The selection of the elements of the bases in an RNS modular multiplication method is crucial and has a great impact in the overall performance.This work proposes specific sets of optimal RNS moduli with elements of Hamming weight three whose inverses used in the MRS reconstruction have very small Hamming weight. This property is exploited in RNS bases conversions, to completely remove and replace the products by few additions/subtractions and shifts, reducing the time complexity of modular multiplication.These bases are specially crafted to computation with operands of sizes 256 or more and are suitable for cryptographic applications such as the ECC protocols. Disciplines Physical Sciences and Mathematics

Abstract. In SAC 2003, J. Chung and A. Hasan introduced a new class of specific moduli for cryptography, called the more generalized Mersenne numbers, in reference to J. Solinas' generalized Mersenne numbers proposed in 1999. This paper pursues the quest. The main idea is a new representation, called Modular Number System (MNS), which allows efficient implementation of the modular arithmetic operations required in cryptography. We propose a modular multiplication which only requires n 2 multiplications and 3(2n 2 − n + 1) additions, where n is the size (in words) of the operands. Our solution is thus more efficient than Montgomery for a very large class of numbers that do not belong to the large Mersenne family.

We propose a new number representation, and the corresponding arithmetic, for the elements of the ring of integers modulo p. The so-called Polynomial Modular Number System (PMNS) allows for fast polynomial arithmetic and easy parallelization. The most important contribution of this paper is certainly the fundamental theorem of a Modular Number System which gives up an upper-bound on the coefficients of the polynomials used to represent the set of non-negative integers {0, . . . , p − 1}. However, we also propose a complete set of algorithms for the arithmetic operations over a PMNS, which make this system of practical interest for people concerned about fast implementation of modular arithmetic.