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This paper focuses on hull dimensional codes obtained by the intersection of linear codes and their dual. These codes were introduced by Assmus and Key and have been the subject of significant theoretical and practical research over the years, gaining increased attention in recent years. Let F q denote the finite field with q elements, and let G be a finite Abelian group of order n. The paper investigates Abelian codes defined as ideals of the group algebra F q G with coefficients in F q . Specifically, it delves into Abelian hull dimensional codes in the group algebra F q G , where G is a finite Abelian group of order n with gcd (n , q) = 1 . Specifically, we first examine general hull Abelian codes and then narrow its focus to Abelian one-dimensional hull codes. Next, we focus on Abelian one-dimensional hull codes and present some necessary and sufficient conditions for characterizing them. Consequently, we generalize a recent result on Abelian codes and show that no binary or ternary Abelian codes with one-dimensional hulls exist. Furthermore, we construct Abelian codes with one-dimensional hulls by generating idempotents, derive optimal ones with one-dimensional hulls, and establish several existing results of Abelian codes with one-dimensional hulls. Finally, we develop enumeration results through a simple formula that counts Abelian codes with one-dimensional hulls in F q G . These achievements exploit the rich algebraic structure of those Abelian codes and enhance and increase our knowledge of them by considering their hull dimensions, reducing the gap between their interests and our understanding of them. [ABSTRACT FROM AUTHOR]

Subspace codes have important applications in random network coding. It is a classical problem to construct subspace codes where both their size and their minimum distance are as large as possible. In particular, cyclic constant dimension subspace codes have additional properties which can be used to make encoding and decoding more efficient. In this paper, we construct large cyclic constant dimension subspace codes with minimum distances 2 k - 2 and 2k. These codes are contained in G q (n , k) , where G q (n , k) denotes the set of all k-dimensional subspaces of the finite filed F q n of q n elements (q a prime power). Consequently, some results in [7, 15], and [23] are extended. [ABSTRACT FROM AUTHOR]

Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on extensions of finite fields, especially for the cases of quadratic and cubic extensions. Our achievements are obtained by investigating absolute irreducibility of some polynomials in two indeterminates. [ABSTRACT FROM AUTHOR]

Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in multiple access communication systems in code division. Firstly, this paper studies the Jacobi sums over Galois rings of arbitrary characteristics and completely determines their absolute values. This extends the work by Li et al. (Sci China 56(7):1457–1465, 2013), where the Jacobi sums over some Galois rings with characteristics of a prime square were discussed. It is worth mentioning that the generalization is not trivial, as the Galois rings of arbitrary characteristics have a more complicated structure. Our deterministic construction of codebooks is based on Jacobi sums over Galois rings of arbitrary characteristics, producing asymptotically optimal codebooks for the Welch bound. Finally, compared to the literature, this article proposes for the first time design of codebooks based on Jacobi sums over Galois rings. In addition, the parameters of the presented codebooks are new. [ABSTRACT FROM AUTHOR]

Boosted by cryptography and coding theory applications and rich connections to objects from geometry and combinatorics, bent functions and related functions developed into a lively research area. In the mid-seventies, Rothaus initially introduced bent functions in the Boolean case, but later, they extended in the p-ary case where p is any prime. Such an extension brought more rich connections to bent functions. In addition to the theory of Fourier transform, handling p-ary bent functions (where p is an odd prime), analyzing their properties and designing them require using the theory of cyclotomic fields. Such a family is classified into weakly regular bent and regular bent. Compared to the Boolean case, the class of p-ary bent functions inherits a much larger variety of properties. For example, it contains the class of dual-bent functions as a proper subclass, which again includes weakly regular bent functions as a proper subclass. Weakly regular bent functions can also be employed in many domains. In particular, they have been widely used in designing good linear codes for several applications (such as secret sharing and two-party computation), association schemes, and strongly regular graphs. More exploration is still needed despite a lot of interest and attention in this topic to increase our knowledge of bent functions and design them. In particular, only a few constructions of weakly regular bent functions have been presented in the literature. Nice construction methods of these functions have been given by Tang et al (IEEE Trans Inf Theory 62(3):1166–1176, 2016). All the known infinite families of weakly regular bent functions belong to a class (denoted by Tang et al. in the previous reference) R F . This paper is devoted to weakly regular bent functions, and its objective is twofold. First, it aims to generate new infinite families of weakly regular bent functions living outside R F and secondly, to exploit the constructed functions to design new families of p-ary linear codes and investigate their use for some standard application after studying this minimality based on their weight distributions. More specifically, we present several classes of weakly regular bent functions obtained from monomial bent functions by modifying the values of some known weakly regular bent functions on some subsets of the finite field F p n (n is a positive integer). Our weakly regular bent functions are of degree either p or p + 1 2 . We also explicitly determine their corresponding dual functions. Finally, we exploit our constructions to derive four new classes of linear codes with three to seven weights. We also show that two of them lead to minimal codes, which are more appropriate for several concrete applications. [ABSTRACT FROM AUTHOR]

Locally recoverable codes (LRCs) have been introduced as a family of erasure codes that support the repair of a failed storage node by contacting a small number of other nodes in the cluster. Boosted by their applications in distributed storage, LRCs have attracted a lot of attention in recent literature since the concept of codes with locality r was introduced by Gopalan et al. in 2012. Aiming to recover the data from several concurrent node failures, the concept of r-locality was later generalized as (r , δ) -locality by Prakash et al. An (r , δ) -LRCs in which every code symbol has (r , δ) -locality is said to be optimal if it achieves the Singleton-like bound with equality. In present paper, we are interested in optimal (r , δ) -LRCs over small fields, more precisely, over quaternary field. We study their parity-check matrices or generator matrices, using the properties of projective space. The classification of optimal quaternary (r , δ) -LRCs and their explicit code constructions are proposed by examining all possible parameters. [ABSTRACT FROM AUTHOR]

Binary linear codes with few weights have wide applications in communication, secret sharing schemes, authentication codes, association schemes, strongly regular graphs, etc. Projective binary linear codes are among the most important subclasses of binary linear codes for practical applications. In this paper, motivated by the two excellent recent papers (Li et al. in IEEE Trans Inf Theory 67(7):4263–4275, 2021) and (Wang et al. in IEEE Trans Inf Theory 67(8):5133–5148, 2021), several new families of few-weight projective binary linear codes are constructed from the defining sets, and then their Hamming weight distributions are determined by employing the Walsh transform of the corresponding two-to-one functions over finite fields with the even characteristic. Our constructions can produce binary linear codes with new parameters. Some of the constructed binary linear codes are optimal or almost optimal according to the online Database of Grassl, and the duals of some of them are distance-optimal with respect to the sphere packing bound. This paper also shows once again that the two-to-one functions initially studied in (Mesnager and Qu in IEEE Trans Inf Theory 65(12):7884–7895, 2019) are also promising objects in coding theory. Although our derived codes use objects considered in the very recent literature, the analysis of our designed codes involves functions having different algebraic structures (and, therefore, other Walsh transform distribution) and requires solving new systems of equations over finite fields, which is an essential step in determining the weight distribution of our constructed codes. As applications, some of the codes presented in this paper can be used to construct association schemes and secret sharing schemes with interesting access structures. [ABSTRACT FROM AUTHOR]

Motivated by recent works on the butterfly structure, particularly by its generalization introduced by Canteaut et al. (IEEE Trans Inf Theory 63(11):7575–7591, 2017), we first push further the study of permutation polynomials over binary finite fields by completely characterizing those permutations f ϵ ̲ defined over the finite field F Q 2 (of order Q 2 ) having the following shape: f ϵ ̲ (X) : = ϵ 1 X ¯ q + 1 + ϵ 2 X ¯ q X + ϵ 3 X ¯ X q + ϵ 4 X q + 1 where q = 2 k , Q = 2 m , m is odd, gcd (m , k) = 1 , X ¯ = X Q and ϵ ̲ = (ϵ 1 , ϵ 2 , ϵ 3 , ϵ 4) ∈ F Q 4 . We shall provide an approach to handle the bijectivity of f ϵ ̲ for any k ≥ 1 . Notably, we show that the problem of finding conditions for bijectivity of the quadrinomial f ϵ ̲ is closely related to the study of the famous equation X q + 1 + X + a = 0 (*). We then reduce the initial problem into the problem of finding conditions for which an equation of the form (*) has a unique solution in F Q for every a ∈ F Q . In addition, as a crucial direct consequence our result, we prove the validity of the conjecture (Conjecture 19) proposed by Li et al. (Des Codes Cryptogr 89:737–761, 2021). We emphasize that our positive answer completely characterizes permutations with boomerang uniformity 4 from the butterfly structure, which leads to the view of the quadrinomial f ϵ ̲ as excellent candidates to design block ciphers in symmetric cryptography. Despite a lot of attention regarding the considered conjecture, it remains unsolved on its whole when the coefficients lie in F Q . However, this article is the first which propose an approach that solves the enter conjecture by handling both sides of it involving equivalence simultaneously. We believe that our novel approach and its strength could benefit from proving the bijectivity of other families of polynomials over finite fields. [ABSTRACT FROM AUTHOR]

Two-dimensional (2-D) Z-complementary array pair (ZCAP) is a pair of 2-D arrays, whose 2-D autocorrelation sum gives zero value at all time shifts in a zone around the (0, 0) time shift, except the (0, 0) time shift. The zone is called a zero correlation zone (ZCZ). 2-D ZCAPs include 2-D Golay complementary array pairs (GCAPs) as special cases, and can be applicable in 2-D synchronization. In this paper, we focus on designing new 2-D ZCAPs by exploring two promising approaches. The first construction of 2-D ZCAPs uses 1-D ZCPs as the initial stage, such that any binary ZCP and q-phase ZCP can produce a q-phase 2-D ZCAP. The second construction of 2-D ZCAPs is based on 2-D generalized Boolean functions (GBFs), and the resulting 2-D ZCAPs can have the largest 2-D ZCZ ratio 6/7, compared with known 2-D ZCAPs but not 2-D GCAPs in the literature. Here the ZCZ ratio is defined as the ratio of the ZCZ size over the array size. [ABSTRACT FROM AUTHOR]

The substitution box (S-box) is an important nonlinear component in most symmetric cryptosystems and thus should have good properties. Its difference distribution table (DDT) and linear approximation table (LAT) affect the security of the cipher against differential and linear cryptanalysis. In most previous work, differential uniformity and linearity of an S-box are two primary cryptographic properties to impact the resistance against differential and linear attacks. In some cases, the branch number and fixed point are also be considered. However, other important cryptographic properties such as the frequency of differential uniformity (resp. linearity) and the number of Bad Input and Bad Output (BIBO) patterns in DDT (resp. LAT) are often ignored. These properties substantially affect lightweight cryptography based on substitution bit permutation networks (SbPN) such as PRESENT, GIFT and RECTANGLE. This paper introduces a new method to search for S-boxes satisfying all above criteria simultaneously. In our strategy, we transform the process of searching for S-boxes under certain constraints on cryptographic properties into a satisfiability (SAT) problem. As applications, we use our new approach to search out 4-bit and 5-bit S-boxes with the same or better cryptographic properties compared with the S-boxes from well-known ciphers. Finally, we also utilize our method to verify a conjecture proposed by Boura et al. in the case of all 3-bit and 4-bit S-boxes. We propose a proposition and two corollaries to reduce the search space in this verification. [ABSTRACT FROM AUTHOR]

For a long time, the literature has demonstrated that designs and codes are exciting topics for combinatorics and coding theory. Linear codes and t-designs are, in fact, closely related. In recent years, significant results have been derived in the connection framework between codes and combinatorial designs. The most relevant recent contribution is the 71-year breakthrough in discovering by C. Ding and C. Tang of an infinite family of linear codes supporting an infinite family of 4-designs. This paper deals with codes from designs. It considers a class of cyclic codes from the support designs of affine invariant codes and studies ternary cyclic codes. These codes hold 2-designs, contain original affine invariant codes and many other affine-invariant subcodes. The dimension and a lower bound of these ternary cyclic codes are determined. [ABSTRACT FROM AUTHOR]

According to a magnific method due to I. Tamo and A. Barg, a class of polynomials over finite fields, called good polynomials, was introduced and used to construct optimal Locally Recoverable Codes (LRC), which have been developed and exploited in distributed storage. An important derived algebraic problem is, for a given finite field F q and a fixed integer r, to find a polynomial of degree r + 1 that is constant on as many subsets of F q as possible of size r + 1 . Compared to the literature on this topic, our main contribution is introducing a new parameter that measures how "good" a polynomial is in the sense of LRC. Our new approach allows us to characterize completely good polynomials of a low degree over finite fields and, next, to derive new constructions of such polynomials, leading to optimal LRC with new flexible localities. Specifically, several good polynomials of degree at most 6 are studied and described precisely in this paper. [ABSTRACT FROM AUTHOR]

Using recent results on solving the equation X 2 k + 1 + X + a = 0 over a finite field F 2 n provided by the second and the third authors, we address an open question raised by the first author in WAIFI 2014 concerning the APN-ness of the Kasami functions x ↦ x 2 2 k - 2 k + 1 with gcd (k , n) = 1 , x ∈ F 2 n . [ABSTRACT FROM AUTHOR]

Rotation symmetric Boolean functions incorporate a super-class of symmetric functions which represent an attractive corpus for computer investigation. These functions have been investigated from the viewpoints of bentness and correlation immunity and have also played a role in the study of nonlinearity. In the literature, many constructions of balanced odd-variable rotation symmetric Boolean functions with optimal algebraic immunity have been derived. While it seems that the construction of balanced even-variable rotation symmetric Boolean functions with optimal algebraic immunity is very hard work to breakthrough. In this paper, we present for the first time a construction of balanced rotation symmetric Boolean functions on an arbitrary even number of variables with optimal algebraic immunity by modifying the support of the majority function. The nonlinearity of the newly constructed rotation symmetric Boolean functions is also derived. [ABSTRACT FROM AUTHOR]

At Eurocrypt'18, Cid, Huang, Peyrin, Sasaki, and Song introduced a new tool called Boomerang Connectivity Table (BCT) for measuring the resistance of a block cipher against the boomerang attack which is an important cryptanalysis technique introduced by Wagner in 1999 against block ciphers. Next, Boura and Canteaut introduced an important parameter related to the BCT for cryptographic S-boxes called boomerang uniformity. The purpose of this paper is to present a brief state-of-the-art on the notion of boomerang uniformity of vectorial Boolean functions (or S-boxes) and provide new results. More specifically, we present a slightly different but more convenient formulation of the boomerang uniformity and prove some new identities. Moreover, we focus on quadratic permutations in even dimension and obtain general criteria by which they have optimal BCT. As a consequence of the new criteria, two previously known results can be derived, and many new quadratic permutations with optimal BCT (optimal means that the maximal value in the Boomerang Connectivity Table equals the lowest known differential uniformity) can be found. In particular, we show that the boomerang uniformity of the binomial differentially 4-uniform permutations presented by Bracken, Tan, and Tan equals 4. Furthermore, we show a link between the boomerang uniformity and the nonlinearity for some special quadratic permutations. Finally, we present a characterization of quadratic permutations with boomerang uniformity 4. With this characterization, we show that the boomerang uniformity of a quadratic permutation with boomerang uniformity 4 is preserved by the extended affine (EA) equivalence. [ABSTRACT FROM AUTHOR]

In 2014, Tamo and Barg have presented in a very remarkable paper a family of optimal linear locally recoverable codes (LRC codes) that attain the maximum possible distance (given code length, cardinality, and locality). The key ingredients for constructing such optimal linear LRC codes are the so-called r-good polynomials, where r is equal to the locality of the LRC code. In 2018, Liu et al. presented two general methods of designing r-good polynomials by using function composition, which led to three new constructions of r-good polynomials. Next, Micheli provided a Galois theoretical framework which allows to construct r-good polynomials. The well-known Dickson polynomials form an important class of polynomials which have been extensively investigated in recent years in different contexts. In this paper, we provide new methods of designing r-good polynomials based on Dickson polynomials. Such r-good polynomials provide new constructions of optimal LRC codes. [ABSTRACT FROM AUTHOR]

BCH codes with efficient encoding and decoding algorithms have many applications in communications, cryptography and combinatorial design. This paper studies a class of linear codes of length q m - 1 2 over F q with special trace representation, where q is an odd prime power. With the help of the inner distributions of some subsets of association schemes of quadratic forms, we determine the weight enumerators of these codes. Determining some cyclotomic coset leaders δ i of cyclotomic cosets modulo q m - 1 2 , we prove that narrow-sense BCH codes of length q m - 1 2 with designed distance δ i = q m - q m - 1 2 - 1 - q ⌊ m - 3 2 ⌋ + i - 1 2 have the corresponding trace representation, and have the minimal distance d = δ i and the Bose distance d B = δ i , where 1 ≤ i ≤ ⌊ m + 11 6 ⌋ . [ABSTRACT FROM AUTHOR]

Lightweight cryptography is an important tool for building strong security solutions for pervasive devices with limited resources. Due to the stringent cost constraints inherent in extremely large applications, the efficient implementation of cryptographic hardware and software algorithms is of utmost importance to realize the vision of generalized computing. In CRYPTO 2016, Beierle, Kranz and Leander have considered lightweight multiplication in F 2 n . Specifically, they have considered the fundamental question of optimizing finite field multiplications with one fixed element and investigated which field representation, that is which choice of basis, allows for an optimal implementation. They have left open a conjecture related to an XOR-count of two. Using the theory of linear algebra, we prove in the present paper that their conjecture is correct. Consequently, this proved conjecture can be used as a reference for further developing and implementing cryptography algorithms in lightweight devices. [ABSTRACT FROM AUTHOR]

The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The hull plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. It has been shown that these algorithms are very effective in general if the size of the hull is small. It is clear that the linear codes with the smallest hull are LCD codes and with the second smallest hull are those with one-dimensional hull. In this paper, we employ character sums in semi-primitive case to construct LCD codes and linear codes with one-dimensional hull from cyclotomic fields and multiplicative subgroups of finite fields. Some sufficient and necessary conditions for these codes are obtained, where prime ideal decompositions of prime p in cyclotomic fields play a key role. In addition, we show the non-existence of these codes in some cases. [ABSTRACT FROM AUTHOR]

Side-channel attacks and fault injection attacks are nowadays important cryptanalysis methods on the implementations of block ciphers, which represent huge threats. Direct sum masking (DSM) has been proposed to protect the sensitive data stored in registers against both SCA and FIA. It uses two linear codes C and D whose sum is direct and equals F q n . The resulting security parameter is the pair (d (C) - 1 , d (D ⊥) - 1) . For being able to protect not only the sensitive input data stored in registers against SCA and FIA but the whole algorithm (which is required at least in software applications), it is useful to change C and D into C ′ , which has the same minimum distance as C , and D ′ , which may have smaller dual distance than D . Precisely, D ′ is the linear code obtained by appending on the right of its generator matrix the identity matrix with the same number of rows. It is then highly desired to construct linear codes D such that d ( D ′ ⊥) is very close to d (D ⊥) . In such case, we say that D is almost optimally extendable (and is optimally extendable if d ( D ′ ⊥) = d (D ⊥) ). In general, it is notoriously difficult to determine the minimum distances of the codes D ⊥ and D ′ ⊥ simultaneously. In this paper, we mainly investigate constructions of (almost) optimally extendable linear codes from irreducible cyclic codes and from the first-order Reed–Muller codes. The minimum distances of the codes D , D ′ , D ⊥ , and D ′ ⊥ are determined explicitly and their weight enumerators are also given. Furthermore, several families of optimally extendable codes are found (for the second time) among such linear codes. [ABSTRACT FROM AUTHOR]

In 2018, Pott et al. have studied in (IEEE Trans Inf Theory 64(1):403-411, 2018) the maximum number of bent components of vectorial functions. They have presented many nice results and suggested several open problems in this context. This paper is in the continuation of their study in which we solve two open problems raised by Pott et al. and partially solve an open problem raised by the same authors. Firstly, we prove that for a vectorial function, the property of having the maximum number of bent components is invariant under the so-called CCZ equivalence. Secondly, we prove the non-existence of APN plateaued functions having the maximum number of bent components. In particular, quadratic APN functions cannot have the maximum number of bent components. Finally, we present some sufficient conditions that the vectorial function defined from F 2 2 k to F 2 2 k by its univariate representation: α x 2 i x + x 2 k + ∑ j = 1 ρ γ (j) x 2 t j + ∑ j = 1 ρ γ (j) x 2 t j + k has the maximum number of bent components, where ρ ≤ k . Further, we show that the differential spectrum of the function x 2 i (x + x 2 k + x 2 t 1 + x 2 t 1 + k + x 2 t 2 + x 2 t 2 + k ) (where i , t 1 , t 2 satisfy some conditions) is different from the binomial function F i (x) = x 2 i (x + x 2 k) presented in the article of Pott et al. [ABSTRACT FROM AUTHOR]

Boolean functions satisfying good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the recent FLIP stream cipher (Méaux et al.: in Lecture Notes in Computer Science, vol. 9665, pp. 311–343, Springer, Berlin, 2016). In this paper, we propose a large class of weightwise perfectly balanced (WPB) functions, which is 2-rotation symmetric. This new class of WPB functions is not extended affinely equivalent to the known constructions. We also discuss the weightwise nonlinearity profile of these functions, and present general lower bounds on k-weightwise nonlinearity, where k is a power of 2. Moreover, we exhibit a subclass of the family. By a recursive lower bound, we show that these subclass of WPB functions have very high weightwise nonlinearity profile. [ABSTRACT FROM AUTHOR]

Linear codes, the most significant class of codes in coding theory, have diverse applications in secret sharing schemes, authentication codes, communication, data storage devices and consumer electronics. The main objectives of this paper are twofold: to construct three-weight linear codes from plateaued functions over finite fields, and to analyze the constructed linear codes for secret sharing schemes. To do this, we generalize the recent contribution of Mesnager given in (Cryptogr Commun 9(1):71-84, 2017). We first introduce the notion of (non)-weakly regular plateaued functions over Fp, with p being an odd prime. We next construct three-weight linear p-ary (resp. binary) codes from weakly regular p-ary plateaued (resp. Boolean plateaued) functions and determine their weight distributions. We finally observe that the constructed linear codes are minimal for almost all cases, which implies that they can be directly used to construct secret sharing schemes with nice access structures. To the best of our knowledge, the construction of linear codes from plateaued functions over Fp, with p being an odd prime, is studied in this paper for the first time in the literature. [ABSTRACT FROM AUTHOR]

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) are of much interest from many viewpoints due to their theoretical and practical properties. However, little work has been done on LCD MDS codes. In particular, determining the existence of q-ary [n, k] LCD MDS codes for various lengths n and dimensions k is a basic and interesting problem. In this paper, we firstly study the problem of the existence of q-ary [n, k] LCD MDS codes and solve it for the Euclidean case. More specifically, we show that for q>3 there exists a q-ary [n, k] Euclidean LCD MDS code, where 0≤k≤n≤q+1, or, q=2m, n=q+2 and k=3orq-1. Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes. [ABSTRACT FROM AUTHOR]

Plateaued functions play a significant role in cryptography, sequences for communications, and the related combinatorics and designs. Comparing to their importance, those functions have not been studied in detail in a general framework. Our motivation is to bring further results on the characterizations of bent and plateaued functions, and to introduce new tools which allow us firstly a better understanding of their structure and secondly to get methods for handling and designing such functions. We first characterize bent functions in terms of all even moments of the Walsh transform, and then plateaued (vectorial) functions in terms of the value distribution of the second-order derivatives. Moreover, we devote to cubic functions the characterization of plateaued functions in terms of the value distribution of the second-order derivatives, and hence this reveals non-existence of homogeneous cubic bent (and also (homogeneous) cubic plateaued for some cases) functions in odd characteristic. We use a rank notion which generalizes the rank notion of quadratic functions. This rank notion reveals new results about (homogeneous) cubic plateaued functions. Furthermore, we observe non-existence of a function whose absolute Walsh transform takes exactly 3 distinct values (one being zero). We finally provide a new class of functions whose absolute Walsh transform takes exactly 4 distinct values (one being zero). [ABSTRACT FROM AUTHOR]

In this survey, we revisit the Rothaus paper and the chapter of Dillon's thesis dedicated to bent functions, and we describe the main results obtained on these functions during these last 40 years. We also cover more briefly super-classes of Boolean functions, vectorial bent functions and bent functions in odd characteristic. [ABSTRACT FROM AUTHOR]

The main topics and interconnections arising in this paper are symmetric cryptography (S-boxes), coding theory (linear codes) and finite projective geometry (hyperovals). The paper describes connections between the two main areas of information theory on the one side and finite geometry on the other side. Bent vectorial functions are maximally nonlinear multi-output Boolean functions. They contribute to an optimal resistance to both linear and differential attacks of those symmetric cryptosystems in which they are involved as substitution boxes (S-boxes). We firstly exhibit new connections between bent vectorial functions and the hyperovals of the projective plane, extending the recent link between bent Boolean functions and the hyperovals. Such a link provides several new classes of optimal vectorial bent functions. Secondly, we exhibit surprisingly a connection between the hyperovals of the projective plane in even characteristic and $$q$$ -ary simplex codes. To this end, we present a general construction of classes of linear codes from o-polynomials and study their weight distribution proving that all of them are constant weight codes. We show that the hyperovals of $$PG_{2}(2^m)$$ from finite projective geometry provide new minimal codes (used in particular in secret sharing schemes, to model the access structures) and give rise to multiples of $$2^r$$ -ary ( $$r$$ being a divisor of $$m$$ ) simplex linear codes (whose duals are the perfect $$2^r$$ -ary Hamming codes) over an extension field $${\mathbb F}_{2^{r}}$$ of $${\mathbb F}_{2^{}}$$ . The following diagram gives an indication of the main topics and interconnections arising in this paper. [Figure not available: see fulltext.] [ABSTRACT FROM AUTHOR]

In any connected, undirected graph G = ( V, E), the distance d( x, y) between two vertices x and y of G is the minimum number of edges in a path linking x to y in G. A sphere in G is a set of the form S( x) = { y ∈ V : d( x, y) = r}, where x is a vertex and r is a nonnegative integer called the radius of the sphere. We first address in this paper the following question: What is the minimum number of spheres with fixed radius r ≥ 0 required to cover all the vertices of a finite, connected, undirected graph G? We then turn our attention to the Hamming Hypercube of dimension n, and we show that the minimum number of spheres with any radii required to cover this graph is either n or n + 1, depending on the parity of n. We also relate the two above problems to other questions in combinatorics, in particular to identifying codes. [ABSTRACT FROM AUTHOR]

Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. This paper is a contribution to the construction of bent functions over $${\mathbb{F}_{2^{n}}}$$ ( n = 2 m) having the form $${f(x) = tr_{o(s_1)} (a x^ {s_1}) + tr_{o(s_2)} (b x^{s_2})}$$ where o( s) denotes the cardinality of the cyclotomic class of 2 modulo 2 − 1 which contains s and whose coefficients a and b are, respectively in $${F_{2^{o(s_1)}}}$$ and $${F_{2^{o(s_2)}}}$$. Many constructions of monomial bent functions are presented in the literature but very few are known even in the binomial case. We prove that the exponents s = 2 − 1 and $${s_2={\frac {2^n-1}3}}$$, where $${a\in\mathbb{F}_{2^{n}}}$$ ( a ≠ 0) and $${b\in\mathbb{F}_{4}}$$ provide a construction of bent functions over $${\mathbb{F}_{2^{n}}}$$ with optimum algebraic degree. For m odd, we give an explicit characterization of the bentness of these functions, in terms of the Kloosterman sums. We generalize the result for functions whose exponent s is of the form r(2 − 1) where r is co-prime with 2 + 1. The corresponding bent functions are also hyper-bent. For m even, we give a necessary condition of bentness in terms of these Kloosterman sums. [ABSTRACT FROM AUTHOR]

School of Mathematics, Sun Yat-sen University, Guangzhou 510275, People's Republic of China, 5. School of Mathematics, Sun Yat-sen University, Guangzhou 510275, People's Republic of China and 2. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, People's Republic of China and 6. [Extracted from the article]