Paper 2019/767

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

Jaeseong Jeong, Chang Heon Kim, Namhun Koo, 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 $d=2^{n/2}+1$

Metadata
Available format(s)
PDF
Category
Secret-key cryptography
Publication info
Published elsewhere. Minor revision. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
DOI
10.1587/transfun.2021EAP1167
Keywords
Permutation PolynomialsDifferential UniformityBoomerang UniformityBoomerang SpectrumExtended Walsh Spectrum
Contact author(s)
komaton @ skku edu
History
2022-02-25: revised
2019-07-02: received
See all versions
Short URL
https://ia.cr/2019/767
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2019/767,
      author = {Jaeseong Jeong and Chang Heon Kim and Namhun Koo and Soonhak Kwon and Sumin Lee},
      title = {On cryptographic parameters of permutation polynomials of the form $x^rh(x^{(q-1)/d})$},
      howpublished = {Cryptology {ePrint} Archive, Paper 2019/767},
      year = {2019},
      doi = {10.1587/transfun.2021EAP1167},
      url = {https://eprint.iacr.org/2019/767}
}
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