Efficient Lifting of Discrete Logarithms Modulo Prime Powers

2025-05-12

AI Summary

This work addresses the efficient lifting of discrete logarithm solutions from modulo a prime $p$ to modulo a prime power $p^k$, where $k geq 1$ is fixed. We propose the first deterministic Hensel-lifting algorithm for solving $a^x equiv b pmod{p^k}$, given a solution modulo $p$. Our method combines refined iterative lifting, optimized modular exponentiation, and precise integer arithmetic analysis. The algorithm achieves worst-case multiplicative complexity $k(lceil log_2 p ceil + 2) + O(log p)$, improving upon prior approaches by at least a factor of eight. Crucially, it attains asymptotically optimal linear dependence on $k$—the first such result—thereby significantly enhancing the efficiency of discrete logarithm computation modulo prime powers.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $p mid a$, 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 $k(lceil log_2 p ceil +2)+O(log p)$ multiplications modulo $p^k$ in the worst case, improving upon prior lifting methods by at least a factor of 8.

Problem

• Efficiently lift discrete logarithms modulo prime powers
• Solve discrete logarithm problem for any fixed exponent
• Improve prior lifting methods by factor of 8

Innovation

• Deterministic algorithm for discrete logarithms
• Efficient lifting to higher prime powers
• Reduces multiplications by factor of 8