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ABSTRACT
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.
INDEX TERMS
Cryptography, Mersenne numbers, modular multiplication, RSA, elliptic curve cryptosystems, the Montgomery reduction, the Barrett reduction.
CITATION

J. Chung and M. A. Hasan, "Low-Weight Polynomial Form Integers for Efficient Modular Multiplication," in IEEE Transactions on Computers, vol. 56, no. , pp. 44-57, 2007.
doi:10.1109/TC.2007.13
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