New Hash Function Base On Braid Subgroups

AES algorithm has replaced the traditional DES algorithm as encryption standard of information security field. In particular, since the outstanding job of cryptography analysis done by Xiaoyun Wang, the traditional hash function MD5 and SHA1 are no longer safe, People need more secure and new hash functions. In recent years, a number of scholars respectively built a number of cryptographic algorithms base on braid groups. However, it was found that some computational difficult problems of cryptography sys-tems based on braid groups tend to be translated into concrete object search problems. And the corresponding search computation can be calculated effectively. Thus, there are huge threats to the securities of these algorithms. For these reasons, this paper put forward Provable security cryptographic algorithm base on braid groups.First of all, Paper introduces the good nature of braid group. one hand, the word problem is decidable in a quadratic polynomial time, each element of a braid group has a unique normal form to be computed in quadratic polynomial time, and the product of each pair elements of a braid group and the inverse of each element of a braid group can be calculated in polynomial time, etc. On the other hand, braid groups also enjoy com-putational hard problems. secondly, group theory workers found that braid groups have some subgroups enjoy unsolvable subgroup members problems-Mihailova subgroup. Xi-aofeng Wang et al have given the structure of a Mihailova subgroup of the direct product of free group. Considering that Colins has proved some subgroups of a braid group which are isomorphic to the group F2× F2. Therefore, we know that a braid group contain Mihailova subgroups. At last, we give all generators and defining relations of Mihailova subgroups of a braid group by finding the isomorphism from the group F2× F2 to the subgroup of a braid group. People wound find that there are 56 generators, and the computation of the expressions of these generators is so huge. Then, by using the unsolv-able of subgroup members problem of Mihailova subgroups of braid groups, combined with AES algorithm through multiple iterations we establish a new hash function. We show that the proposed hashfunction is a one-way and collision free function. We also give some applications of this new hash function in data integration, message or entity authentication, and digital signature. The structure of Mihailova subgroups of a braid group can be also used for people to set up much safer cryptographical mechanism.