Obtaining more Karatsuba-like formulae over the binary field

The aim of this paper is to find more Karatsuba-like formulae for a fixed set of moduli polynomials in GF (2)[x]. To this end, a theoretical framework is established. We first generalize the division algorithm, and then present a generalized definition of the remainder of integer division. Finally, a previously generalized Chinese remainder theorem is used to achieve our initial goal. As a by-product of the generalized remainder of integer division, we rediscover Montgomery’s N -residue and present a systematic interpretation of definitions of Montgomery’s multiplication and addition operations.