Secret sharing was introduced by Shamir to distribute a secret among a group of participants. Later, Massey generalized this approach using minimal linear codes. Recent research has extensively explored the basic properties, code parameters, and constructions of minimal linear codes from algebraic, combinatorial, and geometric perspectives. In this paper, we further generalize Massey’s scheme to multi-secret sharing by introducinghigher-order minimal linear codes. We present some fundamental properties of these codes, including two necessary and sufficient conditions forrth-order minimality. These conditions generalize the Heng–Ding–Zhou and the Lu–Wu–Cao conditions for ordinary minimal linear codes, respectively. We also derive bounds on the parameters of higher-order minimal linear codes. Importantly, we prove that these codes are asymptotically good, exhibiting both non-vanishing rate and relative distance as the code length tends to infinity. A geometric characterization establishes an equivalence betweenrth-order minimal linear codes and cuttingr-blocking sets in projective geometry. Finally, we provide higher-order minimal linear codes using punctured simplex codes, Kronecker products and subfield subcodes, respectively.

The problem of designing explicit bent and plateaued functions has been researched for several decades. However, finding new bent functions outside the well-known completed Maiorana-McFarland class$\mathcal {M}^{\#}$is still a challenge. Plateaued functions have been characterized in many different ways, but there is no general and rigorous mathematical method to generate them directly, except for the ones in the spirit of the well-known Maiorana-McFarland constructions or those obtained through adaptations of the secondary constructions of bent functions. Jeong and Lee recently made significant advances regarding algorithms for constructing balanced plateaued functions with maximal algebraic degrees in [IEEE Trans. Inf. Theory, 70(2), 1408-1421, 2024]. Due to the gap between our significant interest in the notion of plateaued functions and the knowledge we have on it, our motivation is to bring further results on the constructions of plateaued functions that allow us to understand their structure better. This article creates a framework of new generic constructions of bent and plateaued functions by studying Boolean functions of the form$h(x)=f(x)+F(f_{1}(x),\ldots, f_{r}(x))$, where$f_{i}(x)=f(x)+f(x+\mu _{i})$for each$1\leq i\leq r$. We firstly prove that h and f have the same extended Walsh-Hadamard spectrum if$D_{\mu _{i}}D_{\mu _{j}}f=0$for any$1\leq i\lt j\leq r$. This result extends a previous construction of bent functions to any Boolean functions. The strength of such a result is that it allows us to obtain several plateaued functions of high algebraic degrees from known ones with low algebraic degrees, which was a significant and challenging problem raised in the literature. Such a result is a real challenge and breaks a deadlock since no mathematical method allows the general constructions of plateaued functions. We next give an extended affine equivalent form of the function h, which provides us with another compelling perspective to design new bent functions (including those which are outside$\mathcal {M}^{\#}$from certain known ones inside$\mathcal {M}^{\#}$) and plateaued functions. Finally, we present four generic constructions of bent functions outside$\mathcal {M}^{\#}$from generalized Maiorana-McFarland functions.

Recently, much progress has been made to construct minimal linear codes due to their preference in secret sharing schemes and secure two-party computation. In this paper, we put forward a new method to construct minimal linear codes by using vectorial Boolean functions. Firstly, we give a necessary and sufficient condition for a generic class of linear codes from vectorial Boolean functions to be minimal. Based on that, we derive some new three-weight minimal linear codes and determine their weight distributions. Secondly, by studying deeply the construction of linear codes in this paper, we find a necessary and sufficient condition of the linear codes to be minimal and to be violated the AB condition. As a result, we get three infinite families of minimal linear codes violating the AB condition. To the best of our knowledge, this is the first time that minimal liner codes are constructed from vectorial Boolean functions. Compared the parameters with other known ones, in general the minimal liner codes obtained in this paper have higher dimensions.

This article is devoted to Boolean and vectorial bent functions and their duals. Our ultimate objective is to increase such functions’ corpus by designing new ones covering many previous bent functions’ constructions. To this end, we provide several new infinite families of bent functions, including idempotent bent functions of any algebraic degree, bent functions in univariate trace form, and self-dual bent functions. Those bent functions are of great theoretical and practical interest because of their special structures and relationship with self-dual codes. In particular, many well-known bent functions are special cases of our bent functions. Moreover, we extend our results to vectorial bent functions and obtain three new infinite classes of vectorial bent functions of any possible degree by determining the explicit duals of three classes of well-known bent functions.

Let$n=2m$. In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered a family of quadrinomials over$\mathbb {F}_{2^{n}}$of the form$x^{3}+a(x^{2^{s}+1})^{2^{k}}+bx^{3\cdot 2^{m}}+c(x^{2^{s+m}+2^{m}})^{2^{k}}$. 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$\mathbb {F}_{2^{n}} $with the form$f(x)=a{\mathrm{ Tr}}^{n}_{m}(F(x))+a^{2^{m}}{\mathrm{ Tr}}^{n}_{m}(G(x))$, where$a \in \mathbb {F}_{2^{n}}\backslash \mathbb {F}_{2^{m}} $, and both$F$and$G$are quadratic functions over$\mathbb {F}_{2^{n}}$. We characterize when$f(x) $is APN. With the help of our characterization, letting$F(x)=bx^{2^{i}+1} $and$G(x)=cx^{2^{s}+1}$with$b, c\in \mathbb {F}_{2^{n}} $, we obtain an infinite family of APN functions of the form$f(x) $when${\mathrm{ gcd}}(2,m)=1 $and verify that for$n=10 $two APN instances from this infinite family are CCZ-inequivalent to each other, and to any APN function over$\mathbb {F}_{2^{10}} $from the previously known infinite families.

Abstract In 2012, Carlet et al. developed two secondary constructions of bent functions (Advances in Mathematics of Communications, 6: 305‐314) and proposed some applications for their constructions. However, the duals of bent functions in their constructions were not presented. In order to find more general applications to these constructions and obtain new classes of bent functions, an open problem was proposed by Carlet in 2014. Hence, in this study, a class of vectorial bent functions for answering that open problem, which also addresses another open problem on vectorial bent functions proposed by Mesnager in 2014, is constructed. In addition, a new secondary construction of bent functions that generalises one of Carlet et al.'s constructions in 2012 is presented. Based on that, two new classes of bent functions were obtained and their duals were presented explicitly. In particular, some self‐dual bent functions are constructed. Moreover, it can be proved that our bent functions can be EA‐inequivalent to those constructed by Carlet et al. in 2012.

Given a positive even integer n, it is found that the weight distribution of any n-variable symmetric Boolean function with maximum algebraic immunity (AI) n/2 is determined by the binary expansion of n . Based on the foregoing, all n-variable symmetric Boolean functions with maximum AI are constructed. The amount is (2 wt(n)+1)2[log2n].

In this paper, we put forward an efficient method to study the symmetric Boolean functions with high algebraic immunity on even number of variables. We obtain some powerful necessary conditions for symmetric Boolean functions to achieve high algebraic immunity by studying the weight support of some specific types of Boolean functions of low degrees. With these results, we prove that the algebraic immunity of a large class of symmetric correlation immune Boolean functions, namely the symmetric palindromic functions, is not high. Besides, we construct all symmetric Boolean functions with maximum algebraic immunity and give a description for those with submaximum algebraic immunity. We also determine the Hamming weight, degrees and nonlinearity of the symmetric Boolean functions with maximum algebraic immunity.

It is known that Boolean functions used in stream and block ciphers should have good cryptographic properties to resist algebraic attacks. Up until now, there have been several constructions of Boolean functions achieving optimum algebraic immunity. However, most of their nonlinearities are very low. Carlet and Feng studied a class of Boolean functions with optimum algebraic immunity and deduced the lower bound of its nonlinearity, which is good, but not very high. Moreover, the main practical problem with this construction is that it cannot be implemented efficiently. In this paper, we put forward a new method to construct cryptographically significant Boolean functions by using primitive polynomials, and construct three infinite classes of Boolean functions with good cryptographic properties: balancedness, optimum algebraic degree, optimum algebraic immunity, and a high nonlinearity.