Cyclic Codes: Low-Weight Codewords and Locators

Error correcting codes has become an integral part of the design of reliable data transmissions and storage systems. They are also playing an increasingly important role for other applications such as the analysis of pseudorandom sequences and the design of secure cryptosystems. Cyclic codes form a large class of widely used error correcting codes, including important codes such as the Bose-Chaudhuri-Hocquenghem (BCH) codes, quadratic residue (QR) codes and Golay codes. In this thesis I tackle two problems related to cyclic codes: finding low-weight codewords and decoding. Computing efficiently low-weight codewords of a cyclic code is often a key ingredient of correlation attacks to LFSR-based stream ciphers. The best general purpose algorithm is based on the generalized birthday problem. In the first part of the thesis I show an alternative approach based on discrete logarithms which has - in some cases relevant for applications - much lower memory complexity requirements and a comparable time complexity. The second part of the thesis is devoted to some results concerning efficient bounded-distance decoding for cyclic codes.