Improving Width-3 Joint Sparse Form To Attain Asymptotically Optimal Complexity On Average Case

In this paper, we improve a width-3 joint sparse form proposed by Okeya, Katoh, and Nogami. After the improvement, the representation can attain an asymtotically optimal complexity found in our previous work. Although claimed as optimal by the authors, the average computation time of multi-scalar multiplication obtained by the representation is 563/1574n + o(n) approximate to 0.3577n + o(n). That number is larger than the optimal complexity 281/786n + o(n) approximate to 0.3575n + o(n) found in our previous work. To optimize the width-3 joint sparse form, we add more cases to the representation. After the addition, we can show that the complexity is updated to 281/786n + o(n) approximate to 0.3575n + o(n), which implies that the modified representation is asymptotically optimal. Compared to our optimal algorithm in the previous work, the modified width-3 joint sparse form uses less dynamic memory, but it consumes more static memory.