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Title:
On cryptographic parameters of permutation polynomials of the form x^rh(x^{(q-1)/d})
Authors: Jaeseong Jeong, Chang Heon Kim, Namhun Koo, Soonhak Kwon, Sumin Lee
Abstract:The differential uniformity, the boomerang uniformity, and the extended Walsh spectrum etc are important parameters to evaluate the security of S(substitution)-box. In this paper, we introduce efficient formulas to compute these cryptographic parameters of permutation polynomials of the form x^rh(x^{(q-1)/d}) over a finite field of q=2^n elements, where r is a positive integer and d is a positive divisor of q-1. The computational cost of those formulas is proportional to d. We investigate differentially 4-uniform permutation polynomials of the form x^rh(x^{(q-1)/3}) and compute the boomerang spectrum and the extended Walsh spectrum of them using the suggested formulas when 4\le n\le 10 is even, where d=3 is the smallest nontrivial d for even n. We also investigate the differential uniformity of some permutation polynomials introduced in some recent papers for the case d=2^{n/2}+1
ePrint: https://eprint.iacr.org/2019/767
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