An Improved Integer Modular Multiplicative Inverse (modulo $2^w$)

TL;DR

This paper introduces a highly efficient algorithm for computing the integer modular inverse modulo 2^w, optimized for modern microprocessors and closely related to Newton's method, with proven formulas for small exponents.

Contribution

It presents a novel, optimized algorithm for integer modular inverse modulo 2^w, improving efficiency and generality over previous methods, with formal proofs for key formulas.

Findings

Algorithm completes in the fewest cycles on modern microprocessors.

Provides formal proofs for inverse formulas with small exponents.

Shows close relation to Newton's method for inverse calculation.

Abstract

This paper presents an algorithm for the integer multiplicative inverse (mod $2^w$) which completes in the fewest cycles known for modern microprocessors, when using the native bit width $w$ for the modulus $2^w$. The algorithm is a modification of a method by Dumas, and for computers it slightly increases generality and efficiency. A proof is given, and the algorithm is shown to be closely related to the better known Newton's method algorithm for the inverse. Simple direct formulas, which are needed by this algorithm and by Newton's method, are reviewed and proven for the integer inverse modulo $2^k$ with $k$ = 1, 2, 3, 4, or 5, providing the first proof of the preferred formula with $k$=4 or 5.