Fixed-Base Comb with Window-Non-Adjacent Form (NAF) Method for Scalar Multiplication

Elliptic curve cryptography (ECC) is one of the most promising public-key techniques in terms of short key size and various crypto protocols. For this reason, many studies on the implementation of ECC on resource-constrained devices within a practical execution time have been conducted. To this end, we must focus on scalar multiplication, which is the most expensive operation in ECC. A number of studies have proposed pre-computation and advanced scalar multiplication using a non-adjacent form (NAF) representation, and more sophisticated approaches have employed a width-w NAF representation and a modified pre-computation table. In this paper, we propose a new pre-computation method in which zero occurrences are much more frequent than in previous methods. This method can be applied to ordinary group scalar multiplication, but it requires large pre-computation table, so we combined the previous method with ours for practical purposes. This novel structure establishes a new feature that adjusts speed performance and table size finely, so we can customize the pre-computation table for our own purposes. Finally, we can establish a customized look-up table for embedded microprocessors.