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In 1999, Solinas introduced families of moduli called the generalized Mersenne numbers (GMNs), which are expressed in low-weight polynomial form, p = f(t), where t is limited to a power of 2. GMNs are very useful in elliptic curve cryptosystems over prime fields since modular reduction by a GMN requires only integer additions and subtractions. However, since there are not many GMNs and each GMN requires a dedicated implementation, GMNs are hardly useful for other cryptosystems. Here, we modify GMN by removing restriction on the choice of t and restricting the coefficients of f(t) to 0 and \pm1. We call such families of moduli low-weight polynomial form integers (LWPFIs). We show an efficient modular multiplication method using LWPFI moduli. LWPFIs allow general implementation and there exist many LWPFI moduli. One may consider LWPFIs as a trade-off between general integers and GMNs.
Cryptography, Mersenne numbers, modular multiplication, RSA, elliptic curve cryptosystems, the Montgomery reduction, the Barrett reduction.
Jaewook Chung, M. Anwar Hasan, "Low-Weight Polynomial Form Integers for Efficient Modular Multiplication", IEEE Transactions on Computers, vol. 56, no. , pp. 44-57, January 2007, doi:10.1109/TC.2007.13
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