**Some New Results on Binary Polynomial Multiplication**

*Murat Cenk and M. Anwar Hasan*

**Abstract: **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.

**Category / Keywords: **implementation / Polynomial multiplication, binary fields

**Date: **received 9 Feb 2015, last revised 12 Feb 2015

**Contact author: **mcenk at metu edu tr

**Available format(s): **PDF | BibTeX Citation

**Version: **20150223:214206 (All versions of this report)

**Short URL: **ia.cr/2015/094

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