Some Analysis of Radix-r Representations

We deal with the radix-r representation used for the scalar multiplication of pairing-based cryptosystems with characteristic r. Our goal of this paper is to present some invariant properties about the signed radix-r representation; (1) approximation formulae for the average significant length and the average hamming weight of gNAF and wrNAF representation, (2) some classification formulae of equivalent classes called as Cutting Lemma, Collision Lemma, and Search Space Theorem. We also analyze the security of signed radix-r representations in the sense of side channel attacks, and to this end we propose a secure countermeasure.