In this paper we propose an efficient hardware implementation of RSA based on the Residue Number System (RNS) which allows for fast parallel arithmetic. We propose RNS versions of Montgomery multiplication and exponentiation algorithms and illustrate the efficiency of our approach with two implementations of RSA. For the very first time a very attractive conversion-free RSA encryption/decryption scheme is proposed. Compared to previously proposed methods our solution requires less elementary operations and is very promising.

Abstract. In this paper, we propose a efficient and secure point multiplication algorithm, based on double-base chains. This is achieved by taking advantage of the sparseness and the ternary nature of the socalled double-base number system (DBNS). The speed-ups are the results of fewer point additions and improved formulae for point triplings and quadruplings in both even and odd characteristic. Our algorithms can be protected against simple and differential side-channel analysis by using side-channel atomicity and classical randomization techniques. Our numerical experiments show that our approach leads to speed-ups compared to windowing methods, even with window size equal to 4, and other SCA resistant algorithms.

Abstract. In this paper we show how the usage of Residue Number Systems (RNS) can easily be turned into a natural defense against many side-channel attacks (SCA). We introduce a Leak Resistant Arithmetic (LRA), and present its capacities to defeat timing, power (SPA, DPA) and electromagnetic (EMA) attacks.

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.

We investigate the impact of larger digit sets on the length of Double-Base Number system (DBNS) expansions. We present a new representation system called extended DBNS whose expansions can be extremely sparse. When compared with double-base chains, the average length of extended DBNS expansions of integers of size in the range 200-500 bits is approximately reduced by 20% using one precomputed point, 30% using two, and 38% using four. We also discuss a new approach to approximate an integer n by d2 a 3 b where d belongs to a given digit set. This method, which requires some precomputations as well, leads to realistic DBNS implementations. Finally, a left-to-right scalar multiplication relying on extended DBNS is given. On an elliptic curve where operations are performed in Jacobian coordinates, improvements of up to 13% overall can be expected with this approach when compared to window NAF methods using the same number of precomputed points. In this context, it is therefore the fastest method known to date to compute a scalar multiplication on a generic elliptic curve.

Abstract. We describe an algorithm for point multiplication on generic elliptic curves, based on a representation of the scalar as a sum of mixed powers of 2 and 3. The sparseness of this so-called double-base number system, combined with some efficient point tripling formulae, lead to efficient point multiplication algorithms for curves defined over both prime and binary fields. Side-channel resistance is provided thanks to side-channel atomicity.

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