More About the Corpus of Involutions From Two-to-One Mappings and Related Cryptographic S-Boxes

Permutation polynomials have been extensively studied for their applications in cryptography, coding theory, combinatorial design, etc. An important subfamily of permutations is the class of involutions (those permutations are equal to their compositional inverse). Elements of this class have been u...

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Veröffentlicht in:IEEE transactions on information theory Jg. 69; H. 2; S. 1315 - 1327
Hauptverfasser: Mesnager, Sihem, Yuan, Mu, Zheng, Dabin
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.02.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithms
Boolean functions
Ciphers
Codes
Coding
Coding theory
Combinatorial analysis
Cryptography
Dickson polynomial
differentially 4-uniform
Encryption
Fields (mathematics)
Fixed points (mathematics)
graph indicator
involution
Mathematical analysis
Mathematics
permutation polynomial
Permutations
Polynomials
Resists
resultant
S-box
Systematics
two-to-one mapping
Vectorial function
ISSN:0018-9448, 1557-9654
Online-Zugang:Volltext
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Zusammenfassung:Permutation polynomials have been extensively studied for their applications in cryptography, coding theory, combinatorial design, etc. An important subfamily of permutations is the class of involutions (those permutations are equal to their compositional inverse). Elements of this class have been used frequently for block cipher designs and coding theory. In this article, we further investigate this corpus using new approaches, specifically from two-to-one (2-to-1) functions and (in some cases) using the graph indicators introduced by Carlet in 2020. In our constructions of involutions over the finite field <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{2^{n}} </tex-math></inline-formula> of order <inline-formula> <tex-math notation="LaTeX">2^{n} </tex-math></inline-formula>, we shall intensively use 2-to-1 mappings over <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{2^{n}} </tex-math></inline-formula>. More specifically, we present a new constructive method to design involutions from 2-to-1 mappings through their graph indicator and derive new involutions from known 2-to-1 mappings. Besides, we also propose several new classes of 2-to-1 mappings, including 2-to-1 hexanomials, 2-to-1 mappings of the form <inline-formula> <tex-math notation="LaTeX">(x^{2^{k}}+x+\delta)^{s_{1}}+(x^{2^{k}}+x+\delta)^{s_{2}}+cx </tex-math></inline-formula>, and 2-to-1 mappings from linear 2-to-1 mappings. We also exhibit the corresponding involutions of the constructed 2-to-1 mappings. Furthermore, an infinite family of involutions with differential uniformity at most 4 (EA-inequivalent to the inverse function) is obtained. Finally, we highlight that all our derived families of involutions have no fixed point, further accentuating their cryptographic interest.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2022.3211329