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The construction of (minimal) linear codes from functions has received much attention in the literature. In this paper, we derive several minimal codes from the sets of pre-images of weakly regular plateaued and bent functions over the odd characteristic finite fields. Based on the recent construction method, we obtain six-weight and seven-weight minimal codes from these functions. The Hamming weights in proposed codes are calculated from the character sums of the sets based on the Walsh spectrum of the employed functions, while the weight distributions of the codes are determined from both the sizes of the employed sets and the Walsh distributions of the employed functions. [ABSTRACT FROM AUTHOR]

Vectorial Boolean functions and codes are closely related and interconnected. On the one hand, various requirements of binary linear codes are needed for their theoretical interests but, more importantly, for their practical applications (such as few-weight codes or minimal codes for secret sharing, locally recoverable codes for storage, etc.). On the other hand, various criteria and tables have been introduced to analyse the security of S-boxes that are related to vectorial Boolean functions, such as the Differential Distribution Table (DDT), the Boomerang Connectivity Table (BCT), and the Differential-Linear Connectivity Table (DLCT). In previous years, two new tables have been proposed for which the literature was pretty abundant: the c-DDT to extend the DDT and the c-BCT to extend the BCT. In the same vein, we propose extended concepts to study further the security of vectorial Boolean functions, especially the c-Walsh transform, the c-autocorrelation, and the c-differential-linear uniformity and its accompanying table, the c-Differential-Linear Connectivity Table (c-DLCT). We study the properties of these novel functions at their optimal level concerning these concepts and describe the c-DLCT of the crucial inverse vectorial (Boolean) function case. Finally, we draw new ideas for future research toward linear code designs. [ABSTRACT FROM AUTHOR]

This paper concerns a classical subject regarding the structural properties of abelian number fields. The primitive elements, integral bases, conductors and discriminants are important for studying abelian number fields, and this paper emphasizes that they are closely related to the associated Dirichlet characters. Then by evaluating the Gauss sums, we can give explicit forms of abelian fields of small degree. [ABSTRACT FROM AUTHOR]

We obtain the Boomerang Connectivity Table of power permutations F (x) = x 2 m − 1 of F 2 n with m ∈ { 3 , n − 1 2 , n + 1 2 , n − 2 }. In particular, we obtain the Boomerang uniformity and the Boomerang uniformity set of F (x) at b ∈ F 2 n ∖ F 2. Moreover we determine the complete Boomerang distribution spectrum of F(x) using the number of rational points of certain concrete algebraic curves over F 2 n . We also determine the distribution spectra of Boomerang uniformities explicitly. [ABSTRACT FROM AUTHOR]

Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In their articles (Ding and Zhou 2014 [11]) and (Ding, 2013 [7]), Ding and Zhou constructed several classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. This article focuses on these exciting works by investigating new insights in this research direction. Specifically, its objective is twofold. The first is to provide a complement with some former results and present correct proofs and statements on some known ones on the cyclic codes from the APN functions. The second is studying the cyclic codes from some known functions processing low differential uniformity. Along with this article, we shall answer some open problems presented in the literature. The first concerns Open Problem 1, proposed by Ding and Zhou (2014) [11]. The others are Open Problems 5.16 and 5.25, raised by Ding (2013) [7]. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a simple graph with vertex set as V and edge set as E. Within the framework of graph-based binary coding, a graph G is a threshold graph if and only if it can be generated by a binary code of the type 0 s 1 1 t 1 ... 0 s k 1 t k , where s i and t i are positive integers with 1 ≤ i ≤ k. This type of binary code is called a binary generating code of G. In this paper, we address a problem originally posed by Harary and Schwenk in [10] titled "Which graphs have integral spectra?" in which the main focus of our investigations lies in the Seidel Laplacian spectrum of G generated by 0 s 1 1 t 1 ... 0 s k 1 t k . We establish that all eigenvalues of G are integers and multiplicities of eigenvalues are equal to powers of bits 0 and 1 (sizes of strings of 0's and 1's) involved in a binary generating code of G. We exhibit a necessary and sufficient condition in terms of eigenvalues of the matrix associated with a graph-based G binary code to be Seidel Laplacian integral. Furthermore, for any prime p , we explore the Seidel Laplacian spectrum of zero-divisor graphs associated with the ring Z p m (where m is a positive integer) and the polynomial ring Z p [ x ] / (x p). [ABSTRACT FROM AUTHOR]

We introduce a class of permutation polynomial over F q n that can be written in the form L (x) x q + 1 or L (x q + 1) x for some q -linear polynomial L over F q n . Specifically, we present those permutation polynomials explicitly as well as their inverses. In addition, more permutation polynomials can be derived in a more general form. [ABSTRACT FROM AUTHOR]

This paper constructs a class of lattices whose discrete analogues are variable-length non-linear codes. The well-known discrete analogue of lattices and linear codes inspires our approach. We next design a variable length binary non-linear code called automorphism orbit code from a finite abelian p -group of rank greater than 1, where p is a prime. For each finite abelian p -group, codewords of the automorphism orbit code are variable-length codewords called automorphism orbit codewords. The homomorphisms between groups determine homomorphism codes , whereas automorphism orbit codes are specified by partitions of a number, orbits of group action, homomorphisms and automorphisms of groups. For some groups G and H , we shall use elements of H o m (G , H) to create a cover relation for bit strings of codewords of an automorphism orbit code and formulate a lattice of variable length non-linear codes. [ABSTRACT FROM AUTHOR]

Minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. When they are defined over finite fields, those codes have been intensively studied, especially in recent years, but they have been firstly partially characterized by Ashikhmin and Barg since 1998. Next, they were completely characterized in 2018 by Ding, Heng, and Zhou in terms of the minimum and maximum nonzero weights in the corresponding codes. Since then, many construction methods for minimal linear codes over finite fields throughout algebraic and geometric approaches have been proposed in the literature. In particular, the algebraic approach gives rise to minimal codes from (cryptographic) functions. Linear codes over finite fields have been expanded into the collection of acceptable alphabets for codes and study codes over finite commutative rings. A natural way to extend the known results available in the literature is to consider minimal linear codes over commutative rings with unity. In extending coding theory to codes over rings, several essential principles must be considered. Particularly extending the minimality property from finite fields to rings and creating such codes is not simple. Such an extension offers more flexibility in the construction of minimal codes. The present article investigates one-dimensional minimal linear codes over the rings $\mathbb {Z}_{p^{n}}$ (where $p$ is a prime) and $\mathbb {Z}_{p^{m}q^{n}}$ (where $p < q$ are distinct primes and $m\leq n$). Our ultimate objective is to characterize such codes’ minimality and design minimal linear codes over the considered rings. Given our objective, we first introduced the notion of minimal codes over (commutative) rings and succeeded in deriving simple characterization of one-dimensional minimal linear codes over the underlying rings mentioned above. Our new algebraic approach allows designing new minimal linear codes. Almost minimal codes over rings are also presented. To the best of our knowledge, the present paper offers a wide variety of minimal codes over (commutative) rings for the first time. Novel perspectives and developments in this direction are expected in the future. [ABSTRACT FROM AUTHOR]

Reed-Muller codes are error-correcting codes used in many areas related to coding theory, such as electrical engineering and computer science. The binary $r^{th}$ -order Reed-Muller code $RM(r,n)$ can be viewed as the set of all $n$ -variable Boolean functions of algebraic degree at most $r$. Despite the intense work on these codes, many problems are known to be hard (notably, determining their covering radius) and remain open to this day. Fourteen years ago, Carlet and Mesnager improved in [IEEE Transactions on Information Theory, “Improving the Upper Bounds on the Covering Radii of Binary Reed-Muller Codes”, 53(1), 2007] the upper bound on the covering radius of the Reed-Muller code of order 2, and they deduced improved upper bounds on the covering radii of the Reed-Muller codes of higher orders. Until 2021, these upper bounds remained the best ones in the literature. The Reed-Muller code $RM(n-3,n)$ , which corresponds to the dual of the Reed-Muller code $RM(2,n)$ , has attracted much attention. One of the main reasons is that it is precisely the code that has been considered to get the upper bounds derived by Carlet and Mesnager. Those upper bounds have been obtained thanks to the characterization of the codewords of the Reed-Muller code, whose Hamming weights are strictly less than 2.5 times the minimum distance $2^{n-r}$ due to Kasami, Tokura, and Azumi. Despite their impressive work in the seventieth, a more refined study and profound description of those codewords of ${RM}({n-3},{n})$ whose Hamming weight equals 16, and especially 18, seem necessary, as it could help us significantly in improving the covering radius of Reed-Muller codes. In this paper, we push further the known results on the Reed-Muller codes by focusing on the Reed-Muller code ${RM}({n-3},{n})$. We provide a classification of the codewords of weights 16 and 18 of the Reed-Muller code $RM(n-3,n)$. Our algebraic descriptions allow us to count the number of such codewords and to enumerate all of them explicitly. [ABSTRACT FROM AUTHOR]

The differential spectrum of a cryptographic function is of significant interest for estimating the resistance of the involved vectorial function to some variances of differential cryptanalysis. It is well-known that it is difficult to determine a power function's differential spectrum completely. In the present article, we concentrate on studying the differential and the c -differential uniformity (for some c ∈ F p n ∖ { 0 , 1 }) and their related differential spectrum (resp. c -differential spectrum) of the power functions F (x) = x d over the finite field F p n of order p n (where p is an odd prime) for d = p n − 3 2. We emphasize that by focusing on the power functions x d with even d over F p n (p odd), the considered functions are APN, that is, of the lowest differential uniformity and the nontrivial differential spectrum. By investigating some system of equations and specific character sums over F p n , the differential spectrum of F is completely determined. Moreover, we examine the extension of the so-called c -differential uniformity by investigating the c -differential properties of F. Specifically, an upper bound of the c -differential uniformity of F is given, and its c -differential spectrum is considered in the case where c = − 1. Finally, we highlight that, throughout our study of the differential spectrum of the considered power functions, we provide methods for evaluating sums of specific characters with connections to elliptic curves and determining the number of solutions of specific systems of equations over finite fields. [ABSTRACT FROM AUTHOR]

Planar functions, introduced by Dembowski and Ostrom, have attracted much attention in the last decade. As shown in this paper, we present a new class of planar functions of the form Tr (a x q + 1) + ℓ (x 2) on an extension of the finite field F q n / F q. Specifically, we investigate those functions on F q 2 / F q and construct several typical kinds of planar functions. We also completely characterize them on F q 3 / F q. When the degree of extension is higher, it will be proved that such planar functions do not exist given certain conditions. [ABSTRACT FROM AUTHOR]

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over F q 2 , whose numerators are so-called q -quadratic polynomials. To this end, we will first determine the exact number of zeros of a special q -quadratic polynomial in F q 2 , by calculating character sums related to quadratic forms of F q 2 / F q. Then given some rational function, we can demonstrate whether it induces a permutation of F q 2 . [ABSTRACT FROM AUTHOR]

The notion of correlation immune functions has been introduced by Siegenthaler (1984) in symmetric cryptography in the framework of stream ciphers. At the conference CRYPTO’91 by Camion et al., it has been pointed out that this notion existed in statistics and combinatorics. It has recently been highlighted that such functions also play an important role in a new framework related to side-channel attack counter-measures. Since then, the interest in correlation immune Boolean functions has been renewed, and new challenges regarding these functions have appeared. Specifically, low Hamming weight correlation immune functions have been selected as useful for counter-measures to side-channel attacks. Despite their importance, the literature is not abundant in this research direction. Two very interesting articles in which such correlation immune functions were nicely explored, given this novel use of them. Carlet initiated the first one in 2013, and the second one is due to Carlet and Chen (2018). This paper deals with correlation immune Boolean functions aiming to produce more candidates of those processing low Hamming weights. We shall focus on correlation immune Boolean functions with Hamming weights power of 2 (which offer a flexibility to control the correlation immunity aspects) and present several methods of designing them. Some design methods are efficient and could be employed to derive such functions. Consequently, given two positive integers $n$ and $m$ , we derive new effective constructions of correlation immune Boolean functions with Hamming weight power of 2. Furthermore, an upper bound on the correlation immunity of the newly constructed $n$ -variable Boolean functions with Hamming weight $2^{m}$ was determined for $n-m\ge 0$. Besides, exact values and lower bounds on the maximum correlation immunity of those functions are explored and discussed, mainly when the values of $n$ and $m$ are very close. This paper also exhibits explicit examples of those correlation immune functions that illustrate our methods. [ABSTRACT FROM AUTHOR]

In a prior paper (Ellingsen et al., 2020), two of us, along with P. Ellingsen, P. Felke, and A. Tkachenko, defined a new (output) multiplicative differential and the corresponding $c$ -differential uniformity, which has the potential of extending differential cryptanalysis. Here, we continue the work by looking at some APN functions through the mentioned concept and showing that their $c$ -differential uniformity increases significantly in some cases. [ABSTRACT FROM AUTHOR]

Plateaued (vectorial) functions over finite fields have diverse applications in symmetric cryptography, coding theory, and sequence theory. Constructing these functions is an attractive research topic in the literature. We can distinguish two kinds of constructions of plateaued functions: secondary constructions and primary constructions. The first method uses already known functions to obtain new functions while the latter do not need to use previously constructed functions to obtain new functions. In this work, the first secondary constructions of (non)weakly regular plateaued (vectorial) functions are presented over the finite fields of odd characteristics. We also introduce some recursive constructions of (non)weakly regular plateaued p-ary functions by using already known such functions. We obtain nontrivial plateaued functions from the previously known trivial plateaued (partially bent) functions in the proposed construction methods. [ABSTRACT FROM AUTHOR]

Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure a Boolean function’s resistance to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. Some of the binary LCD codes presented in this paper are optimal. These binary LCD codes have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems. [ABSTRACT FROM AUTHOR]

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 F2n using 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 + β xℓ + α ∈ F2n[x] with n ≤ 7. We derive twelve classes of 2-to-1 quadrinomials with trivial coefficients over F2n. [ABSTRACT FROM AUTHOR]

Linear codes with few weights have applications in secret sharing schemes, authentication codes, association schemes and strongly regular graphs. In this paper, we introduce a new flexible defining set, and consequently, we obtain several classes of t -weight linear codes over F p , where t = 4 , 5 , 6 and p is an odd prime. The parameters and weight distributions of the generated linear codes are fully determined using quadratic Gauss sums over finite fields. Moreover, two classes of optimal two-weight codes meeting the Griesmer bound are obtained from our constructions. Our results are heavily based on evaluating some exponential sums over finite fields. Computer experiments using MAGMA programs show that many (almost) optimal codes can be derived from our results. [ABSTRACT FROM AUTHOR]

Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of $\mathbb {F}_{q}$ to construct minimal linear codes. By properties of characteristic functions, we can obtain more minimal binary linear codes from known minimal binary linear codes, which generalizes results of Ding et al. [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536–6545, 2018]. By characteristic functions corresponding to some subspaces of $\mathbb {F}_{q}$ , we obtain many minimal linear codes, which generalizes results of [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536–6545, 2018] and [IEEE Trans. Inf. Theory, vol. 65, no. 11, pp. 7067–7078, 2019]. Finally, we use characteristic functions to present a characterization of minimal linear codes from the defining set method and present a class of minimal linear codes. [ABSTRACT FROM AUTHOR]

Minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. There are several methods to construct linear codes, one of which is based on functions over finite fields. Recently, many construction methods for linear codes from functions have been proposed in the literature. In this paper, we generalize the recent construction methods given by Tang et al. in [IEEE Transactions on Information Theory, 62(3), 1166-1176, 2016] to weakly regular plateaued functions over finite fields of odd characteristic. We first construct three-weight linear codes from weakly regular plateaued functions based on the second generic construction and then determine their weight distributions. We also give a punctured version and subcode of each constructed code. We note that they may be (almost) optimal codes and can be directly employed to obtain (democratic) secret sharing schemes, which have diverse applications in the industry. We next observe that the constructed codes are minimal for almost all cases and finally describe the access structures of the secret sharing schemes based on their dual codes. [ABSTRACT FROM AUTHOR]

Minimal linear codes with few weights have significant applications in secure two-party computation and secret sharing schemes. In this paper, we construct two-weight and three-weight minimal linear codes by using weakly regular plateaued functions in the well-known construction method based on the second generic construction. We also give punctured codes and subcodes for some constructed minimal codes. We finally obtain secret sharing schemes with high democracy from the dual codes of our minimal codes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Plateaued and bent functions play a significant role in cryptography, sequence theory, coding theory and combinatorics. In 1997, Coulter and Matthews redefined bent functions over any finite field F q where q is a prime power, and established their properties. The objective of this work is to redefine the notion of plateaued functions over F q , and to present several explicit characterizations of those functions. We first give, over F q , the notion of q -ary plateaued functions, which relies on the concept of the Walsh–Hadamard transform in terms of canonical additive character of F q. We then give a concrete example of q -ary plateaued function, that is not vectorial p -ary plateaued function. This suggests that the study of plateaued-ness is also significant for q -ary functions over F q. We finally characterize q -ary plateaued functions in terms of derivatives, Walsh power moments and autocorrelation functions. [ABSTRACT FROM AUTHOR]

Linear complementary pairs (LCPs) of codes play an important role in armoring implementations against side-channel attacks and fault injection attacks. One of the most common ways to construct LCP of codes is to use Euclidean linear complementary dual (LCD) codes. In this paper, we first introduce the concept of linear codes with $\sigma $ complementary dual ($\sigma $ -LCD), which includes known Euclidean LCD codes, Hermitian LCD codes, and Galois LCD codes. Like Euclidean LCD codes, $\sigma $ -LCD codes can also be used to construct LCP of codes. We show that for $q > 2$ , all $q$ -ary linear codes are $\sigma $ -LCD, and for every binary linear code $\mathcal C$ , the code $\{0\}\times \mathcal C$ is $\sigma $ -LCD. Furthermore, we study deeply $\sigma $ -LCD generalized quasi-cyclic (GQC) codes. In particular, we provide the characterizations of $\sigma $ -LCD GQC codes, self-orthogonal GQC codes, and self-dual GQC codes, respectively. Moreover, we provide the constructions of asymptotically good $\sigma $ -LCD GQC codes. Finally, we focus on $\sigma $ -LCD abelian codes and prove that all abelian codes in a semi-simple group algebra are $\sigma $ -LCD. The results derived in this paper extend those on the classical LCD codes and show that $\sigma $ -LCD codes allow the construction of LCP of codes more easily and with more flexibility. [ABSTRACT FROM AUTHOR]

Linear complementary dual (LCD) cyclic codes were referred historically to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In particular, it has been shown that binary LCD codes play an important role in implementations against side-channel attacks and fault injection attacks. In this paper, we first present a new characterization of binary LCD codes in terms of their orthogonal or symplectic basis. Using such a characterization, we solve a conjecture proposed by Galvez et al. on the minimum distance of binary LCD codes. Next, we consider the action of the orthogonal group on the set of all LCD codes, determine all possible orbits of this action, derive simple closed formulas of the size of the orbits, and present some asymptotic results on the size of the corresponding orbits. Our results show that almost all binary LCD codes are odd-like codes with odd-like duals, and about half of $q$ -ary LCD codes have orthonormal basis, where $q$ is a power of an odd prime. [ABSTRACT FROM AUTHOR]

Linear codes over finite chain rings meeting the Griesmer bound are particularly interesting since they are closely related to minihyper in projective Hjelmslev spaces. In this paper, motivated by the nice article (Li et al., 2018 [23]), two classes of linear codes with few non-zero Hamming weights over the finite chain ring R = F q + u F q are constructed, and one of them meets the Griesmer bound over finite commutative local rings. By a Gray map, we obtain that the Gray images of the constructed codes over the ring R are linear codes with two weights over the finite field F q with q elements, and these codes are optimal with respect to the well-known Griesmer bound under certain conditions. Our results give an affirmative answer to two (related) open questions raised in the article mentioned above. Remarkably, some of the constructed linear codes over R presented in this paper correspond to minihypers in projective Hjelmslev spaces over finite chain rings. Besides, their Gray images can be used to construct secret sharing schemes with nice access structures and other applications thanks to the minimality property that possesses our constructed codes. [ABSTRACT FROM AUTHOR]

This paper contributes to increase our knowledge on generalized bent functions (including generalized bent Boolean functions and generalized $p$ -ary bent functions with odd prime $p$ ) by bringing new results on their characterization and construction in arbitrary characteristic. More specifically, we first investigate relations between generalized bent functions and bent functions by the decomposition of generalized bent functions. This enables us to completely characterize generalized bent functions and $\mathbb Z_{p^{k}}$ -bent functions by some affine space associated with the generalized bent functions. We also present the relationship between generalized bent Boolean functions with an odd number of variables and generalized bent Boolean functions with an even number of variables. Based on the well-known Maiorana-McFarland class of Boolean functions, we present some infinite classes of generalized bent Boolean functions. In addition, we introduce a class of generalized hyperbent functions that can be seen as generalized Dillon’s $PS$ functions. Finally, we solve an open problem related to the description of the dual function of a weakly regular generalized bent Boolean function with an odd number of variables via the Walsh–Hadamard transform of their component functions, and we generalize these results to the case of odd prime. [ABSTRACT FROM AUTHOR]

Linear complementary dual (LCD) codes are a class of linear codes introduced by Massey in 1964. LCD codes have been extensively studied in literature recently. In addition to their applications in data storage, communications systems, and consumer electronics, LCD codes have been employed in cryptography. More specifically, it has been shown that LCD codes can also help improve the security of the information processed by sensitive devices, especially against so-called side-channel attacks (SCA) and fault non-invasive attacks. In this paper, we are interested in the construction of particular algebraic geometry LCD codes which could be good candidates to be resistant against SCA. We firstly provide a construction scheme for obtaining LCD codes from any algebraic curve. Then, some explicit LCD codes from elliptic curves are presented. Maximum distance separable (MDS) codes are of the most importance in coding theory due to their theoretical significance and practical interests. In this paper, all the constructed LCD codes from elliptic curves are MDS or almost MDS. Some infinite classes of LCD codes from elliptic curves are optimal due to the Griesmer bound. Finally, we also derive some explicit LCD codes from hyperelliptic curves and Hermitian curves. [ABSTRACT FROM AUTHOR]

In recent literature, a family of optimal linear locally recoverable codes (LRC codes) that attain the maximum possible distance (given code length, cardinality, and locality) is presented. The key ingredient for constructing such optimal linear LRC codes is the so-called $r$ -good polynomials, where r$ is equal to the locality of the LRC code. However, given a prime p$ , known constructions of r exist only for some special integers r$ , and the problem of constructing optimal LRC codes over small field for any given locality is still open. In this paper, by using function composition, we present two general methods of designing good polynomials, which lead to three new constructions of r$ -good polynomials. Such polynomials bring new constructions of optimal LRC codes. In particular, our constructed polynomials as well as the power functions yield optimal (n,k,r) for all positive integers $r$ as localities, where $q$ is near the code length $n$ . [ABSTRACT FROM AUTHOR]

Plateaued functions are very important crypto- graphic functions due to their various desirable cryptographic characteristics. We point out that plateaued functions are more general than bent functions (that is, functions with maximum nonlinearity). Some Boolean plateaued functions have large nonlinearity, which provides protection against fast correlation attacks when they are used as combiners or filters in stream ciphers, and contributes, when they are the component functions of the substitution boxes in block ciphers, to protection against linear cryptanalysis. P-ary plateaued functions have attracted recently some attention in the literature, and many activities on generalized $p$ -ary functions have been carried out. This paper increases our knowledge on plateaued functions in the general context of generalized p$ -ary functions. We first introduce two new versions of plateaued functions, which we shall call generalized plateaued functions and admissible plateaued functions. The generalized plateaued functions extend the standard notion of plateaued p$ -ary functions to those whose outputs are in the ring . Next, we study the generalized plateaued functions and use admissible plateaued functions to characterize the generalized plateaued functions by means of their components. Finally, we provide for the first time two constructions of generalized plateaued functions. In particular, we generalize a known secondary construction of binary generalized bent functions and derive constructions of binary generalized plateaued functions with different amplitudes. [ABSTRACT FROM PUBLISHER]

Let S 2 be the set of b 's such that the equation X 2 3 n + 2 2 n + 2 n − 1 + (X + 1) 2 3 n + 2 2 n + 2 n − 1 = b has two solutions in F 2 4 n . This set was originally defined in the article ["Solving X 2 3 n + 2 2 n + 2 n − 1 + (X + 1) 2 3 n + 2 2 n + 2 n − 1 = b in F 2 4 n ″ ([1]) and an alternative proof of a conjecture on the differential spectrum of the related monomial functions. Finite Fields and Their Applications , 83, 102086, 2022]. We show that the definition of S 2 (initially considered from our computational experiments for n ≤ 8) in [1] was, unfortunately, introduced incorrectly. We, therefore, provide in this note the necessary corresponding corrections. More precisely, we present the correct versions of Theorem 2 and Corollary 11, respectively. [ABSTRACT FROM AUTHOR]

In the present article, we provide a complete characterization of permutations on the finite field F 4 m of shape f ϵ _ (X) : = ϵ 1 X ‾ q + 1 + ϵ 2 X ‾ q X + ϵ 3 X ‾ X q + ϵ 4 X q + 1 , definitively (where q = 2 k , Q = 2 m , m is odd, (m , k) = 1 , X ‾ = X Q). Our achievement firstly extends the very recent literature (a long series of interesting recent (2019-2022) results derived in (at least) seven articles) about the bijectivity on F Q 2 of f ϵ _ . It also gives a complete proof of the conjecture (Conjecture 19) posed in the recent literature by Li et al. (2021) [24] about the characterization of permutations f ϵ _ with 4-uniform BCT (which can be viewed, thanks to their nice cryptographic properties, as promising candidates as S-boxes to be used in designing secure block ciphers in symmetric cryptography). To derive our results about the quadrinomials f ϵ _ , we shall keep the essence of our successful approach initiated by Kim et al. in [23] by relating the problem of characterization of the bijectivity of f ϵ _ over F Q 2 to the equation X q + 1 + X + a = 0 but also by performing algebraic developments with new nature by considering some rational functions over finite fields. Besides, we shall employ novel algebraic techniques and derive new auxiliary results (some of them have their independent interest) going beyond the recent results given in [23]. Notably, we present a complete proof of the bijectivity of f ϵ _ over F Q 2 without any restriction. Specifically, our results are valid when the coefficients ϵ i (i ∈ { 1 , 2 , 3 , 4 }) lie in F Q 2 and for any value of k ≥ 1 (at the opposite of the results in [23] valid with the condition that the coefficients ϵ i (i ∈ { 1 , 2 , 3 , 4 }) lie in the finite field F Q while covering all values of k ≥ 1). Finally, we emphasize that very recently, Göloğlu has presented in (Göloğlu, 2022 [13]), an alternative novel approach providing a complete description of permutations of the form mentioned above using biprojective polynomials. [ABSTRACT FROM AUTHOR]

Carlet provides two bounds on the second-order nonlinearity of Boolean functions. We construct a family of Boolean functions where the first bound (the presumed weaker bound) is tight and the second bound is strictly worse than the first bound. We show that the difference between the two bounds can be made arbitrarily large. [ABSTRACT FROM AUTHOR]

This article determines all the solutions in the finite field F 2 4 n of the equation x 2 3 n + 2 2 n + 2 n − 1 + (x + 1) 2 3 n + 2 2 n + 2 n − 1 = b. Specifically, we explicitly determine the set of b 's for which the equation has i solutions for any positive integer i. Such sets, which depend on the number of solutions i , are given explicitly and expressed nicely, employing the absolute trace function over F 2 n , the norm function over F 2 4 n relatively to F 2 n and the set of (2 n + 1) st roots of unity in F 2 4 n . The equation considered in this paper comes from an article by Budaghyan et al. ([2]) in which the authors have investigated novel approaches for obtaining alternative representations for functions from the known infinite APN families. In particular, they have been interested in determining the differential spectrum of some power functions among them is the one F (x) = x 2 3 n + 2 2 n + 2 n − 1 defined over F 2 4 n . The problem of the determination of such spectrum has led to a conjecture (Conjecture 27 in the preprint (2020) [2] for which an updated version will appear in 2022 at the IEEE Transactions Information Theory) stated by Budaghyan et al. As an immediate consequence of our results, we prove that the above equation has 2 2 n solutions for one value of b , 2 2 n − 2 n solutions for 2 n values of b in F 2 4 n and has at most two solutions for all remaining points b , leading to complete proof of the conjecture raised by Budaghyan et al. We highlight that the recent work of Li et al., in [9] gives the complete differential spectrum of F and also gives an affirmative answer to the conjecture of Budaghyan et al. However, we emphasize that our approach is interesting and promising by being different from Li et al. Indeed, on the opposite to their article, our technique allows to determine ultimately the set of b 's for which the considered equation has solutions as well as the solutions of the equation for any b in F 2 4 n . [ABSTRACT FROM AUTHOR]

Because of the recent applications to distributed storage systems, researchers have introduced a new class of block codes, i.e., locally recoverable (LRC) codes. LRC codes can recover information from erasure(s) by accessing a small number of erasure-free code symbols and increasing the efficiency of repair processes in large-scale distributed storage systems. In this context, Tamo and Barg first gave a breakthrough by cleverly introducing a good polynomial notion. Constructing good polynomials for locally recoverable codes achieving Singleton-type bound (called optimal codes) is challenging and has attracted significant attention in recent years. This article aims to increase our knowledge of good polynomials for optimal LRC codes. Using tools from algebraic function fields and Galois theory, we continue investigating those polynomials and studying them by developing the Galois theoretical approach initiated by Micheli in 2019. Specifically, we push further the study of a crucial parameter G (f) (of a given polynomial f), which measures how much a polynomial is "good" in the sense of LRC codes. We provide some characterizations of polynomials with minimal Galois groups and prove some properties of finite fields where polynomials exist with a specific size of Galois groups. We also present some explicit shapes of polynomials with small Galois groups. For some particular polynomials f , we give the exact formula of G (f). [ABSTRACT FROM AUTHOR]

An involution is a permutation, such that its inverse is itself (i.e., cycle length ≤ 2). Due to this property, involutions have been used in many applications, including cryptography and coding theory. In this paper, we provide a systematic study of involutions that are defined over a finite field of characteristic 2. We characterize the involution property of several classes of polynomials and propose several constructions. Furthermore, we study the number of fixed points of involutions, which is a pertinent question related to permutations with short cycle. In this paper, we mostly have used combinatorial techniques. [ABSTRACT FROM AUTHOR]

Bent functions are optimal combinatorial objects. Since their introduction, substantial efforts have been directed toward their study in the last three decades. A complete classification of bent functions is elusive and looks hopeless today, therefore, not only their characterization, but also their generation are challenging problems. This paper is devoted to the construction of bent functions. First, we provide several new effective constructions of bent functions, self-dual bent functions, and antiself-dual bent functions. Second, we provide seven new infinite families of bent functions by explicitly calculating their dual. [ABSTRACT FROM AUTHOR]

The connection between exponential sums and algebraic varieties has been known for at least six decades. Recently, Lisoněk exploited it to reformulate the Charpin-Gong characterization of a large class of hyper-bent functions in terms of numbers of points on hyperelliptic curves. As a consequence, he obtained a polynomial time and space algorithm for certain subclasses of functions in the Charpin-Gong family. In this paper, we settle a more general framework, together with detailed proofs, for such an approach and show that it applies naturally to a distinct family of functions proposed by Mesnager. Doing so, a polynomial time and space test for the hyper-bentness of functions in this family is obtained as well. Nonetheless, a straightforward application of such results does not provide a satisfactory criterion for explicit generation of functions in the Mesnager family. To address this issue, we show how to obtain a more efficient test leading to a substantial practical gain. We finally elaborate on an open problem about hyperelliptic curves related to a family of Boolean functions studied by Charpin and Gong. [ABSTRACT FROM AUTHOR]

We show that any Boolean function, in even dimension, equal to the sum of a Boolean function g which is constant on each element of a spread and of a Boolean function h whose restrictions to these elements are all linear, is semibent if and only if g and h are both bent. We deduce a large number of infinite classes of semibent functions in explicit bivariate (respectively, univariate) polynomial form. [ABSTRACT FROM PUBLISHER]

Kloosterman sums have recently become the focus of much research, most notably due to their applications in cryptography and coding theory. In this paper, we extensively investigate the link between the semibentness property of functions in univariate forms obtained via Dillon and Niho functions and Kloosterman sums. In particular, we show that zeros and the value four of binary Kloosterman sums give rise to semibent functions in even dimension with maximum degree. Moreover, we study the semibentness property of functions in polynomial forms with multiple trace terms and exhibit criteria involving Dickson polynomials. [ABSTRACT FROM AUTHOR]

Abstract: One of the classes of bent Boolean functions introduced by John Dillon in his thesis is family H. While this class corresponds to a nice original construction of bent functions in bivariate form, Dillon could exhibit in it only functions which already belonged to the well-known Maiorana–McFarland class. We first notice that H can be extended to a slightly larger class that we denote by . We observe that the bent functions constructed via Niho power functions, for which four examples are known due to Dobbertin et al. and to Leander and Kholosha, are the univariate form of the functions of class . Their restrictions to the vector spaces , , are linear. We also characterize the bent functions whose restrictions to the ʼs are affine. We answer the open question raised by Dobbertin et al. (2006) in on whether the duals of the Niho bent functions introduced in the paper are affinely equivalent to them, by explicitly calculating the dual of one of these functions. We observe that this Niho function also belongs to the Maiorana–McFarland class, which brings us back to the problem of knowing whether H (or ) is a subclass of the Maiorana–McFarland completed class. We then show that the condition for a function in bivariate form to belong to class is equivalent to the fact that a polynomial directly related to its definition is an o-polynomial (also called oval polynomial, a notion from finite geometry). Thanks to the existence in the literature of 8 classes of nonlinear o-polynomials, we deduce a large number of new cases of bent functions in , which are potentially affinely inequivalent to known bent functions (in particular, to Maiorana–McFarlandʼs functions). [Copyright &y& Elsevier]

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They were introduced by Rothaus in 1976. For their own sake as interesting combinatorial objects, but also because of their relations to coding theory (Reed–Muller codes) and applications in cryptography (design of stream ciphers), they have attracted a lot of research, specially in the last 15 years. The class of bent functions contains a subclass of functions, introduced by Youssef and Gong in 2001, the so-called hyper-bent functions, whose properties are still stronger and whose elements are still rarer than bent functions. Bent and hyper-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. So, it is important to design constructions in order to know as many of (hyper)-bent functions as possible. This paper is devoted to the constructions of bent and hyper-bent Boolean functions in polynomial forms. We survey and present an overview of the constructions discovered recently. We extensively investigate the link between the bentness property of such functions and some exponential sums (involving Dickson polynomials) and give some conjectures that lead to constructions of new hyper-bent functions. [ABSTRACT FROM AUTHOR]

The recent algebraic attacks have received a lot of attention in cryptographic literature. The algebraic immunity of a Boolean function quantifies its resistance to the standard algebraic attacks of the pseudorandom generators using it as a nonlinear filtering or combining function. Very few results have been found concerning its relation with the other cryptographic parameters or with the rth-order nonlinearity. As recalled by Carlet at CRYPTO'06, many papers have illustrated the importance of the rth-order nonlinearity profile (which includes the first-order nonlinearity). The role of this parameter relatively to the currently known attacks has been also shown for block ciphers. Recently, two lower bounds involving the algebraic immunity on the rth-order nonlinearity have been shown by Carlet et al.. None of them improves upon the other one in all situations. In this paper, we prove a new lower bound on the rth-order nonlinearity profile of Boolean functions, given their algebraic immunity, that improves significantly upon one of these lower bounds for all orders and upon the other one for low orders. [ABSTRACT FROM AUTHOR]

By deriving bounds on character sums of Boolean functions and by using the characterizations, due to Kasami et al., of those elements of the Reed-Muller codes whose Hamming weights are smaller than twice and a half the minimum distance, we derive an improved upper bound on the covering radius of the Reed-Muller code of order 2, and we deduce improved upper bounds on the covering radii of the Reed-Muller codes of higher orders. [ABSTRACT FROM AUTHOR]

In this paper, based on a characterization of epimorphisms of R-algebras given by Roby [15], we bring an algorithm testing whether a given finitely generated morphism f : A → B, where A and B are finitely presented affine algebras over the same Nœtherian commutative ring R, is an epimorphism of R-algebras or not. We require two computational conditions on R, which we call a computational ring. [ABSTRACT FROM AUTHOR]

The construction of the normalization of an affine domain over a field is a classical problem solved since sixteen''s by Stolzenberg (1968) and Seidenberg (1970–1975) thanks to classical algebraic methods and more recently by Vasconcelos (1991–1998) and de Jong (1998) thanks to homological methods. The aim of this paper is to explain how to use such a construction to obtain effectively the integral closure of such a domain in any finite extension of its quotient field, thanks to Dieudonné characterization of such an integral closure. As application of our construction, we explain how to obtain an effective decomposition of a quasi-finite and dominant morphism from a normal affine irreducible variety to an affine irreducible variety as a product of an open immersion and a finite morphism, conformly to the classical Grothendieck''s version of Zariski''s main theorem. [Copyright &y& Elsevier]

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

Let l and k be two integers such that l | k. Define T l k (X) : = X + X p l + ⋯ + X p k − 2 l + X p k − l and S l k (X) : = X − X p l + ⋯ + (− 1) (k / l − 1) X p k − l , where p is any prime. This paper gives explicit representations of all solutions in F p n to the affine equations T l k (X) = a and S l k (X) = a , a ∈ F p n . The case p = 2 was solved very recently in [10]. The results of this paper reveal another solution. [ABSTRACT FROM AUTHOR]

Blockchain systems store transaction data in the form of a distributed ledger where each node stores a copy of all data, which gives rise to storage issues. It is well-known that the tremendous storage and distribution of the block data are common problems in blockchain systems. In the literature, some types of secret sharing schemes are employed to overcome these problems. The secret sharing method is one of the most significant cryptographic protocols used to ensure the privacy of the data. The main purpose of this paper is to improve the recent distributed storage blockchain systems by proposing an alternative secret sharing method. We first propose a secure threshold verifiable multi-secret sharing scheme that has the verification and private communication steps based on post-quantum lattice-based hard problems. We then apply the proposed threshold scheme to the distributed storage blockchain (DSB) system to share transaction data at each block. In the proposed DSB system, we encrypt the data block with the AES-256 encryption algorithm before distributing it among nodes at each block, and both its secret key and the hash value of the block are privately shared among nodes simultaneously by the proposed scheme. Thereafter, in the DSB system, the encrypted data block is encoded by the Reed–Solomon code, and it is shared among nodes. We finally analyze the storage and recovery communication costs and the robustness of the proposed DSB system. We observe that our approach improves effectively the recovery communication cost and makes it more robust compared to the previous DSB systems. It also improves extremely the storage cost of the traditional blockchain systems. Furthermore, the proposed scheme brings to the DSB system the desirable properties such as verification process and secret communication without private channels in addition to the known properties of the schemes used in the previous DSB systems. As a result of the flexibility on the threshold parameter of the scheme, a diverse range of qualified subsets of nodes in the DSB system can privately recover the secret values. [ABSTRACT FROM AUTHOR]