Secure Montgomery Multiplication and Repeated Squares for Modular Exponentiation

TL;DR

This paper develops efficient methods for secure modular multiplication and repeated squaring using residue number systems, optimizing ciphertext costs for cryptographic applications.

Contribution

It introduces novel techniques for secure arithmetic in residue number systems, extending existing methods to improve efficiency in modular exponentiation.

Findings

Reduced ciphertext costs for modular squaring in large moduli

Compared performance of new methods with existing techniques

Demonstrated practical efficiency improvements in cryptographic computations

Abstract

The BMR16 circuit garbling scheme introduces gadgets that allow for ciphertext-free modular addition, while the multiplication of private inputs modulo a prime p can be done with 2(p - 1) ciphertexts as described in Malkin, Pastro, and Shelat's An algebraic approach to garbling. By using a residue number system (RNS), we can construct a circuit to handle the squaring and multiplication of inputs modulo a large N via the methods described in Hollman and Gorissen's multi-layer residue number system. We expand on the existing techniques for arithmetic modulo p to develop methods to handle arithmetic in a positional, base-p number system. We evaluate the ciphertext cost of both of these methods and compare their performance for squaring in various large moduli.