**On cryptographic parameters of permutation polynomials of the form $x^rh(x^{(q-1)/d})$**

*Jaeseong Jeong and Chang Heon Kim and Namhun Koo and Soonhak Kwon and 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 $d=2^{n/2}+1$

**Category / Keywords: **secret-key cryptography / Permutation Polynomials, Differential Uniformity, Boomerang Uniformity, Boomerang Spectrum, Extended Walsh Spectrum, Differentially 4-Uniform Permutation Polynomials

**Original Publication**** (with minor differences): **IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
**DOI: **10.1587/transfun.2021EAP1167

**Date: **received 1 Jul 2019, last revised 25 Feb 2022

**Contact author: **komaton at skku edu

**Available format(s): **PDF | BibTeX Citation

**Version: **20220225:115756 (All versions of this report)

**Short URL: **ia.cr/2019/767

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