**Faster Characteristic Three Polynomial Multiplication and Its Application to NTRU Prime Decapsulation**

*Esra Yeniaras and Murat Cenk*

**Abstract: **Efficient computation of polynomial multiplication for characteristic three fields, $\mathbb{F}_{3^{n}}$ for $n\geq1$, is an important attribute for many cryptographic protocols. In this paper, we propose three new polynomial multiplication algorithms over $\mathbb{F}_{3}[x]$ and show that they are more efficient than the current state-of-the-art algorithms. We first examine through the well-known multiplication algorithms in $\mathbb{F}_{3}[x]$ including Karatsuba-2-way and 3-way split formulas along with the recent enhancements. Then, we propose a new 4-way split polynomial multiplication algorithm and an improved version of it which are both derived by using interpolation in $\mathbb{F}_{9}$. Moreover, we propose a 5-way split multiplication algorithm, and then compare the efficiencies of these algorithms altogether. We apply the proposed algorithms to the NTRU Prime protocol, a post-quantum key encapsulation mechanism (KEM), submitted to the NIST PQC Competition by Bernstein et al., performing polynomial multiplication in characteristic three fields in its decapsulation phase. We observe that the new hybrid algorithms provide a $12.9\%$ reduction in the arithmetic complexity. Furthermore, we implement these new hybrid methods on Intel (R) Core (TM) i7-9750H architecture using C and obtain a $37.3\%$ reduction in the implementation cycle count.

**Category / Keywords: **implementation / Polynomial multiplication, Karatsuba, Characteristic three fields, Key encapsulation, NTRU Prime, Lattice-based cryptography, Post-quantum cryptography

**Date: **received 24 Oct 2020

**Contact author: **yeniaras esra at metu edu tr,mcenk@metu edu tr

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

**Version: **20201026:092415 (All versions of this report)

**Short URL: **ia.cr/2020/1336

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