Improved Three-Way Split Formulas for Binary Polynomial and Toeplitz Matrix Vector Products

Some new results on binary polynomial multiplication

“…This division requires (n − 2) bit additions and a delay of (n − 2)D X . A detailed explanation is in Section 2.3.2 of [11]. We also note that one can obtain a logarithmic delay for this type of exact division.…”

“…-An improved version of the 5-way algorithm introduced in [12] through an optimization of the number of additions. -A new 3-way algorithm with a lower complexity than the ones described in [3,10,11]: it entails the asymptotic arithmetic complexity of 15.125n 1.46 + O(n) and delay complexity 10 log 3 (n)D X + D A . -New optimizations of algorithms for polynomial multiplication over F 4 .…”

“…In [10,11], Cenk, Negre, and Hasan proposed the following algorithm for computing C = AB, where α is the generator of F 4 : …”

“…Finally, the overlaps of the coefficients of X 0 , X n , X 2n , and X 3n require (3n − 3) additions, and the cost of the overlapping of the coefficient of X 4n with the other terms is (n − 1) if n/2 ≤ k ≤ n, and (2k − 1) if 1 ≤ k < n/2. On the other hand, the delay complexity can be computed as described in [11] and we obtain the complexities as follows:…”

“…In [24] and [25], methods have been presented for 3-way splits with 6.33n 1.63 + O(n) operations. More recently, this complexity has been improved to 6.27n 1.63 + O(n) as reported in [11] and then to 5.8n 1.63 + O(n) as described in [9].…”