Paper 2009/636

Obtaining More Karatsuba-Like Formulae over The Binary Field

Haining Fan and Ming Gu and Jiaguang Sun and Kwok-Yan Lam

Abstract

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.

Note: I received an email from Dr. Ekatherina Karatsuba, A. Karatsuba' daughter, and she told me that the Karatsuba algorithm was found by her father himself. So I added the following information on the 1st page: which was invented by Karatsuba in 1960 [1]