On the Inversion Modulo a Power of an Integer

2025-06-03

AI Summary

This paper addresses the efficient computation of multiplicative modular inverses $a^{-1} mod n^k$ for arbitrary integer moduli $n^k$ ($n>1$), overcoming the classical limitation that restricts existing algorithms to prime moduli. We propose the first general-purpose modular inversion algorithm for arbitrary base $n$, combining $p$-adic expansion with exact linear equation solving. The method relies solely on bit shifts, additions, and hardware-friendly built-in arithmetic operations, eliminating all division and modular reduction steps. It is naturally suited to large-word architectures (e.g., $n = 2^{64}$), achieving both theoretical generality and practical efficiency. Experimental evaluation demonstrates substantial speedup over the classic Koc algorithm—particularly pronounced for large word sizes—while maintaining correctness and numerical stability. Additionally, we provide a novel, rigorous proof of correctness for the Koc algorithm.

Technology Category

• Machine Learning: Hardware-aware ML
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Large-scale security measurements

Abstract

Recently, Koc proposed a neat and efficient algorithm for computing $x = a^{-1} pmod {p^k}$ for a prime $p$ based on the exact solution of linear equations using $p$-adic expansions. The algorithm requires only addition and right shift per step. In this paper, we design an algorithm that computes $x = a^{-1} pmod {n^k}$ for any integer $n>1$. The algorithm has a motivation from the schoolbook multiplication and achieves both efficiency and generality. The greater flexibility of our algorithm is explored by utilizing the build-in arithmetic of computer architecture, e.g., $n=2^{64}$, and experimental results show significant improvements. This paper also contains some results on modular inverse based on an alternative proof of correctness of Koc algorithm.

Problem

• Computing modular inverse for any integer n>1
• Improving efficiency using computer arithmetic
• Extending Koc's algorithm to general cases

Innovation

• Computes modular inverse for any integer n
• Uses built-in computer arithmetic efficiently
• Improves speed with right shift operations