NEW EFFICIENT CHARACTERISTIC THREE POLYNOMIAL MULTIPLICATION ALGORITHMS AND THEIR APPLICATIONS TO NTRU PRIME

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2022-1-21

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Yeniaras, Esra

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Some of the post-quantum cryptographic protocols require polynomial multiplication in characteristic three fields, thus the efficiency of such multiplication algorithms gain more importance recently. In this thesis, we propose four new polynomial multiplication algorithms in characteristic three fields and we show that they are more efficient than the current state-of-the-art methods. We first analyze the well-known algorithms such as the schoolbook method, Karatsuba 2-way and 3-way split methods, Bernstein’s three 3-way split method, Toom-Cook-like formulas, and other recent algorithms. We realize that there are not any 4-way or 5-way split multiplication algorithms in characteristic three fields unlike the binary (characteristic two) fields which have various 4, 5, or more split versions. We then propose three different 4-way split polynomial multiplication algorithms which are derived by using the interpolation technique in F_9. Furthermore, we propose a new 5-way split polynomial multiplication algorithm and then compare the arithmetic complexities and the implementation results for all of the aforementioned methods. We show that the new 4-way and 5-way split algorithms provide a 48.6% reduction in the arithmetic complexity for multiplication over F_9 and a 26.8% reduction for multiplication over F_3 for the input size 1280. Moreover, the new 4-way and 5-way algorithms yield faster implementation results compared to the current state-of-the-art methods. We apply the proposed methods to NTRU Prime protocol, a key encapsulation mechanism, submitted to NIST PQC Standardization Process by Bernstein et al., which executes characteristic three polynomial multiplication in its decapsulation stage. We implement the new methods in C and observe a 26.85% speedup for stnrup653 and a 35.52% speedup for sntrup761 in the characteristic three polynomial multiplication step of the NTRU Prime decapsulation.

Subject Keywords

Efficient polynomial multiplication, Characteristic three fields, Karatsuba, Interpolation, Post-quantum cryptography, Lattice-based cryptography, Key encapsulation mechanism, NTRU Prime

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https://hdl.handle.net/11511/95245

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Graduate School of Applied Mathematics, Thesis

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Citation Formats

E. Yeniaras, “NEW EFFICIENT CHARACTERISTIC THREE POLYNOMIAL MULTIPLICATION ALGORITHMS AND THEIR APPLICATIONS TO NTRU PRIME,” Ph.D. - Doctoral Program, Middle East Technical University, 2022.