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

A Trace Based GF(2n)GF(2^n) Inversion Algorithm

By associating Fermat\u27s Little Theorem based $GF(2^n)$ inversion algorithms with the multiplicative Norm function, we present an additive Trace based $GF(2^n)$ inversion algorithm. For elements with Trace value 0, it needs 1 less multiplication operation than Fermat\u27s Little Theorem based algorithms in some $GF(2^n)$s

Low Complexity MDS Matrices Using GF(2n)GF(2^n) SPB or GPB

While $GF(2^n)$ polynomial bases are widely used in symmetric-key components, e.g. MDS matrices, we show that even low time/space complexities can be achieved by using $GF(2^n)$ shifted polynomial bases (SPB) or generalized polynomial bases (GPB)