Efficient Lifting of Discrete Logarithms Modulo Prime Powers

TL;DR

This paper introduces a deterministic algorithm that efficiently lifts solutions of discrete logarithms from modulo a prime to higher powers, significantly improving computational efficiency over previous methods.

Contribution

The paper presents a new deterministic lifting algorithm for discrete logarithms modulo prime powers, reducing the number of multiplications needed compared to prior techniques.

Findings

Performs $k(oxed{ ext{log}_2 p}+2)+O( ext{log} p)$ multiplications in worst case

Achieves at least an 8-fold improvement over previous methods

Works efficiently for any fixed $k \\geq 1$

Abstract

We present a deterministic algorithm that, given a prime $p$ and a solution $x \in \mathbb Z$ to the discrete logarithm problem $a^x \equiv b \pmod p$ with $p\nmid a$, efficiently lifts it to a solution modulo $p^k$, i.e., $a^x \equiv b \pmod {p^k}$, for any fixed $k \geq 1$. The algorithm performs $k(\lceil \log_2 p\rceil +2)+O(\log p)$ multiplications modulo $p^k$ in the worst case, improving upon prior lifting methods by at least a factor of 8.