This article deals with an algorithm for the generation of impregnable substitution boxes (S-box). The proposed scheme involves the application of right translation and Gray codes over the original Advanced Encryption Standard (AES) S-box. Regular representation of Galois field GF(2 8 ) is used to produce the translational effect in the AES S-box. The translated AES S-box is then subjected to Gray codes for the enhancement in their algebraic complexity. The proposed scheme results 256 different cryptographically strong S-boxes. Several tests such as non-linearity, bit independence, strict avalanche, linear approximation and differential approximation, algebraic complexity, correlation and histogram are implemented on every newly generated S-box to analyze their resistance against computational attacks. Furthermore, the newly generated S-boxes are compared with the existing well-known S-boxes. The simulation results indicate that the resistance of proposed S-boxes against computational attacks including linear, interpolation, differential and algebraic attacks approaches to the optimal values.
For a given binary BCH code C n of length n = 2 s − 1 generated by a polynomial g(x) 2 F 2 [x] of degree r there is no binary BCH code of length (n + 1)n generated by a generalized polynomial g(Z ≥ 0] of degree 2r. However, it does exist a binary cyclic code C (n+1)n of length (n + 1)n such that the binary BCH code C n is embedded in C (n+1)n . Accordingly a high code rate is attained through a binary cyclic code C (n+1)n for a binary BCH code C n . Furthermore, an algorithm proposed facilitates in a decoding of a binary BCH code C n through the decoding of a binary cyclic code C (n+1)n , while the codes C n and C (n+1)n have the same minimum hamming distance.
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