a b s t r a c tWe 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 M q ( ), which denotes the minimum number of multiplications needed in F q in order to obtain the coefficients of the product of two arbitrary -term polynomials modulo x in F q [x]. We study our method for the finite fields F q n , where 2 ≤ n ≤ 18 and q = 2, 3, 4 and we improve or reach the currently best known bilinear complexities. We also give some applications in cryptography.

This paper presents several methods for reducing the number of bit operations for multiplication of polynomials over the binary field. First, a modified Bernstein's 3-way algorithm is introduced, followed by a new 5-way algorithm. Next, a new 3-way algorithm that improves asymptotic arithmetic complexity compared to Bernstein's 3-way algorithm is introduced. This new algorithm uses three multiplications of one-third size polynomials over the binary field and one multiplication of one-third size polynomials over the finite field with four elements. Unlike Bernstein's algorithm, which has a linear delay complexity with respect to input size, the delay complexity of the new algorithm is logarithmic. The number of bit operations for the multiplication of polynomials over the finite field with four elements is also computed. Finally, all these new results are combined to obtain improved complexities.

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