Extending the Signed Non-zero Bit and Sign-Aligned Columns Methods to General Bases for Use in Cryptography.

An efficient scalar multiplication algorithm is a crucial component of elliptic curve cryptosystems. We propose a scalar multiplication algorithm based on scalar recodings that is regular in nature. Our scalar multiplication algorithm is made from two scalar recoding algorithms called Recode and Align. Recode is the generalization of the signed non-zero bit recoding algorithm given by Hedabou, Pinel and Beneteau in 2005. It recodes the k-ary representation of the given scalar into a signed non-zero form by means of a small lookup table. On the other hand, Align is the generalized k-ary version of the sign-aligned columns recoding algorithm given by Faz-Hernandez, Longa and Sanchez in 2014. It recodes the k-ary representation of a scalar in such a way that the sign of each of its digits agrees with a given {1,-1}-valued sequence. When analyzing the choice of k is an element of {2, 3}, we find some theoretical evidence that k = 3 may offer better performance in certain scenarios.