Montgomery Arithmetic

Introduction

In 1985, P.L. Montgomery introduced an efficient algorithm [6] for computing $u=a \cdot b\; ({\rm mod} n)$ where a, b, and n are k-bit binary numbers (see modular arithmetic). The algorithm is particularly suitable for implementation on general-purpose computers (signal processors or microprocessors) which are capable of performing fast arithmetic modulo a power of 2. The Montgomery reduction algorithm computes the resulting k-bit number u without performing a division by the modulus n. Via an ingenious representation of the residue class modulo n, this algorithm replaces division by n with division by a power of 2. The latter operation is easily accomplished on a computer since the numbers are represented in binary form. Assuming the modulus n is a k-bit number, i.e., 2^k-1 ≤ n < 2^k, let r be 2^k. The Montgomery reduction algorithm requires that r and n be relatively prime, i.e., $\gcd(r,n)=\gcd(2^k,n)=1$. This requirement is satisfied if nis odd. In the following,...