Eta pairing computation on general divisors over hyperelliptic curves y² = x^p - x + d

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 efficient methods that work for general divisors to compute the Eta paring over divisor class groups of the hyperelliptic curves H_d : y² = x^p - x + d where p is an odd prime. On the curve H_d of genus 3, we provide two efficient methods: 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. Our analysis shows that the resultant method is faster than the pointwise method, and our implementation result supports the theoretical analysis. We also emphasize that the Eta pairing technique is generalized to the curve y² = x^p - x + d, p ≡ 1 (mod 4). Furthermore, we provide the closed formula for the Eta pairing computation on general divisors by Mumford representation of the curve H_d of genus 2.