Reduction-Free Multiplication for Finite Fields and Polynomial Rings
Arithmetic of Finite Fields: 9th International Workshop, WAIFI 2022, Chengdu, China, August 29 – September 2, 2022, Revised Selected Papers
Sihem Mesnager, Zhengchun Zhou
The complexity of the multiplication operation over polynomial rings and finite fields drastically changes with the selection of the defining polynomial of the respective mathematical structure. Trinomials and pentanomials are the most natural choices for the best arithmetic. In this paper, we first present a study in which a special type of trinomial does not require any reduction steps. We then introduce two new algorithms, FIKO and RF-FIKO, fully interleaved bit-parallel Karatsuba-Ofman multipliers where the latter is only concerned with the three Karatsuba-Ofman terms and is free from the bipartite reduction circuits. All algorithms are implemented in FPGA and ASIC, and detailed implementation results are presented, showing significant improvements to existing methods.
Cryptography, Equally-spaced polynomials, Finite field arithmetic, Interleaved multiplication, Karatsuba-Ofman multiplication, Mersenne polynomials, Montgomery multiplication, Polynomial bi-partite multiplication, Polynomials rings, Pseudoprimes, Reduction-free multiplication, Reduction-free trinomials
Samira Carolina Oliva Madrigal, Gökay Saldamlı, Chen Li, Yue Geng, Jing Tian, Zhongfeng Wang, and Çetin Kaya Koç. "Reduction-Free Multiplication for Finite Fields and Polynomial Rings" Arithmetic of Finite Fields: 9th International Workshop, WAIFI 2022, Chengdu, China, August 29 – September 2, 2022, Revised Selected Papers (2023): 53-78. https://doi.org/10.1007/978-3-031-22944-2_4