Efficient Self-pairing on Ordinary Elliptic Curves

Abstract

A self-pairing e(P,P) is a special bilinear pairing where both points are equal. Self-pairings are used in some cryptographic schemes and protocols, such as ZSS shorter signatures and so on. In this paper, We first generalize a result in [28] to any elliptic curve with more simpler final exponentiation. Then we present a new self-pairing on ordinary elliptic curves with short loop length in Miller’s algorithm. We also provide examples of self-pairing friendly elliptic curves which are of interest for efficient pairing implementations. Finally, we present explicit formulae for Miller’s algorithm to compute self-pairing on ordinary elliptic curves with embedding degree one.