@TECHREPORT{Solinas99generalizedmersenne, author = {Jerome A. Solinas}, title = {Generalized Mersenne Numbers}, institution = {}, year = {1999} }

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. There is a well known shortcut for modular multiplication modulo a Mersenne number, performing modular reduction without integer division. We generalize this technique to a larger class of primes, and discuss parameter choices which are particularly well suited for machine implementation. Keywords: modular arithmetic, elliptic curves. Introduction It has long been known that certain integers are particularly well suited for modular reduction. The best known examples (e.g., [1]) are the Mersenne numbers m = 2 k \Gamma 1. In this case, the integers (mod m) are represented as k-bit integers. When performing modular multiplication, one carries out an integer multiplication followed by a modular reduction. One thus has the problem of reducing modulo m a 2k-bit number. Modular reduction is usually done by integer division, but this is unnecessary in the Mersenne case. Let n ! m 2 be the integer to be reduced (mod m). Let T be the integer represented by the k most significant bits o...

...ementation. Keywords: modular arithmetic, elliptic curves. Introduction It has long been known that certain integers are particularly well suited for modular reduction. The best known examples (e.g., =-=[1]-=-) are the Mersenne numbers m = 2 k \Gamma 1. In this case, the integers (mod m) are represented as k-bit integers. When performing modular multiplication, one carries out an integer multiplication fol...

...g from such k are never primes (see Proposition 13). It is therefore of interest to generalize the above technique to families of numbers containing primes. One such family is due to Richard Crandall =-=[2]-=-, namely, the integers 2 k \Gamma c for c positive and small enough to fit into one word. In this paper, we generalize in a different direction. Although there is some overlap, many of the generalized...