Quantum Error-correction Codes on Abelian Groups

In classical public key cryptography the security of the cryptosystems are based on the difficulty of calculating certain functions. A famous example is the ASP cryptosystem which was based on the assumption that factoring large integers could not be done in polynomial time (on classical computers). The typical situation in these systems is that two parties (Bob and Alice) whish to communicate in secret. Instead of sharing a secrete key in advance (which confront us with the relatively difficult issue of secret key distribution), Bob announces a public key which is used by Alice to encrypt a message, sent to Bob. The encryption is done in a clever way so that if a third party (Eve) wants to decrypt it a non feasible amount of calculation is needed. Bob, however, has a secret key of his own which enables him to do the encryption in real time. Quantum cryptography has a different way of keeping things secret. The difficulty of some calculations is replaced by the impossibility of some calculations according to the laws of quantum mechanics. The first example of the quantum key distribution protocol was published in 1984 by Bennett and Brassard [BB] which is now called BB84 code. The security of this protocol is gauranteed by the impossibility of measuring the state of a quantum system in two conjugate bases simultaneously. A complete proof of security against any possible attack (i.e. any combination of physical operations permitted by the laws of quantum mechanics) was given later [LC], [M], [BBMR]. A simple proof of this fact is proposed by Shor and Preskill in [SP]. They first showed the security of a modified Lo-Chau code which is a entanglement purification