding discrete logarithm modulo prime. Although, proposed method, similar to Pohlig-Hellman reduces the problem to group of orders $p_i$ for each prime factor and hence in worst case scenario (including when p=2q+1 , q being another prime) order of run time remains the same. However in proposed method, as there is no requirement of combining the result using Chinese Remainder Theorem and do the other associated work ,run times are much faster.