Some new results on binary polynomial multiplication
This paper presents several methods for reducing the number of bit operations for multiplication of polynomials over the binary field. First, a modified Bernsteins 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 Bernsteins 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 Bernsteins 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.
polynomial multiplication binary fields elliptic curve scalar multiplication karatsuba toom divide-and-conquer