Bent functions have a wide range of applications in combinatorial designs, error-correcting codes, sequences, and other domains. The class of 2 n -variable bent functions D , defined by functions of the form f ( x , y ) = x ⋅ π ( y ) + 1 E 1 ( x ) 1 E 2 ( y ) , was initially proposed by Carlet (1994) three decades ago as a construction based on the permutation π , but only one explicit subclass D 0 was presented. To date, only three explicit constructions of D class bent functions have been identified. Moreover, for any bent function f in D (excluding D 0 ), the problem of whether f is equivalent to a function in one of the known primary classes of bent functions (such as PS , M ) remains mainly open. In this paper, we investigate the algebraic structure of bent functions in the D class and their relationship to the completed Maiorana–McFarland class M # . Our primary contribution is to establish a complete characterization of D ∩ M # using the algebraic properties of the permutation π and the subspace E 1 under the condition that dim ( E 1 ) < n − 3 . Specifically, we prove that f belongs to M # if the permutation π is affine, thereby resolving an open problem posed in Zhang et al. (2020). In addition, we show that f is outside M # if π is not affine on any ( n − k − 1 ) -dimensional subspace of E 2 , which generalizes a prior result from Kudin et al. (2022). Furthermore, we demonstrate that the probability of a 2 n -variable bent function of the PS a p class also being in D 1 approaches zero as n increases by comparing their algebraic ranks. Finally, as an application, we construct two new infinite families of bent functions in the D 1 class that lie outside M # .

Linear codes have attracted considerable attention in coding theory and cryptography due to their significant applications in secret sharing schemes, secure two-party computation, and Galois geometries, among others. As two special subclasses of linear codes, minimal linear codes and self-orthogonal linear codes are of particular interest. Constructing linear codes that possess both minimality and self-orthogonality is very interesting. The main purpose of this paper is to construct self-orthogonal minimal linear codes that violate the Ashikhmin-Barg (AB for short) condition over the finite field Fp. First, we present several classes of self-orthogonal minimal linear codes violating the AB condition over the finite field F2and determine their weight distributions. Next, for any odd primep, we construct two classes of self-orthogonal linear codes fromp-ary functions, which contain some optimal or almost optimal codes. Finally, based on plateaued functions, we construct two classes of self-orthogonal linear codes that violate the AB condition. Their weight distributions are also provided. To the best of our knowledge, this paper is the first to investigate the constructions of linear codes that violate the AB condition and satisfy self-orthogonality.

In this paper, we try to apply methods from topological data analysis (TDA) to study geometric properties invariant under code equivalence transformation, especially for the linear codes associated with almost perfect nonlinear (APN) functions which offer optimal resistance to differential attacks and are very important in the design of block ciphers in cryptography. By employing persistent homology from TDA and tools from graph theory, we present new CCZ-invariants for APN functions. Some of them are computationally efficient and sufficient to distinguish many known APN functions, includingx3andx9(resp.x33) over F27(resp. F29) for which previously known invariants fail to do so.

We present two infinite families of APN functions in trivariate form over finite fields of the form${\mathbb F}_{2^{3m}}$. We show that the functions from both families are permutations when$m$is odd, and are 3-to-1 functions when$m$is even. In particular, our functions are AB permutations for$m$odd. Furthermore, we observe that for$m = 3$, i.e. for${\mathbb F}_{2^{9}}$, the functions from our families are CCZ-equivalent to the two bijective sporadic APN instances discovered by Beierle and Leander. We thus generalize these sporadic instances into an infinite family of APN functions. We also perform an exhaustive computational search for quadratic APN functions with binary coefficients in trivariate form over${\mathbb F}_{2^{3m}}$with$m \le 5$and report on the results.

In this paper, we present two new families of APN functions. The first family is in bivariate form$\big (x^{3}+xy^{2}+ y^{3}+xy, x^{5}+x^{4}y+y^{5}+xy+x^{2}y^{2} \big)\,\,\vphantom {_{\int _{\int }}}$over${\mathbb F}_{2^{m}}^{2}$. It is obtained by adding certain terms of the form$\sum _{i}(a_{i}x^{2^{i}}y^{2^{i}},b_{i}x^{2^{i}}y^{2^{i}})$to a family of APN functions recently proposed by Gölo&gcaron;lu. The$\vphantom {_{\int _{\int }}}$second family has the form$L(z)^{2^{m}+1}+vz^{2^{m}+1}$over${\mathbb F}_{{2^{3m}}}$, which generalizes a family of APN functions by Bracken et al. from 2011. By calculating the$\Gamma $-rank of the constructed APN functions over${\mathbb F}_{2^{8}}$and${\mathbb F}_{2^{9}}$, we demonstrate that the two families are CCZ-inequivalent to all known families. In addition, the two new families cover two known sporadic APN instances over${\mathbb F}_{2^{8}}$and${\mathbb F}_{2^{9}}$, which were found by Edel and Pott in 2009 and by Beierle and Leander in 2021, respectively.

In this paper, we apply two-to-one functions over b F2nin two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) (x2t+x)ewith gcd(t, n)=gcd(e, 2n-1)=1. Based on the study of the Walsh transforms of those functions or their variants, we present many classes of linear codes with few nonzero weights, including one weight, three weights, four weights, and five weights. The weight distributions of the proposed codes with one weight and with three weights are determined. In addition, we discuss the minimum distance of the dual of the constructed codes and show that some of them achieve the sphere packing bound. Moreover, examples show that some codes in this paper have best-known parameters.

In this paper, by the Hasse-Weil bound, we determine the necessary and sufficient condition on coefficients$a_{1},a_{2},a_{3}\in {\mathbb F} _{2^{n}}$with$n=2m$such that$f(x) = {x}^{3\cdot 2^{m}} + a_{1}x^{2^{m+1}+1} + a_{2} x^{2^{m}+2} + a_{3}x^{3}$is an APN function over${\mathbb F}_{2^{n}}$. Our work together with the follow-up work by Chase and Lisoněk indicates that all such APN quadrinomials$f(x)$are affine equivalent to two instances of Gold functions, which resolves the first half of an open problem by Carlet at the International Workshop on the Arithmetic of Finite Fields, 83-107, 2014.

2-to-1 mappings over finite fields play an important role in symmetric cryptography, particularly in the constructions of APN functions, bent functions, and semi-bent functions. Very recently, Mesnager and Qu [IEEE Trans. Inf. Theory 65 (12): 7884-7895] provided a systematic study of 2-to-1 mappings over finite fields. In particular, they determined all 2-to-1 mappings of degree at most 4 over any finite field. Besides, another research direction is to consider 2-to-1 polynomials with few terms. Some results about 2-to-1 monomials and binomials have been obtained in [IEEE Trans. Inf. Theory 65 (12): 7884-7895]. Motivated by their work, in this present paper, we push further the study of 2-to-1 mappings, particularly over finite fields with characteristic 2 (binary case being the most interesting for applications). Firstly, we completely determine 2-to-1 polynomials with degree 5 over \mathbb F2nusing the well-known Hasse-Weil bound. Besides, we consider 2-to-1 mappings with few terms, mainly trinomials and quadrinomials. Using the multivariate method and the resultant of two polynomials, we present two classes of 2-to-1 trinomials, which explain all the examples of 2-to-1 trinomials of the form xk+βxl+ αx ∈ \mathbb F2n[x] with n ≤ 7. We derive twelve classes of 2-to-1 quadrinomials with trivial coefficients over \mathbb F2n.

Permutation polynomials and their compositional inverses have wide applications in cryptography, coding theory, and combinatorial designs. Motivated by several previous results on finding compositional inverses of permutation polynomials of different forms, we propose a general method for finding these inverses of permutation polynomials constructed by the AGW criterion. As a result, we have reduced the problem of finding the compositional inverse of such a permutation polynomial over a finite field to that of finding the inverse of a bijection over a smaller set. We demonstrate our method by interpreting several recent known results, as well as by providing new explicit results on more classes of permutation polynomials in different types. In addition, we give new criteria for these permutation polynomials being involutions. Explicit constructions are also provided for all involutory criteria.

In this paper, we aim to construct a class of complete permutations$\mathcal F$over$\mathbb F_{q}^{n}$from some polynomials$f_{1},f_{2},\ldots,f_{n}$over$\mathbb F_{q}$. First of all, we determine a necessary and sufficient condition such that$\mathcal F$is complete. Briefly, we transform the completeness of$\mathcal F$into showing the permutation properties of two polynomials over$\mathbb F_{q}$obtained from these$f_{i}$’s. Then, following the wide applications, we investigate the constructions of linear complete permutations over$\mathbb F_{2}^{n}$based on the rotations andXORs. The following two cases are considered: the first one is to use some different circularly left shift transforms$f_{i}$’s and the second one is to assume$f_{i}$’s are of the form$b_{i}f$with a fixed$f$and different$b_{i}$’s in$\mathbb F_{q}$. In both cases, we show that the completeness of the permutation is closely related to the ranks of some matrices with particular forms, which can be determined by the cycle decomposition of the permutation over the$n$branches. Besides, we present several explicit linear complete permutations which might be used in the design as well as the provable security of cryptographic schemes.

In EUROCRYPT 2018, Cid et al. introduced a new concept on the cryptographic property of S-boxes: boomerang connectivity table (BCT for short) for evaluating the subtleties of boomerang-style attacks. Very recently, BCT and the boomerang uniformity, the maximum value in BCT, were further studied by Boura and Canteaut. In this paper, aiming at providing new insights, we show some new results about BCT and the boomerang uniformity of permutations in terms of theory and experiment. First, we present an equivalent technique to compute BCT and the boomerang uniformity, which seems to be much simpler than the original definition by Cid et al. Second, thanks to Carlet's idea, we give a characterization of functions f from F2nto itself with boomerang uniformity δfby means of the Walsh transform. Third, by our method, we consider boomerang uniformities of some specific permutations, mainly the ones with low differential uniformity. Finally, we obtain another class of 4-uniform BCT permutation polynomials over F2n.