Efficient multiplication in characteristic three fields

Characteristic three fields denoted by F3n , where n ≥ 1, are used in curve based cryptography. In this paper, first we improve the well known Karatsuba 2-way and 3-way algorithms for characteristic three fields. Then, we derive a 3-way polynomial multiplication algorithm with five 1/3 size multiplications using interpolation in F9. After computing the arithmetic and delay complexities of the proposed algorithm, we show that it gives better arithmetic complexity results than the known algorithms. More specifically, we show that the recursive use of the 3-way algorithm in the extension field yields about 14% to 22% improvements for multiplication in F36n , which is the main operation in curve based cryptographic pairing computations. To the best of our knowledge, this is the first time it is shown that the recursive use of the 3-way approach with five multiplications gives significant improvement for multiplication in characteristic three fields of cryptographic sizes.