Trivalent Graph isomorphism in polynomial time

It's important to design polynomial time algorithms to test if two graphs are isomorphic at least for some special classes of graphs. An approach to this was presented by Eugene M. Luks(1981) in the work \textit{Isomorphism of Graphs of Bounded Valence Can Be Tested in Polynomial Time}. Unfortunately, it was a theoretical algorithm and was very difficult to put into practice. On the other hand, there is no known implementation of the algorithm, although Galil, Hoffman and Luks(1983) shows an improvement of this algorithm running in $O(n^3 \log n)$. The two main goals of this master thesis are to explain more carefully the algorithm of Luks(1981), including a detailed study of the complexity and, then to provide an efficient implementation in SAGE system. It is divided into four chapters plus an appendix.Comment: 48 pages. It is a Master Thesi

Quantum Fourier sampling, Code Equivalence, and the quantum security of the McEliece and Sidelnikov cryptosystems

The Code Equivalence problem is that of determining whether two given linear codes are equivalent to each other up to a permutation of the coordinates. This problem has a direct reduction to a nonabelian hidden subgroup problem (HSP), suggesting a possible quantum algorithm analogous to Shor's algorithms for factoring or discrete log. However, we recently showed that in many cases of interest---including Goppa codes---solving this case of the HSP requires rich, entangled measurements. Thus, solving these cases of Code Equivalence via Fourier sampling appears to be out of reach of current families of quantum algorithms. Code equivalence is directly related to the security of McEliece-type cryptosystems in the case where the private code is known to the adversary. However, for many codes the support splitting algorithm of Sendrier provides a classical attack in this case. We revisit the claims of our previous article in the light of these classical attacks, and discuss the particular case of the Sidelnikov cryptosystem, which is based on Reed-Muller codes