Building upon the observation that the newly defined Ellingsen, et al. (2020) concept of c-differential uniformity is not invariant under EA or CCZ-equivalence Hasan et al. (2021), we showed in Stănică and Geary (2021) that adding some appropriate linearized monomials increases the c-differential uniformity of the inverse function, significantly, for some c. We continue that investigation here. First

The recent FLIP cipher is an encryption scheme described by Méaux et al. at the conference EUROCRYPT 2016. It is based on a new stream cipher model called the filter permutator and tries to minimize some parameters (including the multiplicative depth). In the filter permutator, the input to the Boolean function has constant Hamming weight equal to the weight of the secret key. As a consequence, Boolean

Bent functions are extremal combinatorial objects with several applications, such as coding theory, maximum length sequences, cryptography, the theory of difference sets, etc. Based on C. Carlet’s secondary construction, S. Mesnager proposed in 2014 an effective method to construct bent functions in their bivariate representation by employing three permutations of the finite field \({\mathbb {F}}_{2^{m}}\)

We show that the single-server computational PIR protocol proposed by Holzbaur, Hollanti and Wachter-Zeh in [6] is not private, in the sense that the server can recover in polynomial time the index of the desired file with very high probability. The attack relies on the following observation. Removing rows of the query matrix corresponding to the desired file yields a large decrease of the dimension

Browning et al. (2010) exhibited almost perfect nonlinear (APN) permutations on \(\mathbb {F}_{2^{6}}\). This was the first example of an APN permutation on an even degree extension of \(\mathbb {F}_{2}\). In their approach of finding an APN permutation, Browning et al. made use of a necessary and sufficient condition based on the Walsh transform. In this paper, we give an algorithm based on a related

Integrated Circuits (ICs) are sensible to a wide range of (passive, active, invasive, non-invasive) physical attacks. In this context, Hardware Trojans (HTs), that are malicious modifications of a circuit by an untrusted manufacturer, are one of the most challenging threats to mitigate. HTs aim to alter the functionality of the infected chip in a malicious way, e.g. under specific conditions known

This paper is devoted to the affine-invariant ternary codes defined by Hermitian functions. We first compute the incidence matrices of the 2-designs supported by the minimum weight codewords of these ternary codes. Then we show that the linear codes spanned by the rows of these incidence matrices are subcodes of the 4-th order generalized Reed-Muller codes and also hold 2-designs. Finally, we determine

Given a binary sequence, its trace representation allows us to reconstruct itself efficiently and to analyze its properties, such as the linear complexity. In this paper, we study a family of the binary sequences derived from Euler quotients modulo pq, where p and q are two distinct odd primes and p divides q − 1. Our main contribution is to give a trace representation of this family within these assumptions

This article presents the construction of binary linear Hadamard codes with parameters (2n,4n,n) over the fields F4 and F8. We generalize this construction to the field \(F_{2^{k}}\) to generate Hadamard codes with parameters \(\left (2^{k-1}n, 2^{k}n, 2^{k-2}n\right )\) for \(k\in \mathbb {N}\). We construct s-PD sets of size s + 1, a special subset of the permutation automorphism group of a code

We provide new families of minimal codes in any characteristic, useful for the construction of secret sharing schemes. Also, an inductive construction of minimal codes is presented.

A fascinating topic of combinatorics is the study of t-designs, which has a very long history. The incidence matrix of a t-design generates a linear code over GF(q) for any prime power q, which is called the linear code of the t-design over GF(q). On the other hand, some linear codes hold t-designs with t ≥ 1. The purpose of this paper is to study the linear codes of t-designs held in the Reed-Muller

Strong external difference families (SEDFs) have applications to cryptography and are rich combinatorial structures in their own right. We extend the definition of SEDF from abelian groups to all finite groups, and introduce the concept of equivalence. We prove new recursive constructions for SEDFs and generalized SEDFs (GSEDFs) in cyclic groups, and present the first family of non-abelian SEDFs. We

Optical emissions from semiconductors have been found to leak important information from embedded security devices. Unfortunately, the required substrate thinning and photon emission microscopy equipment is typically very expensive. Low cost equipment setups are proposed for substrate thinning/polishing and photon emission microscopy. For the first time, a security attack enabling the parallel readout

Construction of Mutually Unbiased Bases (MUBs) is a very challenging combinatorial problem in quantum information theory with several long standing open questions in this domain. With certain relaxations, the object Approximate Mutually Unbiased Bases (AMUBs) is defined in this context. In this paper we provide a method to construct upto \((\sqrt {d} + 1)\) many AMUBs in dimension d = q2, where q is

Let \(R=\mathbb {Z}_{4}+u\mathbb {Z}_{4}\) be a finite non-chain ring, where u2 = 1. In this paper, we consider (σ, δ)-skew quasi-cyclic codes over the ring R, where σ is an automorphism of R and δ is an inner σ-derivation of R. We determine the structure of 1-generator (σ, δ)-skew quasi-cyclic codes over R and give a sufficient condition for 1-generator (σ, δ)-skew quasi-cyclic codes over R to be

While the classical differential uniformity (c = 1) is invariant under the CCZ-equivalence, the newly defined (Ellingsen et al., IEEE Trans. Inf. Theory 66(9), 5781–5789, 2020) concept of c-differential uniformity (cDU), as was observed in Hasan et al. (2020), is not invariant under EA or CCZ-equivalence, for c≠ 1. In this paper, we find an intriguing behavior of the inverse function, namely, that

For a positive integer k and a linearized polynomial L(X), polynomials of the form \(P(X)=G(X)^{k}-L(X) \in {\mathbb F}_{q^{n}}[X]\) are investigated. It is shown that when L has a non-trivial kernel and G is a permutation of \(\mathbb {F}_{q^{n}}\), then P(X) cannot be a permutation if \(\gcd (k,q^{n}-1)>1\). Further, necessary conditions for P(X) to be a permutation of \(\mathbb {F}_{q^{n}}\) are

The original version of this article unfortunately contained some mistakes on the equations at page 4 of the published paper.

An algorithm for computing the weight distribution of a linear [n,k] code over a finite field \(\mathbb {F}_{q}\) is developed. The codes are represented by their characteristic vector with respect to a given generator matrix and a generator matrix of the k-dimensional simplex code \(\mathcal {S}_{q,k}\).

Subspace codes, especially cyclic subspace codes, have attracted a wide attention in the past few decades due to their applications in error correction for random network coding. In 2016, Ben-Sasson et al. gave a systematic approach to constructing cyclic subspace codes by employing subspace polynomials. Inspired by Ben-Sasson’s idea, Chen et al. also provided some constructions of cyclic subspace

The hull of a linear code over finite fields, the intersection of the code and its dual, has been of interest and extensively studied due to its wide applications. For example, it plays a vital role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and for computing the automorphism group of a linear code. People are interested in pursuing linear codes

Based on the applications of codes with few weights, we define the so-called relative four-weight codes and present a method for constructing such codes by using the finite projective geometry method. Also, the t-wise intersection and the trellis of relative four-weight codes are determined.

Recently, subfield codes of some optimal linear codes have been studied. In this paper, we further investigate a class of subfield codes and generalize the results of the subfield codes of the conic codes in Ding and Wang (Finite Fields Appl. 56, 308–331, 2020). The weight distributions of these subfield codes and the parameters of their duals are determined. Some of the presented codes are optimal

DNA storage has emerged as an important area of research. The reliability of a DNA storage system depends on designing those DNA strings (called DNA codes) that are sufficiently dissimilar. In this work, we introduce DNA codes that satisfy the newly introduced constraint, a generalization of the non-homopolymers constraint. In particular, each codeword of the DNA code has the specific property that

In this study we consider Euclidean and Hermitian self-dual codes over the direct product ring \(\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})\) where v2 = v. We obtain some theoretical outcomes about self-dual codes via the generator matrices of free linear codes over \(\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})\). Also, we obtain upper bounds on the minimum distance of

Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, two families of binary subfield codes with a few weights are presented from two special classes of linear codes, and their parameters are explicitly determined. Moreover

Symbol-pair code is a new coding framework proposed to combat pair-errors in symbol-pair read channels. Remarkably, a classical maximum distance separable (MDS) code is also an MDS symbol-pair code. In this paper, we investigate the symbol-pair weight distribution of MDS codes and the symbol b-weight distribution of simplex codes over finite fields respectively. Surprisingly, one can derive that all

Substitution functions (S-boxes) play an important role in the security of AES-like cryptosystems, but the cryptosystems are highly vulnerable against fault injection attacks. Some research has been carried out previously to prevent fault injection attacks on AES, but most of the countermeasures are restricted to the detection of faults only, and they only work at the cost of large hardware needed

In this paper, we study the relationship between algebraic manipulation detection (AMD) codes and highly nonlinear functions. As applications, on one hand, a generic construction for systematic AMD codes is introduced based on highly nonlinear functions. Systematic AMD codes with new parameters can be generated from known highly nonlinear functions. Especially, several infinite classes of optimal systematic

Let λ be a unit of the finite commutative chain ring \(R=\mathbb {F}_{p^{m}}+u\mathbb {F}_{p^{m}}+u^{2}\mathbb {F}_{p^{m}}=\{\alpha +u\upbeta +u^{2}\gamma : \alpha ,\upbeta ,\gamma \in \mathbb { F}_{p^{m}}\}\) with u3 = 0, where p is an odd prime and m is a positive integer. In this paper, we consider any λ-constacyclic codes of length 2ps over R. In the case of square λ = δ2, where δ ∈ R, the algebraic

This paper contributes to compute the 2-adic complexity of two classes of Ding-Helleseth generalized cyclotomic sequences. Results show that the 2-adic complexity of these sequences is good enough to resist the attack by the rational approximation algorithm.

In this paper, we propose a mechanism for the construction of MDS codes with arbitrary dimensions of Euclidean hulls. Precisely, we construct (extended) generalized Reed-Solomon (GRS) codes with assigned dimensions of Euclidean hulls from self-orthogonal GRS codes. It turns out our constructions are more general than previous works on Euclidean hulls of (extended) GRS codes.

In this paper, we consider an interpolation-based decoding algorithm for a large family of maximum rank distance codes, known as the additive generalized twisted Gabidulin codes, over the finite field \(\mathbb {F}_{q^{n}}\) for any prime power q. This paper extends the work of the conference paper Li and Kadir (2019) presented at the International Workshop on Coding and Cryptography 2019, which decoded

We discuss a special class of permutation polynomials over finite fields focusing on some recent work on their factorization. In particular we obtain permutation polynomials with various factorization patterns that are favoured for applications. We also address a wide range of problems of current interest concerning irreducible factors of the terms of sequences and iterations of such permutation polynomials

Multiplicative complexity (MC) is defined as the minimum number of AND gates required to implement a function with a circuit over the basis (AND, XOR, NOT). Boolean functions with MC 1 and 2 have been characterized in Fisher and Peralta (2002), and Find et al. (IJICoT 4(4), 222–236, 2017), respectively. In this work, we identify the affine equivalence classes for functions with MC 3 and 4. In order

In this paper, we present two new constructions of complex codebooks with multiplicative characters, additive characters and trace functions over finite fields, and determin the maximal cross-correlation amplitude of these codebooks. We prove that the codebooks we constructed are asymptotically optimal with respect to the Welch bound. Moreover, in the first construction, we generalize the result in

Bent functions in odd characteristic can be either (weakly) regular or non-weakly regular. Furthermore one can distinguish between dual-bent functions, which are bent functions for which the dual is bent as well, and non-dual bent functions. Whereas a weakly regular bent function always has a bent dual, a non-weakly regular bent function can be either dual-bent or non-dual-bent. The classical constructions

We define affine equivalence of S-boxes with respect to modular addition, and explore its use in cryptanalysis. We have identified classes of small bijective S-boxes with respect to this new equivalence, and experimentally computed their properties.

The Fischer result about distinctness of the Kloosterman sums on F∗p is extended for the finite fields of degrees of extension that are powers of 2. To obtain the desired outcome, we give an elementary proof of the fact that there does not exist a pair of Kloosterman sums on same odd characteristic fields which are opposite to each other.

Recently, Carlet introduced (Carlet Finite. Fields Appl. 53, 226–253 (2018)) the concept of component-wise uniform (CWU) functions which is a stronger notion than being almost perfect nonlinear (APN). Carlet showed that crooked functions (in particular quadratic functions, including Gold functions) and the compositional inverse of a specific Gold function are CWU. Carlet also compiled a table on the

In this paper we define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, 2k-Hadamard, consta-Hadamard and all HN-transforms. We describe the behavior of what we call the root-Hadamard transform for a generalized Boolean function f in terms of the binary components of f. Further, we define a notion of complementarity (in the spirit of the Golay

A special function is a function either of special form or with a special property. Special functions have interesting applications in coding theory and combinatorial t-designs. The main objective of this paper is to survey t-designs constructed from special functions, including quadratic functions, almost perfect nonlinear functions, almost bent functions, bent functions, bent vectorial functions

The boomerang attack, introduced by Wagner in 1999, is a cryptanalysis technique against block ciphers based on differential cryptanalysis. In particular it takes into consideration two differentials, one for the upper part of the cipher and one for the lower part, and it exploits the dependency of these two differentials. At Eurocrypt’18, Cid et al. introduced a new tool, called the Boomerang Connectivity

A bent function is a Boolean function in even number of variables which is on the maximal Hamming distance from the set of affine Boolean functions. It is called self-dual if it coincides with its dual. It is called anti-self-dual if it is equal to the negation of its dual. A mapping of the set of all Boolean functions in n variables to itself is said to be isometric if it preserves the Hamming distance

Let f(x) = xd be a power mapping over \(\mathbb {F}_{n}\) and \(\mathcal {U}_{d}\) the maximum number of solutions \(x\in \mathbb {F}_{n}\) of \({\Delta }_{f,c}(x):=f(x+c)-f(x)=a\text {, where }c,a\in \mathbb {F}_{n}\text { and } c\neq 0\). f is said to be differentially k-uniform if \(\mathcal {U}_{d} =k\). The investigation of power functions with low differential uniformity over finite fields \(\mathbb

Inspired by a recent work of Tang et al. on constructing bent functions [14, IEEE TIT, 63(1): 6149-6157, 2017], we introduce a property (Pτ) of any Boolean function that its second order derivatives vanish at any direction (ui,uj) for some τ-subset {u1,…,uτ} of \(\mathbb {F}_{2^{n}}\), and then establish a link between this property and the construction of Tang et al. (IEEE Trans. Inf. Theory 63(10)

Linear codes have a wide range of applications in the data storage systems, communication systems, consumer electronics products since their algebraic structure can be analyzed and they are easy to implement in hardware. How to construct linear codes with excellent properties to meet the demands of practical systems becomes a research topic, and it is an efficient way to construct linear codes from

We consider n-bit permutations with differential uniformity of 4 and null nonlinearity. We first show that the inverses of Gold functions have the interesting property that one component can be replaced by a linear function such that it still remains a permutation. This directly yields a construction of 4-uniform permutations with trivial nonlinearity in odd dimension. We further show their existence

Let \(\mathbb {F}_{2^{m}}\) be the finite field with 2m elements, where m is a positive integer. Recently, Heng and Ding in (Finite Fields Appl. 56:308–331, 2019) studied the subfield codes of two families of hyperovel codes and determined the weight distribution of the linear code $$ \mathcal{C}_{a,b}=\left\{((\text{Tr}_{1}^{m}(a f(x)+bx)+c)_{x \in \mathbb{F}_{2^{m}}}, \text{Tr}_{1}^{m}(a), \text{Tr}_{1}^{m}(b))

We correct some mistakes in the paper “A mass formula for negacyclic codes of length 2k and some good negacyclic codes over \(\mathbb {Z}_{4}+u\mathbb {Z}_{4}\)” (Bandi et al. Cryptogr. Commun. 9, 241–272, 2017).

A bent4 function is a Boolean function with a flat spectrum with respect to a certain unitary transform \(\mathcal {T}\). It was shown previously that a Boolean function f in an even number of variables is bent4 if and only if f + σ is bent, where σ is a certain quadratic function depending on \(\mathcal {T}\). Hence bent4 functions are also called shifted bent functions. Similarly, a Boolean function

The concept of a locally repairable code (LRC) was introduced to protect the data from disk failures in large-scale storage systems. In this paper, we consider the LRCs with multiple disjoint repair sets and each repair set contains exactly one check symbol. By using several structures from combinatorial design theory, such as balanced incomplete block design, cyclic packing, group divisible design

The original version of this article unfortunately contained a mistake in the main title.

In this paper we study almost p-ary sequences and their autocorrelation coefficients. We first study the number ℓ of distinct out-of-phase autocorrelation coefficients for an almost p-ary sequence of period n + s with s consecutive zero-symbols. We prove an upper bound and a lower bound on ℓ. It is shown that ℓ can not be less than \(\min \limits \{s,p,n\}\). In particular, it is shown that a nearly

Recently Budaghyan et al. (Cryptogr. Commun. 12, 85–100, 2020) introduced a procedure for investigating if CCZ-equivalence can be more general than EA-equivalence together with inverse transformation (when applicable). In this paper, we show that it is possible to use this procedure for classifying, up to EA-equivalence, all known APN functions in dimension 6. We also give some discussion for dimension

For a Distributed Storage System (DSS), the Fractional Repetition (FR) code is a class in which replicas of encoded data packets are stored on distributed chunk servers, where the encoding is done using the Maximum Distance Separable (MDS) code. The FR codes allow for the exact uncoded repair with minimum repair bandwidth. In this paper, FR codes (called Flower codes) are constructed using finite binary

The set of linear structures of most known balanced Boolean functions is non-trivial. In this paper, some balanced Boolean functions whose set of linear structures is trivial are constructed. We show that any APN function in even dimension must have a component whose set of linear structures is trivial. We determine a general form for the number of bent components in quadratic APN functions in even

This paper presents an explicit representation for the solutions of the equation \({\sum }_{i=0}^{\frac kl-1}x^{2^{li}} = a \in \mathbb {F}_{2^{n}}\) for any given positive integers k, l with l|k and n, in the closed field \({\overline {\mathbb {F}_{2}}}\) and in the finite field \(\mathbb {F}_{2^{n}}\). As a by-product of our study, we are able to completely characterize the a’s for which this equation

We consider a special type of sequence of arithmetic progressions, in which consecutive progressions are related by the property: ithterms ofjth, (j + 1)thprogressions of the sequence are multiplicative inverses of each other modulo(i + 1)thterm ofjthprogression. Such a sequence is uniquely defined for any pair of co-prime numbers. A computational problem, defined in the context of such a sequence

In 2018, Ding et al. introduced a new generalisation of the punctured binary Reed-Muller codes to construct LCD codes and 2-designs. They studied the minimum distance of the codes and proposed an open problem about the minimum distance. In this paper, several new results on the minimum distance of the generalised punctured binary Reed- Muller are presented. Particularly, some of the results are a generalisation