Eta Pairing Computation on General Divisors over Hyperelliptic Curves y ² = x ⁷ − x ±1

Abstract

Recent developments on the Tate or Eta pairing computation over hyperelliptic curves by Duursma-Lee and Barreto et al. have focused on degenerate divisors. We present two efficient methods that work for general divisors to compute the Eta paring over divisor class groups of the hyperelliptic curves $H/{{\mathbb F}}_{7^n}:y^2 = x^7 - x \pm 1$ of genus 3. The first method generalizes the method of Barreto et al. so that it holds for general divisors, and we call it the pointwise method. For the second method, we take a novel approach using resultant. We focus on the case that two divisors of the pairing have supporting points in $H({{\mathbb F}}_{7^{3n}}),$ not in $H({{\mathbb F}}_{7^n})$. Our analysis shows that the resultant method is faster than the pointwise method, and our implementation result supports the theoretical analysis. In addition to the fact that the two methods work for general divisors, they also provide very explicit algorithms.