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.

Cette thèse a pour sujet les protocoles de communication peer-to-peer asynchrones. Nous introduisons deux mesures basées sur la théorie de l'information,la Public Information Complexity (PIC) et la Multi-party Information Complexity (MIC), étudions leurs propriétés et leur relation avec d'autres mesures fondamentales en calcul distribué, telles que la communication complexity et la randomness complexity. Nous utilisons ensuite ces deux mesures pour étudier la fonction parité et la fonction disjointness. / This thesis is concerned with the study of multi-party communicationprotocols in the asynchronous message-passing peer-to-peer model. We introducetwo new information measures, the Public Information Complexity(PIC) and the Multi-party Information Complexity (MIC), study their propertiesand how they are related to other fundamental quantities in distributedcomputing such as communication complexity and randomness complexity.We then use these two measures to study the parity function and the disjointness function.