A Research On The Cryptography Based On Elliptic Curve Isogenies

Cryptography is the study of encoding and decoding, from 1976 the invention of public-key cryptography thought and the advent of data encryption standards for counting, Modern cryptography has gone through 30 years of history. Today, cryptography has become an important application of information technology disciplines, and playing an increasingly important role in the field of information security.As well as a research tool than the abstract elliptic curve, elliptic curve isogenies itself can also be used by cryptography. Isogenies computation between the elliptic curves given can be used to construct new cryptography modules and cryptosystems, in particular the problem seems to be hard for solving with a quantum computer.This paper discussions, research and resolve the security problems existing in every aspect of elliptic curve isogenies cryptosystems, especially the security problems of elliptic curve isogenies crypto schemes and their security proofs. The author divided the securities of elliptic curve isogenies cryptosystems into two relatively independent levels:the securities of their mathematic foundations and the securities of the crypto schemes. The author led his emphasis on solving the following difficult but important problems:establishing a security criteria for elliptic curves isogenies, designing a provable secure hash function based on elliptic curve isogenies and proving its security, designing a provable secure encryption scheme based on elliptic curve isogenies and proving its security, designing a provable secure signature scheme based on elliptic curve isogenies and proving its security. The conclusions and produces in this thesis are very valuable to the applications of elliptic curve isogenies cryptosystems.