New Public-Key Cryptosystem Using Braid Groups

Abstract

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.

An Efficient Implementation of Braid Groups

“…Random braids play an important role in braid cryptosystems [5,7]. Since the braid group B n is discrete and infinite, a probability distribution on B n makes no sense.…”

“…In this section we consider the braid cryptosystem proposed in [7], which is a revised version of one in [5]. Let LB n and U B n be the subgroups of B n generated by the Artin generators σ 1 , .…”

“…We remark that if the secret key is of the form (a, a −1 ) and b −1 1 is taken as b 2 in the above encryption procedure, the cryptosystem in [5] is obtained. Hence in performance issues, there is no difference between the cryptosystems in [7] and [5].…”

“…A new cryptosystem using the braid groups was proposed in [5] at Crypto 2000. Since then, there has been no serious attempt to analyze the system besides one given by inventors [7].…”

“…This excellent speed is achieved because the canonical factors are expressed as permutations that can be efficiently and naturally handled by computers. The efficiency of the implementation shows that the braid group is a good source of cryptographic primitives [5,6]. It is hard to think of any other non-commutative groups that can be digitized as efficiently as the braid group.…”