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
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Abstract	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.
AbstractList	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 [Formula Omitted] of order [Formula Omitted], we shall intensively use 2-to-1 mappings over [Formula Omitted]. 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 [Formula Omitted], 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. 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.
Author	Zheng, Dabin Mesnager, Sihem Yuan, Mu
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SubjectTerms	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
Title	More About the Corpus of Involutions From Two-to-One Mappings and Related Cryptographic S-Boxes
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