The Discrete Lambert Map

TL;DR

This paper analyzes the solutions of the discrete Lambert map modulo prime powers, using p-adic methods to count solutions and uncover patterns relevant for cryptographic security.

Contribution

It introduces p-adic techniques to count solutions of the discrete Lambert map and reveals new patterns in these solutions, advancing understanding in cryptographic contexts.

Findings

Count of solutions for xg^x ≡ c mod p^k determined

Patterns in solutions identified

Implications for ElGamal digital signature security

Abstract

The goal of this paper is to analyze the discrete Lambert map x to xg^x modulo a power of a prime p which is important for security and verification of the ElGamal digital signature scheme. We use p-adic methods (p-adic interpolation and Hensel's Lemma) to count the number of solutions x of xg^x congruent to c modulo powers of an odd prime p and c, g are fixed integers. At the same time, we discover special patterns in the solutions.