**A new algorithm for residue multiplication modulo $2^{521}-1$**

*Shoukat Ali, Murat Cenk*

**Abstract: **We present a new algorithm for residue multiplication modulo the Mersenne prime $2^{521}-1$ based on the Toeplitz matrix-vector product. For this modulo, our algorithm yields better result in terms of the total number of operations than the previously known best algorithm of R. Granger and M. Scott presented in Public Key Cryptography - PKC 2015. Although our algorithm has nine more multiplications than Granger-Scott multiplication algorithm, the total number of additions is forty-two less than their algorithm. Even if one takes a ratio of $1:4$ between multiplication and addition our algorithm still has less total number of operations. We also present the test results of both the multiplication algorithms on an Intel Sandy Bridge Corei5-2410M machine, with and without optimization option in GCC.

**Category / Keywords: **implementation / residue multiplication, Toeplitz matrix-vector product, Mersenne prime, elliptic curve cryptography

**Date: **received 15 Feb 2016

**Contact author: **shoukat 1983 at gmail com

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

**Version: **20160216:210948 (All versions of this report)

**Short URL: **ia.cr/2016/138

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