Pairings on Jacobians of Hyperelliptic Curves

TL;DR

This paper characterizes all non-degenerate, bilinear, anti-symmetric, Galois-invariant pairings on the Jacobian of a hyperelliptic genus two curve over finite fields, showing they cannot be computed more efficiently than the Weil pairing.

Contribution

It provides an explicit description of all such pairings under certain conditions and analyzes the Frobenius endomorphism's representation on the l-torsion subgroup.

Findings

No pairing can be computed more efficiently than the Weil pairing.

Frobenius endomorphism is diagonalizable if its characteristic polynomial splits into linear factors modulo l.

Embedding degree equals total embedding degree under specific endomorphism restrictions.

Abstract

Consider the jacobian of a hyperelliptic genus two curve defined over a finite field. Under certain restrictions on the endomorphism ring of the jacobian we give an explicit description all non-degenerate, bilinear, anti-symmetric and Galois-invariant pairings on the jacobian. From this description it follows that no such pairing can be computed more efficiently than the Weil pairing. To establish this result, we need an explicit description of the representation of the Frobenius endomorphism on the l-torsion subgroup of the jacobian. This description is given. In particular, we show that if the characteristic polynomial of the Frobenius endomorphism splits into linear factors modulo l, then the Frobenius is diagonalizable. Finally, under the restriction that the Frobenius element is an element of a certain subring of the endomorphism ring, we prove that if the characteristic polynomial of the Frobenius endomorphism splits into linear factors modulo l, then the embedding degree and the total embedding degree of the jacobian with respect to l are the same number.