Abstract. The braid groups are infinite non-commutative groups naturally arising from geometric braids. The aim of this article is twofold. One is to show that the braid groups can serve as a good source to enrich cryptography. The feature that makes the braid groups useful to cryptography includes the followings: (i) The word problem is solved via a fast algorithm which computes the canonical form which can be efficiently manipulated by computers. (ii) The group operations can be performed efficiently. (iii) The braid groups have many mathematically hard problems that can be utilized to design cryptographic primitives. The other is to propose and implement a new key agreement scheme and public key cryptosystem based on these primitives in the braid groups. The efficiency of our systems is demonstrated by their speed and information rate. The security of our systems is based on topological, combinatorial and group-theoretical problems that are intractible according to our current mathematical knowledge. The foundation of our systems is quite different from widely used cryptosystems based on number theory, but there are some similarities in design.

The contribution of this paper are twofold. First, we present a n efficient computationally secure anonymous channel which has no problem of ciphertext length expansion. The length is irrelevant to the number of MIXes (control centers). It improves the efficiency of Chaum's election scheme based on the MIX net automatically. Second, we show an election scheme which satisfies fairness. That is, if some vote is disrupted, no one obtains any information about all the other votes. Each voter sends O(nk) bits so that the probability of the fairness is 1-2-k , where n is the bit length of the ciphertext.

Abstract. Most public key cryptosystems have been constructed based on abelian groups up to now. We propose a new public key cryptosystem built on finite non abelian groups in this paper. It is convertible to a scheme in which the encryption and decryption are much faster than other well-known public key cryptosystems, even without no message expansion. Furthermore a signature scheme can be easily derived from it, while it is difficult to find a signature scheme using a non abelian group.

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