Radix-r Non-Adjacent Form

Abstract

AbstractRecently, the radix-3 representation of integers is used for the efficient implementation of pairing based cryptosystems. In this paper, we propose non-adjacent form of radix-r representation (rNAF) and efficient algorithms for generating rNAF. The number of non-trivial digits is (r–2)(r+1)/2 and its average density of non-zero digit is asymptotically (r–1)/(2r–1). For r=3, the non-trivial digits are { ± 2, ± 4} and the non-zero density is 0.4. We then investigate the width-w version of rNAF for the general radix-r representation, which is a natural extension of the width-w NAF. Finally we compare the proposed algorithms with the generalized NAF (gNAF) discussed by Joye and Yen. The proposed scheme requires a larger table but its non-zero density is smaller even for large radix. We explain that gNAF is a simple degeneration of rNAF — we can consider that rNAF is a canonical form for the radix-r representation. Therefore, rNAF is a good alternative to gNAF.KeywordsNon-adjacent formradix-r representationsigned window methodelliptic curve cryptosystempairing based cryptosystem