New Efficient Algorithms for Multiplication Over Fields of Characteristic Three

In this paper, we first present an enhancement of the well-known Karatsuba 2-way and 3-way algorithms for characteristic three fields, denoted by 𝔽_3^n where n ≥1. We then derive a 3-way polynomial multiplication algorithm with five 1/3 sized multiplications that use interpolation in 𝔽_9 . Following the computation of the arithmetic and delay complexity of the proposed algorithm, we provide the results of our hardware implementation of polynomial multiplications over 𝔽_3 and 𝔽_9 . The final proposal is a new 3-way polynomial multiplication algorithm over 𝔽_3 that uses three polynomial multiplications of 1/3 of the original size over 𝔽_3 and one polynomial multiplication of 1/3 of the original size over 𝔽_9 . We show that this algorithm represents about 15