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Some new techniques and progress towards the resolution of the conjecture of exceptional apn functions and absolutely irreducibility of a class of polynomials

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DESIGNS CODES AND CRYPTOGRAPHY 2023年第7期91卷 2481-2495页

作者： Delgado, Moises Janwa, Heeralal Agrinsoni, Carlos Univ Puerto Rico Dept Math 205 Calle Antonio R Barcelo Cayey PR 00736 USA Univ Puerto Rico Dept Math 17 Univ Ave Ste 1701 San Juan PR 00925 USA

Almost Perfect Nonlinear (apn) functions are very useful in cryptography, when they are used as S-Boxes due to their good resistance to differential cryptanalysis. Also, some of the curves and surfaces defined by the corresponding nonlinear functions have many rational points and have applications to Algebraic-Geometric (AG) codes (Janwa and Wilson in Applied algebra, algebraic algorithms and error-correcting codes (San Juan, PR, 1993), vol 673, pp 180-194, Springer, Berlin, 1993, in IEEE Trans Inform Theory, accepted). An apn function f : F-2(n) -> F-2(n) is called an exceptional apn if it is apn on infinitely many extensions of F2n. Aubry et al. (Contemp Math 518:23-31, 2010) conjectured that the only exceptional apn functions are the Gold and the Kasami-Welch monomial functions. They established that a polynomial function of odd degree is not an exceptional apn provided the degree is not a Gold number (2(k) + 1) or a Kasami-Welch number (2(2k) - 2(k) + 1). When the degree of the polynomial function is a Gold number, several partial results are known. Here, we prove the relative primeness conjecture of the Gold degree polynomials. This result helps us substantially to make advances toward the resolution of the exceptional apn conjecture in the Gold degree case. We prove that Gold degree polynomials of the form x(2k+1) + h(x), where deg(h) is an odd integer, cannot be exceptional apn (with a few natural exceptions). We also prove that the absolutely irreducible components of the KasamiWelch degree curves intersect transversally at a particular point. Consequently, we prove that Kasami-Welch degree functions of type x(22k-)2(k)+1 + h(x) produce absolutely irreducible surfaces in the case deg(h) = 3 (mod 4);and also deg(h) = 2(m-1) + 1 (mod 2(m)) under certain conditions. As a result, we show that the exceptional apn conjecture is true for these classes as well.

关键词： apn functions Exceptional apn functions Janwa-McGuire-Wilson conjecture Absolutely irreducible polynomials S-boxes Differential cryptanalysis

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On the exceptionality of rational apn functions

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DESIGNS CODES AND CRYPTOGRAPHY 2023年第10期91卷 3167-3186页

作者： Bartoli, Daniele Fatabbi, Giuliana Ghiandoni, Francesco Univ Perugia Dipartimento Matemat & Informat Perugia Italy Univ Perugia Dipartimento Matemat & Informat Via Vanvitelli Perugia Italy Univ Firenze Dipartimento Matemat & Informat Ulisse Dini Florence Italy

We investigate apn functions which can be represented as rational functions and we provide non-existence results exploiting the connection between these functions and specific algebraic varieties over finite fields. T... 详细信息

关键词： apn functions Algebraic varieties Finite fields

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Constructing New apn functions Through Relative Trace functions

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IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第11期68卷 7528-7537页

作者： Zheng, Lijing Kan, Haibin Li, Yanjun Peng, Jie Tang, Deng Univ South China Sch Math & Phys Hengyang 421001 Hunan Peoples R China Fudan Univ Shanghai Key Lab Intelligent Informat Proc Sch Comp Sci Shanghai 200433 Peoples R China Fudan Univ Shanghai Engn Ctr Blockchains Shanghai 200433 Peoples R China Anhui Univ Finance & Econ Inst Stat & Appl Math Bengbu 233030 Anhui Peoples R China Shanghai Normal Univ Math & Sci Coll Shanghai 200234 Peoples R China Shanghai Jiao Tong Univ Sch Elect Informat & Elect Engn Shanghai 200240 Peoples R China

Let n = 2m. In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered a family of quadrinomials over F-2n of the form x(3) + a(x(2s+1))(2k) + bx(3.2m) + c(x(2s+m) + 2(m))2(k). They showed that two infinite classes of almost perfect nonlinear (apn) functions belong to this family when gcd(6, m) = 1. We observe that these two infinite classes of apn quadrinomials and the infinite class of apn polynomials from the Budaghyan-Carlet family belong to a more general family of polynomials over F-2n with the form f (x) = aTr(m)(n)(F (x)) + a(2m) Tr-m(n)(G(x)), where a is an element of F-2n \ F-2m, and both F and G are quadratic functions over F(2)n. We characterize when f (x) is apn. With the help of our characterization, letting F(x) = bx(2i+1) and G(x) = cx(2s)(+1) with b, c is an element of F-2n, we obtain an infinite family of apn functions of the form f (x) when gcd(2, m) = 1 and verify that for n = 10 two apn instances from this infinite family are CCZ-inequivalent to each other, and to any apn function over F-210 from the previously known infinite families.

关键词： apn functions relative trace functions quadratic functions CCZ-equivalence

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On the infiniteness of a family of apn functions

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JOURNAL OF ALGEBRA 2022年 598卷 68-84页

作者： Bartoli, Daniele Calderini, Marco Polverino, Olga Zullo, Ferdinando Univ Perugia Dipartimento Matemat & Informat Perugia Italy Univ Trento Dipartimento Matemat Trento Italy Univ Campania Luigi Vanvitelli Dipartimento Matemat & Fis Caserta Italy

apn functions play a fundamental role in cryptography against attacks on block ciphers. Several families of quadratic apn functions have been proposed in the recent years, whose construction relies on the existence of specific families of polynomials. A key question connected with such constructions is to determine whether such apn functions exist for infinitely many dimensions or do not. In this paper we consider a family of functions recently introduced by Li et al. in 2021 showing that for any dimension m >= 3 there exists an apn function belonging to such a family. Our main result is proved by a combination of different techniques arising from both algebraic varieties over finite fields connected with linearized permutation rational functions and partial vector space partitions, together with investigations on the kernels of linearized polynomials. (c) 2022 Elsevier Inc. All rights reserved.

关键词： apn functions Algebraic varieties Finite fields

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Generalized isotopic shift construction for apn functions

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DESIGNS CODES AND CRYPTOGRAPHY 2021年第1期89卷 19-32页

作者： Budaghyan, Lilya Calderini, Marco Carlet, Claude Coulter, Robert Villa, Irene Univ Bergen Dept Informat PB 7803 N-5020 Bergen Norway Univ Paris 08 LAGA St Denis France Univ Delaware Dept Math Sci Newark DE 19716 USA

In this work we give several generalizations of the isotopic shift construction, introduced recently by Budaghyan et al. (IEEE Trans Inform Theory 66:5299-5309, 2020), when the initial function is a Gold function. In particular, we derive a general construction of apn functions which covers several unclassified apn functions for n = 8 and produces fifteen new apn functions for n = 9.

关键词： apn functions Isotopic shift Vectorial Boolean functions

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On the linear structures of balanced functions and quadratic apn functions

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN functions AND SEQUENCES 2020年第5期12卷 859-880页

作者： Musukwa, A. Sala, M. Mzuzu Univ P Bag 201 Luwinga 2 Mzuzu Malawi Univ Trento Via Sommar 14 I-38123 Povo Trento Italy

The set of linear structures of most known balanced Boolean functions is non-trivial. In this paper, some balanced Boolean functions whose set of linear structures is trivial are constructed. We show that any apn function in even dimension must have a component whose set of linear structures is trivial. We determine a general form for the number of bent components in quadratic apn functions in even dimension and some bounds on the number are produced. We also count bent components in any quadratic power functions.

关键词： Boolean functions Linear space apn functions Bent functions

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On equations of finite fields of characteristic 2 and apn functions

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AKCE INTERNATIONAL JOURNAL OF GRAPHS AND COMBINATORICS 2015年第2-3期12卷 75-93页

作者： Nakagawa, Nobuo Kinki Univ Res Ctr Quantum Comp 3-4-1 Kowakae Higashiosaka Osaka 5778502 Japan

Let F be the finite field GF(2(n)) of characteristic 2 and f a quadratic apn function on F. We construct a n(n-3)/2 dimensional subspace W-f of F boolean AND F, where F boolean AND F is the alternative product of F, that is, the quotient space of the tensor product F circle times F of F by the subspace < x circle times x vertical bar x is an element of F >. We denote by S the set of all subspaces W-f constructed from quadratic apn functions f on F. We prove that a group G isomorphic to GL(n, 2) acts on S and that there exists a one-to-one correspondence between the extended affine equivalence classes of quadratic apn functions and the set of G-orbits on S. The correspondence above was first observed by Yoshiara in Section 5 of Yoshiara (2010) and Edel (2011) and the author Nakagawa (2009). Moreover we prove F boolean AND F is isomorphic to Fn-1/2 (for n odd) or Fn/2-1 x GF(2(n/2)) (for n even) through an explicit isomorphism, and we give practical forms of those subspaces which correspond to the Gold function f (x) = x(2k+1) where gcd(n, k) = 1 and to the function f (x) = x(3) + Tr(x(9)), viewed as subspaces of Fn-1/2 (for n odd) or Fn/2-1 x GF(2(n/2)) (for n even) (see Section 6 in Yoshiara, 2010). We estimate the number of solutions of linear equations x(2e+1) + alpha x(2e) + beta x(2) + (alpha + beta + 1)x = 0 on GF(2(2e)), and then construct some quadratic apn functions. (C) 2015 Kalasalingam University. Production and Hosting by Elsevier B.V.

关键词： Finite fields apn functions Permutation groups

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On the functions which are CCZ-equivalent but not EA-equivalent to quadratic functions over Fpⁿ

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FINITE FIELDS AND THEIR APPLICATIONS 2025年 103卷

作者： Jeong, Jaeseong Koo, Namhun Kwon, Soonhak NIMS Dept Innovat Ctr Ind Math Seongnam South Korea Ewha Womans Univ Inst Math Sci Seoul 03760 South Korea Sungkyunkwan Univ Appl Algebra & Optimizat Res Ctr Suwon 16420 South Korea

For a given function F from F-p(n) to itself, determining whether there exists a function which is CCZ-equivalent but EA-inequivalent to F is a very important and interesting problem. For example, Kolsch [33] showed that there is no function which is CCZ-equivalent but EA-inequivalent to the inverse function. On the other hand, for the cases of Gold function F(x)=x(2i+1) and F(x)=x(3)+Tr(x(9)) over F-2(n), Budaghyan, Carlet and Pott (respectively, Budaghyan, Carlet and Leander) [12], [14] found functions which are CCZ-equivalent but EA-inequivalent to F. In this paper, when a given function F has a component function which has a linear structure, we present functions which are CCZ-equivalent to F, and if suitable conditions are satisfied, the constructed functions are shown to be EA-inequivalent to F. As a consequence, for every quadratic function F on F-2(n) (n >= 4) with nonlinearity greater than 0 and differential uniformity not exceeding 2(n-3), we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to F. Also for every non-planar quadratic function on F-p(n) (p>2,n >= 4) with |W-F|<= p(n-1) and differential uniformity not exceeding p(n-3), we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to F. As an application, for a proper divisor m of n, we present many examples of (n,m)-functions F on F-p(n) such that the CCZ-equivalence class of F is strictly larger than the EA-equivalence class of F. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Boolean function Nonlinearity Walsh transform apn functions CCZ-equivalence

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On Dillon's property of (n, m)-functions

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN functions AND SEQUENCES 2024年第6期16卷 1449-1473页

作者： Abbondati, Matteo Calderini, Marco Villa, Irene Univ Trento Dept Math Via Sommar 14 I-38123 Povo TN Italy Univ Montpellier CNRS LIRMM Montpellier France Univ Genoa Dept Math Via Dodecaneso 35 I-16146 Genoa Italy

Dillon observed that an apn function F over F-2(n) with n greater than 2 must satisfy the condition {F(x) + F(y) + F(z) + F(x + y + z) :x, y, z is an element of F-2(n)} = F-2(n). Recently, Taniguchi (Cryptogr. Commun. 15, 627-647 2023) generalized this condition to functions defined from F-2(n) to F-2(m), with m > n, calling it the D-property. Taniguchi gave some characterizations of apn functions satisfying the D-property and provided some families of apn functions from F-2(n) to F-2(n+1) satisfying this property. In this work, we further study the D-property for (n, m)-functions with m >= n. We give some combinatorial bounds on the dimension m for the existence of such functions. Then, we characterize the D-property in terms of the Walsh transform and for quadratic functions we give a characterization of this property in terms of the ANF. We also give a simplification on checking the D-property for quadratic functions, which permits to extend some of the apn families provided by Taniguchi. We further focus on the class of the plateaued functions, providing conditions for the D-property. To conclude, we show a connection of some results obtained with the higher-order differentiability and the inverse Fourier transform.

关键词： Vectorial Boolean functions D-property apn functions Walsh transform Plateaued functions

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Sidon sets, thin sets, and the nonlinearity of vectorial Boolean functions

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JOURNAL OF COMBINATORIAL THEORY SERIES A 2025年 212卷

作者： Nagy, Gabor P. Tech Univ Budapest Inst Math & Econ Muegyet Rkp 3 H-1111 Budapest Hungary Univ Szeged Bolya Inst Arad Vertanuk Tere 1 H-6720 Szeged Hungary

The vectorial nonlinearity of a vector-valued function is its distance from the set of affine functions. In 2017, Liu, Mesnager, and Chen conjectured a general upper bound for the vectorial linearity. Recently, Carlet established a lower bound in terms of differential uniformity. In this paper, we improve Carlet's lower bound. Our approach is based on the fact that the level sets of a vectorial Boolean function are thin sets. In particular, level sets of apn functions are Sidon sets, hence the Liu-Mesnager-Chen conjecture predicts that in F 2 n , there should be Sidon sets of size at least 2 n / 2 + 1 for all n. This paper provides an overview of the known large Sidon sets in F 2 n , and examines the completeness of the large Sidon sets derived from hyperbolas and ellipses of the finite affine plane. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Sidon sets Vectorial Boolean functions apn functions Hamming distance Error correcting codes

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