Recent Developments in Cryptography

摘要

In this short note, we briefly describe cryptosystems that are believed to be quantum-resistant and focus on isogeny-based cryptosystems. Recent SIDH (Supersingular Isogeny Diffie-Hellman) developments have focused on (2,2)-reducible Jacobians, where addition is executed via the Kummer surface. While elliptic curve isogenies are easy, explicit, and fast to compute thanks to Velús formulas, this is not the case for higher genus curves. The case of (2,2)-isogenies in genus 2 curves are an exception thanks to the work of Richelot. In addition, some explicit work has been completed in the case of (3,3) and (5,5)-isogenies, which are much more complicated than the case of Richelot isogenies. In this paper, we further investigate the case of (4,4)-reducible Jacobians and explicitly compute the locus ℒ4.