Generators of Jacobians of Genus Two Curves

TL;DR

This paper demonstrates that the Frobenius endomorphism on genus two Jacobians is often diagonal, enabling an explicit Weil-pairing description and an efficient probabilistic algorithm for finding l-torsion generators, with cryptographic relevance.

Contribution

It provides a new explicit description of the Weil-pairing and an efficient algorithm for generating l-torsion points on genus two Jacobians, applicable in cryptography.

Findings

Frobenius endomorphism is diagonal in most cryptographic cases.

Explicit Weil-pairing formula derived for l-torsion subgroup.

Probabilistic algorithm efficiently finds generators of l-torsion subgroup.

Abstract

We prove that in most cases relevant to cryptography, the Frobenius endomorphism on the Jacobian of a genus two curve is represented by a diagonal matrix with respect to an appropriate basis of the subgroup of l-torsion points. From this fact we get an explicit description of the Weil-pairing on the subgroup of l-torsion points. Finally, the explicit description of the Weil-pairing provides us with an efficient, probabilistic algorithm to find generators of the subgroup of l-torsion points on the Jacobian of a genus two curve.