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 give many new Singleton type upper bounds on the sizes of codes with given minimum Hamming distances. These upper bounds are stronger than the Griesmer bound when the lengths of codes are large. Some upper bounds on the lengths of general small Singleton defect codes are presented. Our generalized Singleton type upper bounds have wide applications to symbol-pair codes, insertion-deletion codes and locally recoverable codes. The generalized Singleton type upper bounds on symbol-pair codes and insertion-deletion codes are much stronger than the direct Singleton bounds on symbol-pair codes and insertion-deletion codes when the lengths are large and the Hamming minimum distances are small. Upper bounds on the lengths of small dimension optimal locally recoverable codes and small dimension optimal$(r, \delta)$locally recoverable codes with any given minimum distance are also presented.

Identity-based encryption with equality test (IBEET) provides a feasible way for cloud server partitioning or searching on ciphertexts in the cloud. In that case, the server can judge if two different ciphertexts encrypt the same plaintexts or not. In response to threats posed by quantum computers, lattice-based IBEET schemes have been proposed to make cloud service post-quantum secure. However, those schemes are inefficient and can hardly meet the needs of resource-constrained devices. In this article, an efficient post-quantum IBEET scheme is introduced, which achieves testability by embedding the hash value of plaintext into testing trapdoor instead of encrypting it directly and doubling the size of ciphertext. We also prove that, with the Learning With Errors (LWE) problem assumption, the new scheme is one-way secure against selective identity and chosen ciphertext attacks (OW-sID-CCA) in quantum secure model. Furthermore, we evaluate the performance of the new construction and demonstrate its efficiency by showing that it only costs about half storage comparing with other lattice-based IBEET schemes. The execution time of encryption and decryption phrases in the new scheme reduce by 50%, while the computational cost in test algorithm keeps the same. Therefore, the new proposed IBEET is much more practical for working in post-quantum cloud computing scenarios.

In response to the threat posed by quantum computers, Pilaram and Eghlidos proposed the first lattice-based multi-stage secret sharing scheme which is the only post-quantum multi-stage secret sharing scheme. In this paper, we introduce an efficient attack on it and show that any adversary can easily reconstruct unrecovered secrets as long as it collects enough pseudo-secret shares. For the sake of complete, we further list two countermeasures to protect the scheme from such attack.

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.

Combinatorial designs are closely related to linear codes. Recently, some near MDS codes were employed to construct$t$-designs by Ding and Tang, which settles the question as to whether there exists an infinite family of near MDS codes holding an infinite family of$t$-designs for$t \geq 2$. This paper is devoted to the construction of infinite families of 3-designs and 2-designs from special equations over finite fields. First, we present an infinite family of almost MDS codes over${\mathrm{ GF}}(p^{m})$holding an infinite family of 3-designs. We then provide an infinite family of almost MDS codes over${\mathrm{ GF}}(p^{m})$holding an infinite family of 2-designs for any field${\mathrm{ GF}}(q)$. In particular, some of these almost MDS codes are near MDS. Second, we present an infinite family of near MDS codes over${\mathrm{ GF}}(2^{m})$holding an infinite family of 3-designs by considering the number of roots of a special linearized polynomial. Compared to previous constructions of 3-designs or 2-designs from linear codes, the parameters of some of our designs are new and flexible.

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.

Abstract Subversion of cryptography has received wide attentions especially after the Snowden Revelations in 2013. Most of the currently proposed subversion attacks essentially rely on the freedom of randomness choosing in the cryptographic protocol to hide backdoors embedded in the cryptosystems. Despite the fact that significant progresses in this line of research have been made, most of them mainly considered the classical setting, while the research gap regarding subversion attacks against post-quantum cryptography remains tremendous. Inspired by this observation, we investigate a subversion attack against existing protocol that is proved post-quantum secure. Particularly, we show an efficient way to undetectably subvert the well-known lattice-based encryption scheme proposed by Regev (STOC 2005). Our subversion enables the subverted algorithm to stealthily leak arbitrary messages to the outsider who knows the backdoor. Through theoretical analysis and experimental observations, we demonstrate that the subversion attack against the LWE encryption scheme is feasible and practical.

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.

Two-to-one (2-to-1) mappings over finite fields play an important role in symmetric cryptography. In particular they allow to design APN functions, bent functions and semi-bent functions. In this paper we provide a systematic study of two-to-one mappings that are defined over finite fields. We characterize such mappings by means of the Walsh transforms. We also present several constructions, including an AGW-like criterion, constructions with the form of$x^{r}h(x^{(q-1)/d})$, those from permutation polynomials, from linear translators and from APN functions. Then we present 2-to-1 polynomial mappings in classical classes of polynomials: linearized polynomials and monomials, low degree polynomials, Dickson polynomials and Muller-Cohen-Matthews polynomials, etc. Lastly, we show applications of 2-to-1 mappings over finite fields for constructions of bent Boolean and vectorial bent functions, semi-bent functions, planar functions and permutation polynomials. In all those respects, we shall review what is known and provide several new results.

Minimal linear codes are a special subclass of linear codes and have significant applications in secret sharing and secure two-party computation. In this paper, we focus on constructing minimal linear codes with (wmin)/(wmax) ≤ (p-1)/p for any odd prime p based on a generic construction of linear codes, where wminand wmaxdenote the minimum and maximum nonzero weights in a code, respectively. First, we present two new infinite families of minimal linear codes with two or three weights by selecting suitable subcode of linear codes which are not minimal. Second, we also present an infinite family of minimal linear codes by employing partial spreads, which can be viewed as a generalization of the construction of Ding et al. In addition, we determine the weight distributions of all these minimal linear codes.

Planar functions over finite fields give rise to finite projective planes. They were also used in the constructions of DES-like iterated ciphers, error-correcting codes, and codebooks. They were originally defined only in finite fields with odd characteristic, but recently Zhou introduced pesudo-planar functions in even characteristic, which yields similar applications. All known pesudo-planar functions are quadratic, and hence, they give presemifields. In this paper, a new approach to constructing quadratic pseudo-planar functions is given. Then, five explicit families of pseudo-planar functions are constructed, one of which is a binomial, two of which are trinomials, and the other two are quadrinomials. All known pesudo-planar functions are revisited, some of which are generalized. These functions not only lead to projective planes, relative difference sets, and presemifields, but also give optimal codebooks meeting the Levenstein bound, complete sets of mutually unbiased bases and compressed sensing matrices with low coherence.

Permutation polynomials have been a subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are described in the literature so far. In this paper we present a number of permutation trinomials over finite fields, which are of different forms.

In this paper, we investigate the permutation property of the Dickson polynomials $E_n(x, a)$ of the second kind over $\mathbb{Z}_m$. Due to a known result, it suffices to consider permutation polynomials $E_n(x, a)$ over $\mathbb{Z}_{p^t}$, where $p$ is a prime and $t$ is a positive integer. We identify all permutation polynomials of $E_n(x, a)$ over $\mathbb{Z}_{p^t}$ for (I) $p=2$ and (II) $p$ is odd and $a$ is a square over $\mathbb{Z}_p$. For odd $p$ and nonsquares $a$ in $\mathbb{Z}_p$, we determine a large class (if not all) of permutation polynomials $E_n(x, a)$ over $\mathbb{Z}_{p^t}$. A conjecture is also presented in this paper. If this conjecture is true, then all Dickson permutation polynomials $E_n(x, a)$ of the second kind over $\mathbb{Z}_m$ are determined.

In this paper, a new method is presented to compute the 2-adic complexity of pseudo-random sequences. With this method, the 2-adic complexities of all the known sequences with ideal 2-level autocorrelation are determined in a unified way. Results show that their 2-adic complexities equal their periods. In other words, their 2-adic complexities attain the maximum. In addition, 2-adic complexities of two classes of optimal autocorrelation sequences with period N ≡ 1mod4, namely Legendre sequences and Ding-Helleseth-Lam sequences, are investigated. This method also can be used to compute the linear complexity of binary sequences regarded as sequences over other finite fields.

In this study, the minimal polynomials and the linear complexity of interleaved binary sequences are investigated. Both the linear complexity and the minimal polynomials of low correlation zone sequences constructed by Zhou et al. are completely determined. Besides, an open problem proposed by Li and Tang is discussed. At last, a sufficient condition and a necessary condition are presented about when the linear complexity of the interleaved sequences constructed by Tang et al. attains the maximum.

Almost perfect nonlinear (APN) functions on ${\mathbb F}_{2^n}$ are functions achieving the lowest possible differential uniformity. All APN functions discovered until now are either power or quadratic ones, except for one sporadic multinomial nonquadratic example on ${\mathbb F}_{2^6}$ due to Edel and Pott. It is well known that certain binary codes with good properties can be obtained from APN functions, and determining their (Hamming) weight distribution is equivalent to determining the Fourier spectra of the corresponding functions. The Fourier spectra of all known infinite families of quadratic APN functions discovered through 2010 have been determined, and it was found that they are the same as the ones of the Gold APN functions, i.e., a $5$-valued set when $n$ is even and a $3$-valued set when $n$ is odd, while a sporadic example on ${\mathbb F}_{2^6}$ found by Dillon has a $7$-valued Fourier spectrum. In 2011, two new generic constructions of APN functions were presented in [Y. Zhou and A. Pott, Adv. Math., 234 (2013), pp. 43--60] and [C. Carlet, Des. Codes Cryptogr., 59 (2011), pp. 89--109]. In this paper, we determine the Fourier spectra of the APN functions obtained from them and show that their Fourier spectra are again the same as those of the Gold APN functions. Moreover, since the APN functions in [C. Bracken, C. H. Tan, and Y. Tan, On a Class of Quadratic Polynomials with No Zeros and Its Applications to APN Functions, preprint, arXiv:1110.3177v1, 2011], which are demonstrated to exist when $n\equiv 0\mod 4$ and $3\nmid n$, are covered by the construction in [C. Carlet, Des. Codes Cryptogr., 59 (2011), pp. 89--109], a positive answer to the conjecture proposed in the former paper on determining their Fourier spectrum is given in this paper.

Many block ciphers use permutations defined on F(22k) with low differential uniformity, high nonlinearity, and high algebraic degree as their S-boxes to provide confusion. It is well known that, for a function on F(2n), the lowest differential uniformity is 2 and the functions achieving this lower bound are called almost perfect nonlinear (APN) functions. However, due to the lack of knowledge on APN permutations on F(22k), differentially 4-uniform permutations are usually chosen as S-boxes. For example, the currently endorsed Advanced Encryption Standard chooses one such function, the multiplicative inverse function, as its S-box. By a recent survey on differentially 4-uniform permutations over F(22k), there are only five known infinite families of such functions, and most of them have small algebraic degrees. In this paper, we apply the powerful switching method to discover many CCZ-inequivalent infinite families of such functions on F(22k) with optimal algebraic degree, wherekis an arbitrary positive integer. This greatly expands the list of differentially 4-uniform permutations and hence provide more choices for the S-boxes. Furthermore, lower bounds for the nonlinearity of the functions obtained in this paper are presented and they imply that some infinite families have high nonlinearity.

Rotation symmetric Boolean functions (RSBFs) that are invariant under circular translation of indices have been used as components of different cryptosystems. In this paper, even-variable-balanced RSBFs with maximum algebraic immunity (AI) are investigated. At first, we give an original construction of 2m-variable-balanced RSBFs with maximum AI. Then we improve the construction to obtain more 2m-variable-balanced RSBFs with maximum AI, and these new RSBFs have higher non-linearity than all previously obtained RSBFs. Further, we generalise our construction of 2m-variable RSBFs to a new construction that can generate any even-variable RSBFs.

In this paper, the weight distributions of two classes of linear codes based on all known explicit perfect nonlinear functions fromFqmto itself are determined using a unified approach. All the minimal codewords of these codes are characterized according to their weights, which suggests that their covering structures are determined. Finally, all the minimal access sets of the secret sharing schemes based on their dual codes are obtained.

Symmetric Boolean functions with even variables2kand maximum algebraic immunity AI(f)=khave been constructed in Braeken's thesis (2006). In this paper, we show more constructions of such Boolean functions including the generalization of a result and prove a conjecture raised in Braeken's thesis (2006).

In this correspondence, we introduce a method to construct Boolean functions in any number of variables, with optimal algebraic immunity. Remarkably, all functions of this type with an odd number of variables can be obtained in this way. We study some cryptographic properties, such as balancedness, algebraic degree of the constructed functions. Moreover, a lower bound of the number of Boolean functions with optimal algebraic immunity is given.

In this note, it is proved that for each odd positive integer n there are exactly two n-variable symmetric Boolean functions with maximum algebraic immunity.