Counting Solutions to Discrete Non-Algebraic Equations Modulo Prime Powers

TL;DR

This paper investigates counting solutions to generalized discrete logarithm problems modulo prime powers by employing p-adic interpolation, Hensel's lemma, and the Chinese Remainder Theorem, which could impact cryptographic security analysis.

Contribution

It introduces a novel approach to count solutions of discrete logarithm problems modulo prime powers using p-adic methods and lifting techniques.

Findings

Provides a method to count solutions using p-adic interpolation.

Utilizes Hensel's lemma and Chinese Remainder Theorem for solution lifting.

Enhances understanding of the structure of solutions in cryptographic contexts.

Abstract

As society becomes more reliant on computers, cryptographic security becomes increasingly important. Current encryption schemes include the ElGamal signature scheme, which depends on the complexity of the discrete logarithm problem. It is thought that the functions that such schemes use have inverses that are computationally intractable. In relation to this, we are interested in counting the solutions to a generalization of the discrete logarithm problem modulo a prime power. This is achieved by interpolating to p-adic functions, and using Hensel's lemma, or other methods in the case of singular lifting, and the Chinese Remainder Theorem.