Montgomery Reduction for Gaussian Integers

Abstract

Modular arithmetic over integers is required for many cryptography systems. Montgomery reduction is an efficient algorithm for the modulo reduction after a multiplication. Typically, Montgomery reduction is used for rings of ordinary integers. In contrast, we investigate the modular reduction over rings of Gaussian integers. Gaussian integers are complex numbers where the real and imaginary parts are integers. Rings over Gaussian integers are isomorphic to ordinary integer rings. In this work, we show that Montgomery reduction can be applied to Gaussian integer rings. Two algorithms for the precision reduction are presented. We demonstrate that the proposed Montgomery reduction enables an efficient Gaussian integer arithmetic that is suitable for elliptic curve cryptography. In particular, we consider the elliptic curve point multiplication according to the randomized initial point method which is protected against side-channel attacks. The implementation of this protected point multiplication is significantly faster than comparable algorithms over ordinary prime fields.

A New Class of Q-Ary Codes for the McEliece Cryptosystem

“…The syndrome decoding of OMEC codes can be implemented efficiently using the Montgomery arithmetic for Gaussian integers proposed in [27] for the syndrome calculation. The error correction can be implemented using two-dimensional look-up tables for the error positions and error values, where the real and imaginary parts of the syndrome are the arrays' indices.…”