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.
We show that multiplication complexities of n-term Karatsuba-Like formulae of GF (2)[x] (7 < n < 19) presented in the above paper can be further improved using the Chinese Remainder Theorem and the construction multiplication modulo (x − ∞) w. Index Terms Karatsuba algorithm, polynomial multiplication, finite field. I. INTRODUCTION The Karatsuba-Ofman 2-term multiplication algorithm and its extensions, i.e., n-term Karatsubalike formula (n > 2), are often used to design subquadratic complexity GF (2 n) multiplication algorithms. In [1], for 1 < n < 19, Montgomery presents values of the multiplication complexity M (n), which is defined as the minimum number of multiplications needed to multiply two n-term polynomials a(x) = n−1 i=0 a i x i and b(x) = n−1 i=0 b i x i in GF (2)[x]. Applying the Chinese Remainder Theorem (CRT) for the design of polynomial multiplication algorithms is well known in the literature [2], [3], [4] and [5]. In this comment, we use the CRT and the construction multiplication modulo (x − ∞) w to improve values of M (n) (7 < n < 19) obtained in [1]. Unless otherwise stated, we assume that all polynomials considered here are in GF (2)[x]. The CRT for GF (2)[x] states that:
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