Low-weight polynomial form integers for efficient modular multiplication

In 1999, Jerome 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 only integer additions and subtractions are required in modular reductions. 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 ±1. 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