Abstract
In this paper, we explore further the non-linearity and affine equivalence as proposed by Mishra et al. (Non-linearity and affine equivalence of permutations. 2014. http://eprint.iacr.org/2014/974.pdf). We propose an efficient algorithm in order to compute affine equivalent permutation(s) of a given permutation of length n, of complexity \(O(n^4)\) in worst case and \(O(n^2)\) in best case. Also in the affirmative in a special case \(n = p\), prime, it is of complexity \(O(n^3)\). We also propose an upper bound of non-linearity of permutation(s) whose length satisfies a special condition. Further, behaviour of non-linearity on direct sum and skew sum of permutation has been analysed. Also the distance of an affine permutation from the other affine permutations has also been studied. The cryptographic implication of this work is on permutation based stream ciphers like RC4 and its variants. In this paper, we have applied this study on RC4 cipher. The analysis shows that increasing the key size for RC4 does not mean that increase in the security or saturation after a limit but security may falls as key size increases.
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The authors are extremely thankful to the referee for his valuable comments and suggestions. This led to the over all improvement in quality and presentation of the paper.
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Appendix
Appendix
Note: inv at line no. 3 is actually inverse of a / d in \(\mathbb Z_{b/d}^*\). The function \(EX\_GCD(a, b)\) compute GCD of a, b and x, y such that \(d=ax+by\).
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Kumar, Y., Mishra, P.R., Pillai, N.R. et al. Affine equivalence and non-linearity of permutations over \(\mathbb Z_n\) . AAECC 28, 257–279 (2017). https://doi.org/10.1007/s00200-016-0307-6
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DOI: https://doi.org/10.1007/s00200-016-0307-6