The hull of a linear code is defined as the intersection of the code and its dual. This concept was initially introduced to classify finite projective planes. The hull plays a crucial role in determining the complexity of algorithms used to check the permutation equivalence of two linear codes and compute a linear code’s automorphism group. Research has shown that these algorithms are very effective when the hull size is small. Linear complementary dual (LCD) codes have the smallest hulls, while codes with a one-dimensional hull have the second smallest. A recent notable paper that directs our investigation is authored by H. Chen, titled “On the Hull-Variation Problem of Equivalent Linear Codes", published in IEEE Transactions on Information Theory, volume 69, issue 5, in 2023. In this paper, we first explore the one-dimensional hull of a linear code over finite fields. Additionally, we demonstrate that any LCD code over an extended binary field $$ \mathrm{I\!F}_q $$ (where $$ q > 3 $$ ) with a minimum distance of at least 2 is equivalent to the one-dimensional hull of a linear code under a specific weak condition. Furthermore, we provide a construction for creating hulls with $$ \ell + 1 $$ -dimensionality from an $$ \ell $$ -dimensional hull of a linear code, again under a weak condition. This corresponds to a particularly challenging direction, as creating $$ \ell $$ -dimensional hulls from $$ \ell + 1 $$ -dimensional hulls. Finally, we derive several constructions for the $$ \ell $$ -dimensional hulls of linear codes as a consequence of our results.

Minimal linear codes have garnered significant attention in cryptography due to their essential role in secretsharing schemes, multiparty computation (MPC), and secure communication. While earlier studies primarily focused on minimal codes over finite fields, extending these codes to rings offers enhanced security, flexibility, and efficiency for various cryptographic applications. This article deals with minimal linear codes over finite commutative rings. Specifically, we handle a key question in minimal linear codes, which is determining the existence of an [m,k] minimal linear code withk≤m. In 2021, Lu,Wu and Cao proved that for some integerm(k;q), a minimal linear code of lengthmand dimensionkalways exists whenm≥m(k;q), providing upper and lower bounds form(k;q). This paper expands on these results by establishing both upper and lower bounds form(k;pl) andm(k;p1p2) over Zpland Zp1p2, respectively. We also present a necessary and sufficient condition for a linear code to achieve minimality when analyzed over the ring Zn. The fundamental question regarding minimal linear codes is whether a [m,k] minimal linear code exists, withkbeing less than or equal tom. Lu et al. demonstrated that there is a positive integerm(k;q) such that ifm≥m(k;q), a minimal linear code of lengthmand dimensionkover the finite field Fqmust exist (whereqis a prime power). They provided both upper and lower bounds form(k;q). In our study, we investigate the existence of minimal codes within an extended novel framework. We analyze the case of one-dimensional minimal codes over Znin depth and present improved upper bounds onm(k;q) specifically for the case whenk= 1.

The relative hull of a linear codeC1with respect to another linear codeC2is defined as the intersection ofC1and the dual ofC2, i.e.,C1∩C⊥2. Andersonet al. [2] demonstrated that the dimension of the relative hull can be adjusted, either repeatedly increased or decreased by one, until a certain bound is reached by substituting eitherC1orC2with its equivalent code. As a special class of linear codes, BCH codes are both theoretically significant and practically valuable for communication and storage systems due to their good algebraic structures and flexible error-correcting capabilities. This raises an important question: how does the dimension of the relative hull change when C1 and C2 are BCH codes or their equivalent codes? This paper focuses on the relative hull dimensions of narrow-sense primitive BCH codes, building on the insights from [2]. We present several sufficient and necessary conditions based on the designed distances of BCH codes, which ensure that the dimensions of the relative hulls reach the lower or upper bounds for linear codes. Additionally, we study the parameters of the relative hulls of various classes of BCH codes, providing details on their dimensions and developing lower bounds for their minimum distances. Furthermore, we investigate how the dimensions of the relative hulls are affected when the BCH codesC2are replaced by their equivalent codesC′2.

Low ambiguity zone (LAZ) sequences play a crucial role in modern integrated sensing and communication (ISAC) systems. In this paper, we introduce a novel class of functions known as locally perfect nonlinear functions (LPNFs). By utilizing LPNFs and interleaving techniques, we propose three new classes of both periodic and aperiodic LAZ sequence sets with flexible parameters. The proposed periodic and aperiodic LAZ sequence sets are asymptotically optimal with respect to the periodic and aperiodic lower AF bounds, respectively, which were proposed recently in [IEEE J. Sel. Areas Commun. 40 (6): 1809-1822]. Notably, the aperiodic LAZ sequence sets are the first such sequence sets in the literature that satisfy the bound. Finally, we demonstrate that the proposed sequence sets are cyclically distinct.

Quasi complementary sequence sets (QCSSs) are important in modern communication systems as they are capable of supporting more users, which is desired in applications like MC-CDMA nowadays. In this paper, we first derive a tighter bound on the maximum aperiodic correlation among all constituent complementary sequence sets in QCSSs. By proposing a new combinatorial structure called quasi-Florentine rectangles, we obtain a new construction of QCSSs with large set sizes. Using Butson-type Hadamard matrices and quasi-Florentine rectangles, we propose another construction which can construct QCSSs with flexible parameters over any given alphabet size, including small alphabets. All the proposed sequences are optimal with respect to the newly proposed bound. Also, through some of the constructions, the column sequence PMEPR of the proposed QCSSs are upper bounded by 2.

In coding theory, the well-known class of block codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes), form a class of cyclic error-correcting codes constructed using polynomials over a finite field. They are used for various critical practical applications in communication and storage due to their efficient encoding and decoding algorithms. In the past sixty years, significant progress has been made in understanding BCH codes’ dimensions and minimum distances. However, there has been limited research on the minimum distances of the dual codes of BCH codes, making it challenging to determine their actual minimum distances. Therefore, developing accurate lower bounds on the minimum distances of the dual codes of BCH codes is crucial and exciting. In this paper, we primarily use the multiplier technique proposed by Huffman and Pless to investigate the lower bounds on minimum distances of the dual codes$\mathcal {C}_{(q,q^{m}-1,\delta)}^{\perp } $of the primitive narrow-sense BCH codes with designed distance$\delta $. When$q = p^{e}$with$e \ge 2$, we improve the lower bounds on minimum distances of the dual codes$\mathcal {C}_{(q,q^{m}-1,\delta)}^{\perp } $in the ranges$p^{ei}-p^{e-1}+2 \le \delta \le p^{ei+e-1}-p^{e-1}+1$, where$m \ge 2$and$1 \le i \le m-1$. These new lower bounds are much tighter than the previously known bounds in the literature. This technique also applies to the study of binary dual codes$\mathcal {C}_{(2,2^{m}-1,\delta)}^{\perp } $, for which we obtain tight lower bounds for$\delta = 2^{t}$, where$m \ge 5$is odd and$2 \le t \le m-3$is even.

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.

Letqbe an odd prime power with$q\equiv 3\ ({\mathrm {mod}}\,4)$. In this paper, we study the differential and boomerang properties of the function$F_{2,u}(x)=x^{2}\big (1+u\eta (x)\big)$over$\mathbb {F}_{q}$, where$u\in \mathbb {F}_{q}^{*}$and$\eta $is the quadratic character of$\mathbb {F}_{q}$. We determine the differential uniformity of$F_{2,u}$for any$u\in \mathbb {F}_{q}^{*}$, as well as the differential spectra and boomerang uniformity of the locally-APN functions$F_{2,\pm 1}$, thereby disproving a conjecture proposed in Budaghyan and Pal (2024), which states that there exist infinitely many values ofqandusuch that$F_{2,u}$is an APN function.

Self-orthogonal and dual-containing codes are two important subclasses of linear codes in coding theory and have been studied for many years. In this paper, we present several sufficient conditions for self-orthogonal or dual-containing codes when a linear code, cyclic code or BCH codeCis transformed to an equivalent code v ·C. Specifically, we prove that linear codes are equivalent to Euclidean or Hermitian self-orthogonal codes if the dimension is very small. For primitive BCH codes, we prove that when designed distances are small, equivalent Euclidean dual-containing codes always exist. From our method presented in this paper, many self-orthogonal or dual-containing linear, cyclic or BCH codes with good parameters can be constructed explicitly. We also construct some Euclidean dual-containing binary BCH codes with best-known parameters.

Frequency hopping sequences (FHSs) play a crucial role in frequency hopping (FH) communication systems due to their strong anti-interference ability, low interception probability, high confidentiality and strong concealment. The objective of this paper is to construct FHSs for quasi-synchronous frequency-hopping multiple access (FHMA) communication systems that simultaneously achieve optimal no-hit zone (NHZ) length and optimal gap. To accomplish this, the paper first derives tighter upper bounds for the gap size in both periodic and aperiodic scenarios under the assumption that all frequencies within the designated frequency slot set are fully utilized. Subsequently, this paper proposes a class of wide-gap frequency hopping sequences (WGFHSs) and a class of multi-timeslot wide-gap frequency hopping sequences (MTWGFHSs), both of which simultaneously exhibit optimal NHZ length and optimal gap.

The (Schur) squares of linear codes are an interesting research topic in coding theory, and they have important applications in cryptography. Linear complementary dual codes (LCD codes) have been widely applied in data storage, communication systems, consumer electronics, and cryptography. Given these exciting applications of squares and LCD codes, we mainly focus on the squares of LCD cyclic codes in this paper. It will be proved that the square of an LCD cyclic code is still an LCD cyclic code. As a subclass of cyclic codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes) have explicit defining sets that include consecutive integers, which gives an advantage of analyzing the parameters of BCH codes and their related codes. We will investigate the squares$\mathcal {C}^{2}(t)$and$\mathcal {C}^{2}(t)^{c}$of the primitive LCD BCH codes$\mathcal {C}(t)$and their complements$\mathcal {C}(t)^{c}$, respectively, where$\mathcal {C}(t)=\mathcal {C}_{(q,q^{m}-1,2t,-t+1)}$is the BCH code of length$q^{m}-1$over$\mathbb{F}_{q}$with designed distance$2t$. Two sufficient and necessary conditions to guarantee that$\mathcal {C}^{2}(t) \ne \Bbb \{\textbf {0}\}$and$\mathcal {C}^{2}(t)^{c} \ne \mathbb{F}_{q}^{n}$are proposed by giving restrictions on designed distances. Furthermore, the dimensions and lower bounds on minimum distances of$\mathcal {C}^{2}(t)$and$\mathcal {C}^{2}(t)^{c}$are presented in some cases. The parameters of the squares of the complements of the Melas codes$M(q,m)$are also investigated.

Subfield codes of linear codes over finite fields have recently received much attention since they can produce optimal codes, which may have applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction framework of 3-dimensional linear codesCf,gover Fqmparameterized by any two functionsf,gover Fqm, and then study the properties of six types ofCf,g, its punctured codeC*f,gand their corresponding subfield codes over Fq. The classification ofCf,gis based on special choices off,gas trace function, norm function, almost bent function, Boolean bent function or a combination of these functions. For the first two types ofCf,g, we explicitly determine the weight distributions and dualities ofCf,g, C*f,gand their subfield codes over Fq. The remaining four types ofCf,gare restricted toq= 2, and the weight distributions and dualities of the subfields codeC(q)f,gandC*f,g(q)are completely determined. Most of the resultant linear codes (over Fqmor over Fq) have few weights. Some of them are optimal and some have the best-known parameters according to the tables maintained at http://www.codetables.de. In fact, 16 infinite families of optimal linear codes are produced in this paper. As a byproduct, a family of [24m-2, 2m+ 1, 24m-3] quaternary Hermitian self-orthogonal codes are obtained withm≥ 2. As an application, we present several infinite families of 2-designs or 3-designs with some of the codes presented in this paper.

Studying the Schur square of a linear code is an important research topic in coding theory. Schur squares have important applications in cryptography and private information retrieval schemes, notably in secure multiparty computing or designing bilinear multiplication algorithms in finite extensions of finite fields through the notion of supercodes. Thanks to their exciting applications in cryptography, squares and powers of several linear codes have been investigated. In this paper, we will focus on the Schur square of a relevant well-known subclass of cyclic codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes), which have wide applications in communication and storage systems and benefit from explicit defining sets that include consecutive integers, which gives the advantage of analyzing the parameters of BCH codes and their complements. Our main objective is to investigate the parameters of the squares of primitive narrow-sense BCH codes$\mathcal C(\delta)$and their complements$\mathcal C(\delta)^{c}$. We will present two sufficient and necessary conditions to guarantee that$\mathcal C^{2}(\delta) \ne \Bbb F_{q}^{n}$and$\mathcal C^{2}(\delta)^{c} \ne \Bbb F_{q}^{n}$by giving restrictions on designed distance$\delta $, where$2 \le \delta \le n$. Based on these two characterizations, the dimensions and minimum distances of$\mathcal C^{2}(\delta)$and$\mathcal C^{2}(\delta)^{c}$are investigated in some cases. The dimensions of these squares are determined explicitly, and lower bounds on the minimum distance are given.

Minimal linear codes have important applications in secure communications, including in the framework of secret sharing schemes and secure multi-party computation. A lot of research have been carried out to derive codes with few weights (but more importantly, being minimal) using algebraic or geometric approaches. One of the main power and fructify algebraic methods is based on the design of those codes by employing functions over finite fields. Li et al. (2021) have recently identified some binary linear codes with few weights from two classes of two-to-one functions. In this paper, our ultimate objective is to expand the class of codes derived from the paper of Li et al. by proposing larger classes of binary linear codes with few weights via generic constructions involving other known families of two-to-one functions over the finite field$\mathbb {F}_{2^{n}}$of order$2^{n}$. We succeed in constructing such codes, and we also completely determine their weight distributions. The linear codes presented in this paper differ in parameters from those known in the literature. Besides, some of them are optimal concerning the well-known Griesmer bound. Notably, we prove that our codes are either optimal or almost optimal with respect to the online Database of Grassl. We next observe that the derived binary linear codes also have the minimality property for most cases. We then describe the access structures of the secret-sharing schemes based on their dual codes. Finally, we solve two problems left open in the paper by Li et al. (more specifically, a complete solution to Problem 2 and a partial solution to Problem 1).

Permutation polynomials have been extensively studied for their applications in cryptography, coding theory, combinatorial design, etc. An important subfamily of permutations is the class of involutions (those permutations are equal to their compositional inverse). Elements of this class have been used frequently for block cipher designs and coding theory. In this article, we further investigate this corpus using new approaches, specifically from two-to-one (2-to-1) functions and (in some cases) using the graph indicators introduced by Carlet in 2020. In our constructions of involutions over the finite field$\mathbb {F}_{2^{n}}$of order$2^{n}$, we shall intensively use 2-to-1 mappings over$\mathbb {F}_{2^{n}}$. More specifically, we present a new constructive method to design involutions from 2-to-1 mappings through their graph indicator and derive new involutions from known 2-to-1 mappings. Besides, we also propose several new classes of 2-to-1 mappings, including 2-to-1 hexanomials, 2-to-1 mappings of the form$(x^{2^{k}}+x+\delta)^{s_{1}}+(x^{2^{k}}+x+\delta)^{s_{2}}+cx$, and 2-to-1 mappings from linear 2-to-1 mappings. We also exhibit the corresponding involutions of the constructed 2-to-1 mappings. Furthermore, an infinite family of involutions with differential uniformity at most 4 (EA-inequivalent to the inverse function) is obtained. Finally, we highlight that all our derived families of involutions have no fixed point, further accentuating their cryptographic interest.

Complete complementary codes (CCCs) have important applications in communication, radar, and information security. In modern communication and radar systems, certain spectrum is reserved or prohibited from transmission, which leads to the so-called spectrally null constrained (SNC) problem. Compared with vast works on conventional CCCs, relatively little is known about SNC-CCCs. One objective of this paper is to derive several constructions of CCCs with more flexible settings from the graph of extended Boolean functions. This generalizes earlier achievements on CCCs from recent literature. Another objective of this paper is to employ the graphs of extended Boolean functions and polynomial representation of sequences to construct SNC-CCCs.

This article focuses on the so-called locally-APN power functions introduced by Blondeau, Canteaut and Charpin, which generalize the well-known notion of APN functions and possibly more suitable candidates against differential attacks. Specifically, given two coprime positive integers$m$and$k$such that$\gcd (2^{m}+1,2^{k}+1)=1$, we investigate the locally-APN-ness property of the Niho type power function$F(x)=x^{s(2^{m}-1)+1}$over the finite field$\mathbb {F}_{2^{2m}}$for$s=(2^{k}+1)^{-1}$, where$(2^{k}+1)^{-1}$denotes the multiplicative inverse modulo$2^{m}+1$. By employing finer studies of the number of solutions of certain equations over finite fields, we prove that$F(x)$is locally-APN and determine its differential spectrum. We emphasize that computer experiments show that this class of locally-APN power functions covers all Niho type locally-APN power functions for$2\leq m\leq 10$. In addition, we also determine the boomerang spectrum of$F(x)$by using its differential spectrum, which particularly generalizes a recent result by Yan, Zhang and Li.

Permutation polynomials over finite fields are fundamental objects as they are used in various theoretical and practical applications in cryptography, coding theory, combinatorial design, and related topics. This family of polynomials constitutes an active research area in which advances are being made constantly. In particular, constructing infinite classes of permutation polynomials over finite fields with good differential properties (namely, low) remains an exciting problem despite much research in this direction for many years. This article exhibits low differentially uniform power permutations over finite fields of odd characteristics. Specifically, its objective is twofold concerning the power functions$F(x)=x^{\frac {p^{n}+3}{2}}$defined over the finite field${\mathbb {F}}_{p^{n}}$of order$p^{n}$, where$p$is an odd prime, and$n$is a positive integer. The first is to complement some former results initiated by Helleseth and Sandberg in 1997 by solving the open problem left open for more than twenty years concerning the determination of the differential spectrum of$F$when$p^{n}\equiv 3\pmod 4$and$p\neq 3$. The second is to determine the exact value of its differential uniformity. Our achievements are obtained firstly by evaluating some character sums over${\mathbb {F}}_{p^{n}}$(which amounts to evaluating the number of${\mathbb {F}}_{p^{n}}$-rational points on some related curves and secondly by computing the number of solutions in$({\mathbb {F}}_{p^{n}})^{4}$of a system of equations presented by Helleseth, Rong, and Sandberg, naturally appears while determining the differential spectrum of$F$. We show that in the considered case ($p^{n}\equiv 3\pmod 4$and$p\neq 3$),$F$is an APN power permutation when$p^{n}=11$, and a differentially 4-uniform power permutation otherwise.

Cross Z-complementary pairs (CZCPs) are a special kind of Z-complementary pairs (ZCPs) having zero autocorrelation sums around the in-phase and end-shift positions and zero cross-correlation sums around the end-shift positions. CZCPs can be crucial in designing optimal training sequences for broadband spatial modulation (SM) systems over frequency-selective channels. In this paper, we focus on designing new CZCPs with large cross Z-complementary ratio (CZCR). A construction framework of CZCPs with a large ZCZ ratio is proposed using Turyn’s method on some seed CZCPs and GCPs. By choosing suitably the seed CZCPs, we obtain 24 classes of new CZCPs with large CZCR. Especially, if the GCP is strengthened, our resultant CZCPs have the maximum$\mathrm {\mathbf{CZCR}}\approx \frac {M-1}{M}$for$M\in \{6,12,24,28,48,56\}$. We also obtain optimal CZCPs with new parameters (28, 13), (48, 23), (56, 27), (96, 47) and (112, 55), which can be extended to$(96N,47N)$-CZCPs and$(112N,55N)$-CZCPs respectively for any Golay number$N$.

In frequency hopping (FH) communication systems, frequency hopping sequences (FHSs) are crucial in determining the system’s anti-jamming performance. If FHSs can ensure a wide-gap between two adjacent frequency points to avoid the frequency points with high interference probability, it will significantly improve the FH communication system’s anti-interference ability. Moreover, if each frequency point appears at the same number of times in a sequence period, the system’s anti-electromagnetic interference will be enhanced. Therefore, it is desirable to employ FHSs with low Hamming autocorrelation, wide frequency-hopping gap, and good uniformity in practical applications. However, to the best of our knowledge, no such infinite classes of FHSs have been reported in the literature to date. This paper aims to present two constructions of uniform wide-gap frequency-hopping sequences (WGFHSs) by concatenating two or three adequately designed sequences. For the first time, we obtain two infinite classes of WGFHSs, which are optimal with respect to the well-known Lempel-Greenberger bound.

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.

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.

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.

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.

Masking is one of the most popular countermeasures to protect cryptographic implementations against side-channel analysis since it is provably secure and can be deployed at the algorithm level. To strengthen the original Boolean masking scheme, several works have suggested using schemes with high algebraic complexity. The Inner Product Masking (IPM) is one of those. In this paper, we propose a unified framework to quantitatively assess the side-channel security of the IPM in a coding-theoretic approach. Specifically, starting from the expression of IPM in a coded form, we use two defining parameters of the code to characterize its side-channel resistance. In order to validate the framework, we then connect it to two leakage metrics (namely signal-to-noise ratio and mutual information, from an information-theoretic aspect) and one typical attack metric (success rate, from a practical aspect) to build a firm foundation for our framework. As an application, our results provide ultimate explanations on the observations made by Balasch et al. at EUROCRYPT'15 and at ASIACRYPT'17, Wang et al. at CARDIS'16 and Poussier et al. at CARDIS'17 regarding the parameter effects in IPM, like higher security order in bounded moment model. Furthermore, we show how to systematically choose optimal codes (in the sense of a concrete security level) to optimize IPM by using this framework. Eventually, we present a simple but effective algorithm for choosing optimal codes for IPM, which is of special interest for designers when selecting optimal parameters for IPM.

Let m be an even positive integer. A Boolean bent function f on F(2m-1)×F2is called a cyclic bent function if for any a≠b∈F(2m-1) and ε∈F2, f( ax1,x2)+f( bx1,x2+ε) is always bent, where x1∈F(2m-1),x2∈F2. Cyclic bent functions look extremely rare. This paper focuses on cyclic bent functions on F(2m-1)×F2and their applications. The first objective of this paper is to establish a link between quadratic cyclic bent functions and a special type of prequasifields, and construct a class of quadratic cyclic bent functions from the Kantor-Williams prequasifields. The second objective is to use cyclic bent functions to construct families of optimal sequences. The results of this paper show that cyclic bent functions have nice applications in several fields such as coding theory, symmetric cryptography, and CDMA communication.

There are few mathematicians whose contributions go beyond named conjectures and theorems: Vladimir Iosifovich Levenshtein (, 1935–2017) is one such true exception. During the five decades of his active research career, he enriched combinatorics, coding, and information theory with elegant problem formulations, ingenious algorithmic solutions, and highly original proof techniques. However, his work accomplished much more—it paved the way for the creation and advancement of new scientific disciplines, such as natural language processing, metagenomics, sequence alignment, and reference-based genome assembly, as well as DNA-based data storage, to name a few. A crucial concept behind sequence alignment algorithms used in phylogeny, comparative, and cancer genomics, as well as in natural language processing is the Levenshtein (edit) distance and its extension, termed the Damerau–Levenshtein distance between strings. The Levenshtein distance equals the smallest number of insertions, deletions, or substitutions required to convert one string into another. Levenshtein introduced this metric in 1965 [item 1) in the Appendix], followed by the notion of deletion and insertion error-correcting codes that have since been used in a myriad of systems presented with synchronization errors [items 1) and 2) in the Appendix]. Levenshtein’s work also inspired the introduction of the trace reconstruction problem [items 3) and 4) in the Appendix] which has since sparked substantial interest in the field of DNA-based data storage.

Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing minimal binary codes from sets, Boolean functions and vectorial Boolean functions with high algebraic immunity, are proposed. More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented. The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high algebraic immunity. Via these general constructions, infinite families of minimal binary linear codes of dimension m and length less than or equal to m(m+1)/2 are obtained. Besides, a lower bound on the minimum distance of the proposed minimal linear codes is established. Conjectures and open problems are also presented. The results of this paper show that Boolean functions with high algebraic immunity have nice applications in several fields additionally to symmetric cryptography, such as coding theory and secret sharing schemes.

Self-orthogonal codes are an important type of linear codes due to their wide applications in communication and cryptography. The Euclidean (or Hermitian) hull of a linear code is defined to be the intersection of the code and its Euclidean (or Hermitian) dual. It is clear that the hull is self-orthogonal. The main goal of this paper is to obtain self-orthogonal codes by investigating the hulls. Let$\mathcal {C}_{(r,r^{m}-1,\delta,b)}$be the primitive BCH code over$\mathbb {F}_{r}$of length$r^{m}-1$with designed distance$\delta $, where$\mathbb {F}_{r}$is the finite field of order$r$. In this paper, we will present Euclidean (or Hermitian) self-orthogonal codes and determine their parameters by investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes. Several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances. Furthermore, some Hermitian self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code$\mathcal {C}_{(r,r^{2}-1,\delta,1)}$and its hull in both Hermitian and Euclidean cases for$2 \le \delta \le r^{2}-1$. We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension.

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

In a prior paper (Ellingsenet 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.

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.

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.

Self-orthogonal codes are an interesting type of linear codes due to their wide applications in communication and cryptography. It is known that self-orthogonal codes are often used to construct quantum error-correcting codes, which can protect quantum information in quantum computations and quantum communications. Let C be an [n, k] cyclic code over Fq, where Fqis the finite field of order q. The hull of C is defined to be the intersection of the code and its dual. In this paper, we will employ the defining sets of cyclic codes to present two general characterizations of the hulls that have dimension k - 1 or k⊥- 1, where k⊥is the dimension of the dual code C⊥. Several sufficient and necessary conditions for primitive and projective BCH codes to have (k - 1)-dimensional (or (k⊥-1)dimensional) hulls are also developed by presenting lower and upper bounds on their designed distances. Furthermore, several classes of self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. The dimensions and minimum distances of some self-orthogonal codes are determined explicitly. In addition, several optimal codes are obtained.

Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of Fqto 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 Fq, 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.

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.

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.

Linear complementary pairs (LCPs) of codes play an important role in armoring implementations against sidechannel 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 o complementary dual (σ-LCD), which includes known Euclidean LCD codes, Hermitian LCD codes, and Galois LCD codes. Like Euclidean LCD codes, σ-LCD codes can also be used to construct LCP of codes. We show that for q 2, all q-ary linear codes are σ-LCD, and for every binary linear code C, the code {0} × C is σ-LCD. Furthermore, we study deeply σ-LCD generalized quasi-cyclic (GQC) codes. In particular, we provide the characterizations of σ-LCD GQC codes, self-orthogonal GQC codes, and self-dual GQC codes, respectively. Moreover, we provide the constructions of asymptotically good σ-LCD GQC codes. Finally, we focus on σ-LCD abelian codes and prove that all abelian codes in a semisimple group algebra are σ-LCD. The results derived in this paper extend those on the classical LCD codes and show that σ-LCD codes allow the construction of LCP of codes more easily and with more flexibility.

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.

Linear codes with complementary duals (LCD) are linear codes whose intersection with their dual are trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Nonbinary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. In this paper, we introduce a general construction of LCD codes from any linear codes. Further, we show that any linear code over Fq(q > 3) is equivalent to a Euclidean LCD code and any linear code over Fq2(q > 2) is equivalent to a Hermitian LCD code. Consequently an [n, k, d]-linear Euclidean LCD code over Fqwith q > 3 exists if there is an [n, k, d]-linear code over Fqand an [n, k, d]-linear Hermitian LCD code over Fq2with q > 2 exists if there is an [n, k, d]-linear code over Fq2. Hence, when q > 3 (resp. q > 2) q-ary Euclidean (resp. q2-ary Hermitian) LCD codes possess the same asymptotical bound as q-ary linear codes (resp. q2-ary linear codes). This gives a direct proof that every triple of parameters [n, k, d] which is attainable by linear codes over Fqwith q > 3 (resp. over Fq2with q > 2) is attainable by Euclidean LCD codes (resp. by Hermitian LCD codes). In particular there exist families of q-ary Euclidean LCD codes (q > 3) and q2-ary Hermitian LCD codes (q > 2) exceeding the asymptotical Gilbert-Varshamov bound. Further, we give a second proof of these results using the theory of Gröbner bases. Finally, we present a new approach of constructing LCD codes by extending linear codes.

Bent functions are maximally nonlinear Boolean functions. Introduced by Rothaus and first examined by Dillon, these important functions have subsequently been studied by many researchers over the last four decades. Since a complete classification of bent functions appears elusive, many researchers concentrate on methods for constructing bent functions. In this paper, we investigate constructions of bent functions from involutions over finite fields in even characteristic. We present a generic construction technique, study its equivalence issues and show that linear involutions (which are an important class of permutations) over finite fields give rise to bent functions in bivariate representations. In particular, we exhibit new constructions of bent functions involving binomial linear involutions, whose dual functions are directly obtained without computation. The existence of bent functions from involutions relies heavily on solving systems of equations over finite fields.

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-good polynomials over some extension field of Fp 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) LRC codes over Fq for all positive integers r as localities, where q is near the code length n.

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 sidechannel 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.

Let p be an odd prime. Recently, Camarero and Martínez (in “Quasi-perfect Lee codes of radius 2 and arbitrarily large dimension”, IEEE Trans. Inform. Theory, vol. 62, no. 3, 2016) constructed some p-ary 2-quasi-perfect Lee codes for p ≡ ±5 (mod 12). In this paper, some infinite classes of p-ary 2-quasi-perfect Lee codes for any odd prime p with flexible length and dimension are presented. More specifically, we provide a new method for constructing quasi-perfect Lee codes. Our approach uses subsets derived from some quadratic curves over finite fields (in odd characteristic) to obtain two classes of 2-quasi-perfect Lee codes defined in the space Zpnfor n = pk+1/2 (with p ≡ 1, -5 (mod 12) and k is any integer, or p ≡ -1, 5 (mod 12) and k is an even integer) and n = pk-1/2 (with p ≡ -1, 5 (mod 12), k is an odd integer and pk> 12). Our codes encompass the p-ary (p ≡ ±5 (mod 12)) 2-quasiperfect Lee codes constructed by Camarero and Martínez. Furthermore, we prove that the related Cayley graphs are Ramanujan or almost Ramanujan using Kloosterman sums. This generalizes the work of Bibak, Kapron, and Srinivasan (in “The Cayley graphs associated with some quasi-perfect Lee codes are Ramanujan graphs”, IEEE Trans. Inform. Theory, vol. 62, no. 11, 2016) from the case p ≡ 3 (mod 4) and k = 1 to the case of any odd prime p and positive integer k. Finally, we derive some necessary conditions with the exponential sums of all 2-perfect codes and 2-quasi-perfect codes, and present a heuristic algorithm for constructing 2-perfect codes and 2-quasi-perfect codes. Our results show that, in general, the Cayley graphs associated with 2-perfect codes are Ramanujan. From the algorithm, some new 2-quasi-perfect Lee codes different from those constructed from quadratic curves are given. The Lee codes presented in this paper have applications in constrained and partial-response channels, flash memories, and decision diagrams.

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.

This paper is composed of two main parts related to the nonlinearity of vectorial functions. The first part is devoted to maximally nonlinear (n, m) functions (the so-called bent vectorial functions), which contribute to an optimal resistance to both linear and differential attacks on symmetric cryptosystems. They can be used in block ciphers at the cost of additional diffusion/compression/expansion layers, or as building blocks for the construction of substitution boxes (S-boxes), and they are also useful for constructing robust codes and algebraic manipulation detection codes. A main issue on bent vectorial functions is to characterize bent monomial functions Trmn(λxd) from F2nto F2m(where m is a divisor of n) leading to a classification of those bent monomials. We also treat the case of functions with multiple trace terms involving general results and explicit constructions. Furthermore, we investigate some open problems raised by Pasalic et al. and Muratovic-Ribic et al. in a series of papers on vectorial functions. The second part is devoted to the nonlinearity of (n, m)-functions. No tight upper bound is known when n/2 <; m <; n. The covering radius bound is the only known upper bound in this range (the Sidelnikov- Chabaud-Vaudenay bound coincides with it when m = n - 1 and it has no sense when m <; n - 1). Finding better bounds is an open problem since the 1990s. Moreover, no bound has been found during the last 23 years, which improve upon the covering radius bound for a large part of (n, m)-functions. We derive such upper bounds for functions, which are sufficiently unbalanced or which satisfy some conditions. These upper bounds imply some necessary conditions for vectorial functions to have large nonlinearity.

Within the diversity of existing Big Data and data processing solutions, meeting the requirements of privacy and security is becoming a real need. In this paper we tackle the security analysis of a new protocol of data processing in distributed system (PPDS). This protocol is composed of three phases: authentication, node head selection and data linking. This paper deals with its formal validation done using HLPSL language via AVISPA. We provide also its security analysis. Some performance analysis based on its proof of concept are also given in this paper.

Preserving privacy in Big data is one of the most debated topics of computer security. The fast growing volume of data and the need of companies to extract value from that data creates new complicated and serious security challenges. Despite its wide spread, the common use and the popularity of Big data paradigm, significant risks and challenges are inherent to this new concept, especially when we talk about externalized treatment of sensitive data via insecure network. In this paper we tackle the privacy challenge in Big Data. We focus in special case of data processing. Several Bank agencies want to share data processing while protecting the privacy. We propose a new protocol of communication between agencies. This protocol is composed of three phases: authentication, node head selection and data linking. Some proof of concept are also given in this paper.

In this paper, we introduce generalized hyperbent functions from F2nto ℤ2k, and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions f from F2nto ℤ2kconsist of components which are generalized (hyper)bent functions from F2ntoZ2k'for some k' <; k. For even n, most notably we show that the g-hyperbentness of f is equivalent to the hyperbentness of the components of f with some conditions on the Walsh-Hadamard coefficients. For odd n, we show that the Boolean functions associated to a generalized bent function form an affine space of semibent functions. This complements a recent result for even n, where the associated Boolean functions are bent.

Plateaued functions are very important cryptographic 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 Zpk. 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.

Bent functions are maximally nonlinear Boolean functions. They are important functions introduced by Rothaus and studied firstly by Dillon and next by many researchers for four decades. Since the complete classification of bent functions seems elusive, many researchers turn to design constructions of bent functions. In this paper, we show that linear involutions (which are an important class of permutations) over finite fields give rise to bent functions in bivariate representations. In particular, we exhibit new constructions of bent functions involving binomial linear involutions whose dual functions are directly obtained without computation. The existence of bent functions from involutions heavily relies on solving systems of equations over finite fields.

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.

In this paper we study involutions over a finite field of order 2n. We present some classes, several constructions of involutions and we study the set of their fixed points.

Since 2003, algebraic attacks have received a lot of attention in the cryptography literature. In this context, algebraic immunity quantifies the resistance of a Boolean function to the standard algebraic attack of the pseudo-random generators using it as a nonlinear Boolean function. A high value of algebraic immunity is now an absolutely necessary cryptographic criterion for a resistance to algebraic attacks but is not sufficient, because of more general kinds of attacks so-called Fast Algebraic Attacks. In view of these attacks, the study of the set of annihilators of a Boolean function has become very important. We show that studying the annihilators of a Boolean function can be translated into studying the codewords of a linear code. We then explain how to exploit that connection to evaluate or estimate the algebraic immunity of a cryptographic function. Direct links between the theory of annihilators used in algebraic attacks and coding theory are established using an atypical univariate approach.

In this paper, a generic construction of codebooks based on binary codes is introduced. With this generic construction, a few previous constructions of optimal codebooks are extended, and a new class of codebooks almost meeting the Levenshtein bound is presented. Exponentially many codebooks meeting or almost meeting the Levenshtein bound from binary codes are obtained in this paper. The codebooks constructed in this paper have alphabet size 4. As a byproduct, three bounds on the parameters of binary codes are derived.

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.

This paper is devoted to hyperbent functions with multiple trace terms (including binomial functions) via Dillon-like exponents. We show how the approach developed by Mesnager to extend the Charpin–Gong family, which was also used by Wang and coworkers to obtain another similar extension, fits in a much more general setting. To this end, we first explain how the original restriction for Charpin–Gong criterion can be weakened before generalizing the Mesnager approach to arbitrary Dillon-like exponents. Afterward, we tackle the problem of devising infinite families of extension degrees for which a given exponent is valid and apply these results not only to reprove straightforwardly the results of Mesnager and Wang and coworkers, but also to characterize the hyperbentness of several new infinite classes of Boolean functions. We go into full details only for a few of them, but provide an algorithm (and the corresponding software) to apply this approach to an infinity of other new families. Finally, we compare the asymptotic and practical performances of different characterizations, including these in terms of hyperelliptic curves, and actually build hyperbent functions in cases which could not be attained through naive computations of exponential sums.

This paper is devoted to hyper-bent functions with multiple trace terms (including binomial functions) via Dillon-like exponents. We show how the approach developed by Mesnager to extend the Charpin-Gong family, which was also used by Wang et al. to obtain another similar extension, fits in a much more general setting. To this end, we first explain how the original restriction for Charpin-Gong criterion can be weakened before generalizing the Mesnager approach to arbitrary Dillon-like exponents. Afterward, we tackle the problem of devising infinite families of extension degrees for which a given exponent is valid and apply these results not only to reprove straightforwardly the results of Mesnager, and Wang et al., but also to characterize the hyper-bentness of new infinite classes of Boolean functions.

This paper consists of two main contributions. First, the Niho bent function consisting of 2rexponents (discovered by Leander and Kholosha) is studied. The dual of the function is found and it is shown that this new bent function is not of the Niho type. Second, all known univariate representations of Niho bent functions are analyzed for their relation to the completed Maiorana-McFarland classM. In particular, it is proven that two families do not belong to the completed classM. The latter result gives a positive answer to an open problem whether the classHof bent functions introduced by Dillon in his thesis of 1974 differs from the completed classM.

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

Computed is the dual of the Niho bent function consisting of 2rexponents that was found by Leander and Kholosha. The algebraic degree of the dual is calculated and it is shown that this new bent function is not of the Niho type. This note is a follow-up of the recent paper by Carlet and Mesnager.

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.

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.

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).

Secret sharing is an important topic in cryptography and has applications in information security. We use self-dual codes to construct secret-sharing schemes. We use combinatorial properties and invariant theory to understand the access structure of these secret-sharing schemes. We describe two techniques to determine the access structure of the scheme, the first arising from design properties in codes and the second from the Jacobi weight enumerator, and invariant theory.

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 therth-order nonlinearity. As recalled by Carlet at CRYPTO'06, many papers have illustrated the importance of therth-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 therth-order nonlinearity have been shown by Carlet . None of them improves upon the other one in all situations. In this paper, we prove a new lower bound on therth-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.

By deriving bounds on character sums of Boolean functions and by using the characterizations, due to Kasami , 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

By deriving bounds on character sums of Boolean functions and by using the characterizations, due to Kasami and Tokura, of those elements of the Reed-Muller codes whose Hamming weights are smaller than twice 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