On Cryptographic Parameters of Permutation Polynomials of the form xrh(x(2n-1)/d)
Summary :
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 xrh(x(2n-1)/d) over a finite field of q=2n elements, where r is a positive integer and d is a positive divisor of 2n-1. The computational cost of those formulas is proportional to d. We investigate differentially 4-uniform permutation polynomials of the form xrh(x(2n-1)/3) and compute the boomerang spectrum and the extended Walsh spectrum of them using the suggested formulas when 6≤n≤12 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=2n/2+1.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E105-A No.8 pp.1134-1146
- Publication Date
- 2022/08/01
- Publicized
- 2022/02/22
- Online ISSN
- 1745-1337
- DOI
- 10.1587/transfun.2021EAP1167
- Type of Manuscript
- PAPER
- Category
- Cryptography and Information Security
Authors
Jaeseong JEONG
https://orcid.org/0000-0003-0103-2920
Sungkyunkwan University
Chang Heon KIM
https://orcid.org/0000-0002-2234-8572
Sungkyunkwan University
Namhun KOO
https://orcid.org/0000-0003-1678-8480
Ewha Womans University
Soonhak KWON
https://orcid.org/0000-0002-3336-0817
Sungkyunkwan University
Sumin LEE
Sungkyunkwan University
Keyword
Latest Issue
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Jaeseong JEONG, Chang Heon KIM, Namhun KOO, Soonhak KWON, Sumin LEE, "On Cryptographic Parameters of Permutation Polynomials of the form xrh(x(2n-1)/d)" in IEICE TRANSACTIONS on Fundamentals,
vol. E105-A, no. 8, pp. 1134-1146, August 2022, doi: 10.1587/transfun.2021EAP1167.
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 xrh(x(2n-1)/d) over a finite field of q=2n elements, where r is a positive integer and d is a positive divisor of 2n-1. The computational cost of those formulas is proportional to d. We investigate differentially 4-uniform permutation polynomials of the form xrh(x(2n-1)/3) and compute the boomerang spectrum and the extended Walsh spectrum of them using the suggested formulas when 6≤n≤12 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=2n/2+1.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2021EAP1167/_p
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@ARTICLE{e105-a_8_1134,
author={Jaeseong JEONG, Chang Heon KIM, Namhun KOO, Soonhak KWON, Sumin LEE, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={On Cryptographic Parameters of Permutation Polynomials of the form xrh(x(2n-1)/d)},
year={2022},
volume={E105-A},
number={8},
pages={1134-1146},
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 xrh(x(2n-1)/d) over a finite field of q=2n elements, where r is a positive integer and d is a positive divisor of 2n-1. The computational cost of those formulas is proportional to d. We investigate differentially 4-uniform permutation polynomials of the form xrh(x(2n-1)/3) and compute the boomerang spectrum and the extended Walsh spectrum of them using the suggested formulas when 6≤n≤12 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=2n/2+1.},
keywords={},
doi={10.1587/transfun.2021EAP1167},
ISSN={1745-1337},
month={August},}
Copy
TY - JOUR
TI - On Cryptographic Parameters of Permutation Polynomials of the form xrh(x(2n-1)/d)
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1134
EP - 1146
AU - Jaeseong JEONG
AU - Chang Heon KIM
AU - Namhun KOO
AU - Soonhak KWON
AU - Sumin LEE
PY - 2022
DO - 10.1587/transfun.2021EAP1167
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E105-A
IS - 8
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - August 2022
AB - 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 xrh(x(2n-1)/d) over a finite field of q=2n elements, where r is a positive integer and d is a positive divisor of 2n-1. The computational cost of those formulas is proportional to d. We investigate differentially 4-uniform permutation polynomials of the form xrh(x(2n-1)/3) and compute the boomerang spectrum and the extended Walsh spectrum of them using the suggested formulas when 6≤n≤12 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=2n/2+1.
ER -
English
