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Construction of substitution boxes depending on linear block codes

Yıldız, Senay
The construction of a substitution box (S-box) with high nonlinearity and high resiliency is an important research area in cryptography. In this thesis, t-resilient nxm S-box construction methods depending on linear block codes presented in "A Construction of Resilient Functions with High Nonlinearity" by T. Johansson and E. Pasalic in 2000, and two years later in "Linear Codes in Generalized Construction of Resilient Functions with Very High Nonlinearity" by E. Pasalic and S. Maitra are compared and the former one is observed to be more promising in terms of nonlinearity. The first construction method uses a set of nonintersecting [n-d,m,t+1] linear block codes in deriving t-resilient S-boxes of nonlinearity 2̂(n-1)-2̂(n-d-1),where d is a parameter to be maximized for high nonlinearity. For some cases, we have found better results than the results of Johansson and Pasalic, using their construction. As a distinguished reference for nxn S-box construction methods, we study the paper "Differentially Uniform Mappings for Cryptography" presented by K.Nyberg in Eurocrypt 1993. One of the two constructions of this paper, i.e., the inversion mapping described by Nyberg but first noticed in 1957 by L. Carlitz and S. Uchiyama, is used in the S-box of Rijndael, which is chosen as the Advanced Encryption Standard. We complete the details of some theorem and proposition proofs given by Nyberg.