Provably-secure indentificaiton schemes based on conjugacy and ddh problems

Abstract

There are many situations where it is necessary to ``prove`` one``s identity. Typical scenarios are to login to a computer, to get access to an account for electronic banking or to withdraw money from an automatic teller machine. Older methods use passwords or PIN``s to implement user identification. Though successfully used in certain environments, these methods also have weakness. For example, anyone to whom you must give your pass-word to be verified has the ability to use this password and impersonate you. Zero-knowledge (and other) identification schemes provide a new type of user identification. It is possible for you to authenticate yourself without giving to the authenticator any knowledge to impersonate you. This thesis deals with a technique, called an $\emph{identification scheme}$ or entity authentication scheme, which allows one party to gain assurances that the identity of another is as declared, thereby preventing impersonation. Our proposed identification scheme is based on braid groups. Most of cryptosystems are based on commutative groups, but new cryptosystems based on non-commutative groups have been proposed. ($\emph{Braid cryptosystem}$ is one of them.) These systems are very difficult to analyze for their non-commutative properties. In the recent years, beginning with [44], several authors proposed to build secure cryptographical schemes using noncommutative groups, in particular Artin``s braid groups [1, 29, 30, 34], a natural idea as, on the one hand, braid groups are more complicated than Abelian groups, but, on the other hand, they are not too complicated to be worked with. In particular, the conjugacy problem in braid groups is algorithmically difficult, and it consequently provides one-way functions. In this thesis we construct two new interactive identification schemes. One is based on the conjugacy problem and another is based on decision DiffieHellman (DDH) assumption. The first scheme is the primary one based on conjugacy p...