Complexity measures for classes of sequences and cryptographic apllications

Pseudo-random sequences are a crucial component of cryptography, particularly in stream cipher design. In this thesis we will investigate several measures of randomness for certain classes of finitely generated sequences. We will present a heuristic algorithm for calculating the k-error linear complexity of a general sequence, of either finite or infinite length, and results on the closeness of the approximation generated. We will present an linear time algorithm for determining the linear complexity of a sequence whose characteristic polynomial is a power of an irreducible element, again presenting variations for both finite and infinite sequences. This algorithm allows the linear complexity of such sequences to be determined faster than was previously possible. Finally we investigate the stability of m-sequences, in terms of both k-error linear complexity and k-error period. We show that such sequences are inherently stable, but show that some are more stable than others

The k-error Linear Complexity Distribution for Periodic Sequences

This thesis proposes various novel approaches for studying the k-error linear complexity distribution of periodic binary sequences for k > 2, and the second descent point and beyond of k-error linear complexity critical error points. We present a new tool called Cube Theory. Based on Games-Chan algorithm and the cube theory, a constructive approach is presented to construct periodic sequences with the given k-error linear complexity profile. All examples are verified by computer programs