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Braid groups, Artin groups and their applications in cryptography Webster, Catherine


The aim of this article is to show how the braid groups can serve as a good platform to enrich cryptography. Braid groups are useful to cryptography for a number of reasons: (i) the word problem is solved via a fast algorithm which computes the canonical form which can be efficiently handled by computers, (ii) an algorithm which computes an unfaithful matrix representation for a braid exists, which is also efficiently handled by computers. Introduced are two protocols for asymmetric key exchange based on the braid groups. The braid groups are an example of a larger set of groups, namely, the Artin groups. This raises the question as to whether the other Artin groups are useful in public key cryptography, or whether the braid group is unique. Outline: Below is an introduction to public key cryptography and the key agreement protocol proposed in [1]. Chapter 2 is an introduction to the braid group. Chapter 3 discusses the canonical form for words in the braid group, and how this form can be used for key exchange as in [18]. In Chapter 4, the braid cryptosystem based on the coloured Burau matrix, as in [2], is discussed. Chapter 5 gives an introduction to Artin groups and Coxeter groups, and discusses possible cryptosystems using other Artin groups of finite type. Finally, in Chapter 6, I will discuss various attacks to the proposed systems and draw some general conclusions on the braid and Artin groups applications to public key cryptograhy.

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