Modular Reduction Operation Based on Monic Pentanomial

With the development of public key cryptography, especially elliptic curve cryptography, the research on the modular reduction with special moduli, such as Mersenne number, pseudo-Mersenne number, and generalized Mersenne number, has been rekindled. In this paper, we analysis and study modulo reduction operation based on monic polynomial, and obtain the following results: for Mersenne number and pseudo-Mersenne numbers, we obtain modular reduction formulas, and determine exact expressions of the number of modular addition to monic pentanomial. Using these formulas, one can easily compute the number of modular addition of A mod p for any given monic pentanomial