Efficiently computable distortion maps for supersingular curves

Efficiently computable distortion maps are useful in cryptography. Galbraith-Pujolàs-Ritzenthaler-Smith [6] considered them for supersingular curves of genus 2. They showed that there exists a distortion map in a specific set of efficiently computable endomorphisms for every pair of nontrivial divisors under some unproven assumptions for two types of curves. In this paper, we prove that this result holds using a different method without these assumptions for both curves with r > 5 and r > 19 respectively, where r is the prime order of the divisors. In other words, we solve an open problem in [6]. Moreover, we successfully generalize this result to the case C : Y ² = X ^2g+1 + 1 over for any g s.t. 2g+1 is prime. In addition, we provide explicit bases of Jac _C [r] with a new property that seems interesting from the cryptographic viewpoint.