Fractional Windows Revisited: Improved Signed-Digit Representations for Efficient Exponentiation

This paper extends results concerning efficient exponentiation in groups where inversion is easy (e.g. in elliptic curve cryptography). It examines the right-to-left and left-to-right signed fractional window (RL-SFW and LR-SFW) techniques and shows that both RL-SFW and LR-SFW representations have minimal weight among all signed-digit representations with digit set {±1,±3, . . .,±m, 0}. (Fractional windows generalize earlier sliding-window techniques, providing more flexibility for exponentiation algorithms in order to make best use of the memory that is available for storing intermediate results.) Then it considers the length of representations: LR-SFW representations are an improvement over RL-SFW representations in that they tend to be shorter; further length improvements are possible by post-processing the representations.