2007.115: Cryptographic Applications of Non-Commutative Algebraic Structures and Investigations of Nonlinear Recursions

2007.115: George Petrides (2006) Cryptographic Applications of Non-Commutative Algebraic Structures and Investigations of Nonlinear Recursions. PhD thesis, Manchester Institute for Mathematical Sciences, The University of Manchester.

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Abstract

In this thesis we investigate the application of non-commutative algebraic structures and nonlinear recursions in cryptography. To begin with, we demonstrate that the public key cryptosystem based on the word problem on the Grigorchuk groups, as proposed by M. Garzon and Y. Zalcstein [8], is insecure. We do this by exploiting information contained in the public key in order to construct a key which behaves like the private key and allows successful decryption of ciphertexts.

Further on, we present a new block cipher with key-dependent S-boxes, based on the Grigorchuk groups. To the best of our knowledge, it is the first time groups are used in a block cipher, whereas they have been extensively used in public key cryptosystems. The study of the cipher’s properties is, at this stage, purely theoretical.

Finally, we investigate the notion of nonlinear complexity, or maximal order complexity as it was first defined in 1989 [15], for sequences. Our main purpose is to begin classification of periodic binary sequences into nonlinear complexity classes. Previous work on the subject also includes approximation of the size of each class, found in [7]. Once the classification is completed, we can use it to show how to perform checks for short cycles in large nonlinear feedback shift registers using our proposed algorithm.

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