On multiplication in finite fields

We present a method for multiplication in finite fields which gives multiplication algorithms with improved or best known bilinear complexities for certain finite fields. Our method generalizes some earlier methods and combines them with the recently introduced complexity notion M̂q(ℓ), which denotes the minimum number of multiplications needed in FqFq in order to obtain the coefficients of the product of two arbitrary ℓℓ-term polynomials modulo xℓxℓ in Fq[x]Fq[x]. We study our method for the finite fields FqnFqn, where 2≤n≤182≤n≤18 and q=2,3,4q=2,3,4 and we improve or reach the currently best known bilinear complexities. We also give some applications in cryptography.