Public-Key Cryptosystem Based on Isogenies

A new general mathematical problem, suitable for publickey cryptosystems, is proposed: morphism computation in a category of Abelian groups. In connection with elliptic curves over finite fields, the problem becomes the following: compute an isogeny (an algebraic homomorphism) between the elliptic curves given. The problem seems to be hard for solving with a quantum computer. ElGamal public-key encryption and Diffie-Hellman key agreement are proposed for an isogeny cryptosystem. The paper describes theoretical background and a publickey encryption technique, followed by security analysis and consideration of cryptosystem parameters selection. A demonstrative example of encryption is included as well. public-key cryptography, elliptic curve cryptosystem, cryptosystem on isogenies of elliptic curves, isogeny star, isogeny cycle, quantum computer