More Generalized Mersenne Numbers

Abstract

Abstract. In 1999, Jerome Solinas introduced families of moduli called the generalized Mersenne numbers [8]. The generalized Mersenne numbers are expressed in a polynomial form, p = f (t), where t is a power of 2. It is shown that such p's lead to fast modular reduction methods which use only a few integer additions and subtractions. We further generalize this idea by allowing any integer for t. We show that more generalized Mersenne numbers still lead to a significant improvement over well-known modular multiplication techniques. While each generalized Mersenne number requires a dedicated implementation, more generalized Mersenne numbers allow flexible implementations that work for more than one modulus. We also show that it is possible to perform long integer modular arithmetic without using multiple precision operations when t is chosen properly. Moreover, based on our results, we propose efficient arithmetic methods for XTR cryptosystem.

The double-base number system and its application to elliptic curve cryptography

“…they do not take into account the possible special form of the modulus. When implementing ECC/HECC algorithms, it is a good idea to use primes that allow fast modular arithmetic, such as those recommended by the NIST [39], the SEC Group [43], or more generally the primes belonging to what Bajard et al called the Mersenne family [42,11,5]. In these cases, the multiplication becomes much more efficient than the inversion.…”

“…Let P = (x 1 , y 1 ) ∈ E(K) be a point on an elliptic curve E defined by (11). By definition, we have [2]P = (x 2 , y 2 ), where…”

Modular Number Systems: Beyond the Mersenne Family

“…For a large part, the moduli (possibly primes) we are able to generate do not belong neither to Solinas' [8] or Chung and Hasan's generalized Mersenne family [2]. Thus, we only compare our algorithm with Montgomery since it was the best known algorithm available for those numbers.…”

“…In 2003, J. Chung and A. Hasan, in a paper entitled "more generalized Mersenne numbers" [2], extended J. Solinas' concept, by allowing any integer for t.…”

Efficient SIMD Arithmetic Modulo a Mersenne Number

“…These numbers, and a larger family of numbers called generalized Mersenne numbers [20], [21], [22], have found many arithmetic applications ranging from number theoretic transforms [23] to cryptography. In the latter they are used to run calculations concurrently using RNS [24] or to improve the speed of finite field arithmetic in ECC based schemes [20], [25].…”