We investigate APN functions which can be represented as rational functions and we provide non-existence results exploiting the connection between these functions and specific algebraic varieties over finite fields. This approach allows to classify families of functions when previous approaches cannot be applied.

Pairing-based cryptographic protocols are typically vulnerable to small-subgroup attacks in the absence of protective measures. Subgroup membership testing is one of the feasible methods to address this security weakness. However, it generally causes an expensive computational cost on many pairing-friendly curves. Recently, Scott proposed efficient methods of subgroup membership testings for \(\mathbb

Extended Boolean functions (EBFs) are one of the most important tools in cryptography and spreading sequence design in communication systems. In this paper, we use EBFs to design new sets of spreading sequences for non-orthogonal multiple access (NOMA), which is an emerging technique capable of supporting massive machine-type communications (mMTC) in 5 G and beyond. In this work, first p-ary complementary

The Patterson–Wiedemann (PW) construction, which is defined for an odd number n of variables with n being the product of two distinct prime numbers p and q, can be interpreted as idempotent functions which are represented by the (d, r)-interleaved sequences formed by all-zero and all-one columns, where \(r=(2^p-1)(2^q-1)\) and \(d=\frac{(2^n-1)}{r}\). We here study a modified form of the PW construction

A Boolean function f on n variables is said to be a bent function if the absolute value of all its Walsh coefficients is \(2^{n/2}\). Our main result is a new asymptotic lower bound on the number of Boolean bent functions. It is based on a modification of the Maiorana–McFarland family of bent functions and recent progress in the estimation of the number of transversals in latin squares and hypercubes

This paper addresses a number of problems concerning Buekenhout-Tits unitals in \({{\,\textrm{PG}\,}}(2, q^2)\), where \(q = 2^{2e + 1}\) and \(e \ge 1\). We show that all Buekenhout-Tits unitals are equivalent under \({{\,\textrm{PGL}\,}}(3, q^2)\) [addressing an open problem in Barwick and Ebert (Unitals in Projective Planes. Springer Monographs in Mathematics. Springer, New York, 2008)], explicitly

Let \({\mathcal {D}}\) be a nontrivial 3-(v, k, 1) design admitting a block-transitive group G of automorphisms. A recent work of Gan and the second author asserts that G is either affine or almost simple. In this paper, it is proved that if G is almost simple with socle an alternating group, then \({\mathcal {D}}\) is the unique 3-(10, 4, 1) design, and \(G=\textrm{PGL}(2,9)\), \(\textrm{M}_{10}\)

In this paper, several families of irreducible constacyclic codes over finite fields and their duals are studied. The weight distributions of these irreducible constacyclic codes and the parameters of their duals are settled. Several families of irreducible constacyclic codes with a few weights and several families of optimal constacyclic codes are constructed. As by-products, a family of \([2n, (n-1)/2

A perfect code in a graph \(\Gamma \) is a subset C of \(V(\Gamma )\) such that no two vertices in C are adjacent and every vertex in \(V(\Gamma ){\setminus } C\) is adjacent to exactly one vertex in C. Let G be a finite group and C a subset of G. Then C is said to be a perfect code of G if there exists a Cayley graph of G admiting C as a perfect code. It is proved that a subgroup H of G is a perfect

To find the non-standard binary Golay sequences of length \(2^{m}\) or theoretically prove their nonexistence are still open. It has been shown that all the standard q-ary (q even) Golay sequence pairs of length \(2^m\) can be obtained by q-ary Golay array pairs of dimension m and size \(2\times 2 \times \cdots \times 2\) of a particular algebraic normal form. We extend the appellation “standard" from

We present a polynomial time algorithm for breaking NTRU encryption schemes with multiple keys. Our algorithm takes advantage of the specific sampling regime used in NTRU encryption, which samples secret polynomials with a fixed number of coefficients of 1 and \(-1\). By constructing an equation system on the secret keys, we are able to recover the unique secret key when n multiple keys sharing a common

APN power functions are fundamental objects as they are used in various theoretical and practical applications in cryptography, coding theory, combinatorial design, and related topics. Let p be an odd prime and n be a positive integer. Let \(F(x)=x^d\) be a power function over \({\mathbb {F}}_{p^n}\), where \(d=\frac{3p^n-1}{4}\) when \(p^n\equiv 3\pmod 8\) and \(d=\frac{p^n+1}{4}\) when \(p^n\equiv

Graphical designs are the extension of spherical designs to finite graphs from the viewpoint of quadrature formulas. In this paper, we investigate optimal graphical designs on hypercubes, especially the conjecture proposed by Babecki that the Hamming code is an optimal graphical design on the hypercube. We prove that this conjecture is not true using certain binary t-error-correcting BCH codes. We

The existence of large sets of Kirkman triple systems (LKTSs) is one of the best-known open problems in combinatorial design theory. Steiner quadruple systems with resolvable derived designs (RDSQSs) play an important role in the recursive constructions of LKTSs. In this paper, we introduce a special combinatorial structure \(\hbox {RDSQS}^{*}(v)\) and use it to present a construction for RDSQS(4v)

In this paper we prove that the class of Goppa codes whose Goppa polynomial is of the form \(g(x) = \textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}\) where \(\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}\) is a trace polynomial from a field extension of degree \(m \ge 3\) has a better minimum distance than what the Goppa bound \(d \ge 2\deg (g(x))+1\) implies. This

The intersection \(\textbf{C}\cap \textbf{C}^{\perp _H}\) of a linear code \(\textbf{C} \subset \textbf{F}_{q^2}^n\) and its Hermitian dual \(\textbf{C}^{\perp _H}\) is called the Hermitian hull of this code. A linear code \(\textbf{C} \subset \textbf{F}_{q^2}^n\) satisfying \(\textbf{C} \subset \textbf{C}^{\perp _H}\) is called Hermitian self-orthogonal. Many Hermitian self-orthogonal codes were given

Let \(r\ge 3\) be a positive integer and \({\mathbb {F}}_q\) the finite field with q elements. In this paper, we consider the r-regular complete permutation property of maps with the form \(f=\tau \circ \sigma _M\circ \tau ^{-1}\) where \(\tau \) is a PP over an extension field \({\mathbb {F}}_{q^d}\) and \(\sigma _M\) is an invertible linear map over \({\mathbb {F}}_{q^d}\). When \(\tau \) is additive

Orthogonal array is one of the important research subjects in combinatorial design theory and experimental design theory, and it is widely applied to statistics, computer science, coding theory and cryptography. There are many constructions and results for orthogonal array of strength 2, however the results on orthogonal array of strength \(t\ge 3\) are rare. In this paper, we first present two new

In this paper we settle the question of whether a finite-dimensional vector space \({{\mathcal {V}}}\) over \({\mathbb {F}}_p,\) with p an odd prime, and the family of all the k-sets of elements of \({\mathcal {V}}\) summing up to a given element x, form a 1-\((v,k,\lambda _1)\) or a 2-\((v,k,\lambda _2)\) block design, and, in either case, we find a closed form for \(\lambda _i\) and characterize

The hull of a linear code over finite fields is the intersection of the code and its dual code, which has been widely studied due to its wide applications. In this paper, we develop a general method for constructing linear codes with small hulls using the eigenvalues of the generator matrices. Using this method, we construct many optimal Euclidean and Hermitian LCD codes, which improve the previously

Broadcast Encryption (BE) is public-key encryption allowing a sender to encrypt a message by specifing recipients, and only the specified recipients can decrypt the message. In several BE applications, since the privacy of recipients allowed to access the message is often as important as the confidentiality of the message, anonymity is introduced as an additional but important security requirement

Almost Perfect Nonlinear (APN) functions are very useful in cryptography, when they are used as S-Boxes due to their good resistance to differential cryptanalysis. Also, some of the curves and surfaces defined by the corresponding nonlinear functions have many rational points and have applications to Algebraic-Geometric (AG) codes (Janwa and Wilson in Applied algebra, algebraic algorithms and error-correcting

Let C be a four-weight binary code, which has all one vector. Furthermore, we assume that C supports t-designs for all weights obtained from the Assmus–Mattson theorem. We previously showed that \(t\le 5\). In the present paper, we show an analogue of this result in the cases of five and six-weight codes.

The paper deals with t-designs that can be partitioned into s-designs, each missing a point of the underlying set, called point-missing s-resolvable t-designs, with emphasis on their applications in constructing t-designs. The problem considered may be viewed as a generalization of overlarge sets which are defined as a partition of all the \(\left( {\begin{array}{c}v +1\\ k\end{array}}\right) \) k-sets

We give upper bounds on the power moments of the number of fixed points of a family of subset sum pseudorandom number generators, introduced by Rueppel (Analysis and design of stream ciphers, Springer-Verlag, Berlin, 1986).

Let \(\mathcal {S}\) be a finite thick generalized quadrangle, and suppose that G is an automorphism group of \(\mathcal {S}\). If G acts primitively on both the points and lines of \(\mathcal {S}\), then it is known that G must be almost simple. In this paper, we show that if the socle of G is \(\textrm{PSL}(2,q)\) with \(q\ge 4\), then \(q=9\) and \(\mathcal {S}\) is the unique generalized quadrangle

Negacyclic BCH codes are a subclass of neagcyclic codes and are the best linear codes in many cases. However, there have been very few results on negacyclic BCH codes. Let q be an odd prime power and m be a positive integer. The objective of this paper is to study negacyclic BCH codes with length \(\frac{q^m-1}{2}\) and \(\frac{q^m+1}{2}\) over the finite field \({\textrm{GF}}(q)\) and analyse their

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

Permutation codes have received a great attention due to various applications. For different applications, one needs permutation codes under different metrics. The generalized Cayley metric was introduced by Chee and Vu (in: 2014 IEEE international symposium on information theory, Honolulu, June 29–July 4, 2014, pp 2959–2963, 2014) and this metric includes several other metrics as special cases. However

Up to a new invariant \(\mu (b)\), the complete b-symbol weight distribution of a particular kind of two-weight irreducible cyclic codes, was recently obtained by Zhu et al. (Des Codes Cryptogr 90(5):1113–1125, 2022). The purpose of this paper is to simplify and generalize the results of Zhu et al., and obtain the b-symbol weight distributions of all one-weight and two-weight semiprimitive irreducible

In the present paper, we introduce the concept of harmonic Tutte polynomials of matroids and discuss some of their properties. In particular, we generalize Greene’s theorem, thereby expressing harmonic weight enumerators of codes as evaluations of harmonic Tutte polynomials.

We consider the problem of constructing an unconditionally secure cipher for the case when the key length is less than the length of the encrypted message. (Unconditional security means that a computationally unbounded adversary cannot obtain information about the encrypted message without the key). In this article, we propose a cipher based on data compression and randomisation in combination with

It is well-known that the dimension of optimal anticodes in the rank-metric is divisible by the maximum m between the number of rows and columns of the matrices. Moreover, for a fixed k divisible by m, optimal rank-metric anticodes are the codes with least maximum rank, among those of dimension k. In this paper, we study the family of rank-metric codes whose dimension is not divisible by m and whose

In this paper, we provide a recursive method to construct self-orthogonal and self-dual codes of the type \(\{k_1,k_2,\ldots ,k_e\}\) and length n over the quasi-Galois ring \(\mathbb {F}_{2^r}[u]/\) from a self-orthogonal code of the same length n and dimension \(k_1+k_2+\cdots +k_{\lceil \frac{e}{2}\rceil }\) over \(\mathbb {F}_{2^r}\) and vice versa, where \(\mathbb {F}_{2^r}\) is the finite field

Let \({\mathbb K}\) be the Galois field \({\mathbb F}_{q^t}\) of order \(q^t, q=p^e, p\) a prime, \(A={{\,\mathrm{{Aut}}\,}}({\mathbb K})\) be the automorphism group of \({\mathbb K}\) and \(\varvec{\sigma }=(\sigma _0,\ldots , \sigma _{d-1}) \in A^d\), \(d \ge 1\). In this paper the following generalization of the Veronese map is studied: $$\begin{aligned} \nu _{d,\varvec{\sigma }} : \langle v \rangle