On Newton-Raphson iteration for multiplicative inverses modulo prime powers

TL;DR

This paper analyzes and improves algorithms for computing modular inverses modulo prime powers, combining explicit formulas with asymptotic methods to optimize performance across different exponent sizes.

Contribution

It introduces an explicit formula for modular inverses using Newton-Raphson iteration and proposes a hybrid approach that enhances efficiency for both small and large exponents.

Findings

Explicit formula is efficient for small exponents.

Hybrid method outperforms existing algorithms for large exponents.

Constant factor speedup achieved for exponents over 700 bits.

Abstract

We study algorithms for the fast computation of modular inverses. Newton-Raphson iteration over $p$-adic numbers gives a recurrence relation computing modular inverse modulo $p^m$, that is logarithmic in $m$. We solve the recurrence to obtain an explicit formula for the inverse. Then we study different implementation variants of this iteration and show that our explicit formula is interesting for small exponent values but slower or large exponent, say of more than $700$ bits. Overall we thus propose a hybrid combination of our explicit formula and the best asymptotic variants. This hybrid combination yields then a constant factor improvement, also for large exponents.