In order to minimize or reduce the mutual interference, low-hit-zone (LHZ) frequency-hopping sequence (FHS) sets with optimal periodic partial Hamming correlation (PPHC) properties have been well applied in quasi-synchronous (QS) frequency-hopping multiple-access (FHMA) communication systems. Besides, to better meet the needs of different frequency-hopping communication scenarios, the research on LHZ-FHS

Threshold Implementations are known countermeasures defending against side-channel attacks via the use of masking techniques. While sufficient properties are known to defend against first-order side-channel attacks, it is not known how to achieve higher-order security. This work generalizes the Threshold Implementation notion of uniformity and proves it achieves second-order protection. The notion

We define a family of efficiently computable invariants for (n,m)-functions under EA-equivalence, and observe that, unlike the known invariants such as the differential spectrum, algebraic degree, and extended Walsh spectrum, in the case of quadratic APN functions over \(\mathbb {F}_{2^n}\) with n even, these invariants take on many different values for functions belonging to distinct equivalence classes

Let \(R=\mathbb {F}_{q^{2}}+u \mathbb {F}_{q^{2}},\) where \(\mathbb {F}_{q^{2}}\) is the finite field with q2 elements, q is a power of a prime p, and u2 = 0. In this paper, a class of maximal entanglement entanglement-assisted quantum error-correcting codes (EAQECCs) is obtained by employing (1 − u)-constacyclic Hermitian linear complementary dual (LCD) codes of length n over R. First, we give a

Espresso cipher is designed targeting 5G wireless communication systems. To achieve high efficiency, a maximum period Galois NLFSR is used as the only building block. The Galois NLFSR is constructed by a scalable method which converts a maximum LFSR to a Galois NLFSR. Based on this method, a new class of stream ciphers, namely maximum period Galois NLFSR-based stream ciphers can be designed. However

Three constructions of binary sequences with period 4N and optimal autocorrelation value or optimal autocorrelation magnitude have been presented by Tang and Gong based on interleaving technique. In this paper, the 2-adic complexity of the sequences with optimal autocorrelation magnitude constructed from the Legendre sequence pair or the twin-prime sequence pair is investigated. With the method proposed

In the context of side-channel countermeasures, threshold implementations (TI) have been introduced in 2006 by Nikova et al. to defeat attacks which exploit hardware effects called glitches. On several aspects, TI may be seen as an extension of another classical side-channel countermeasure, called masking, which is essentially based on the sharing of any internal state of the processing into independent

Nikova et al. investigated the decomposition problem of power permutations over finite fields \(\mathbb {F}_{2^{n}}\) in (Cryptogr. Commun. 11:379–384, 2019). In particular, they provided an algorithm to give a decomposition of a power permutation into quadratic power permutations. Their algorithm has a precomputation step that finds all cyclotomic classes of \(\mathbb {F}_{2^{n}}\) and then use the

We study the notion of formal self duality in finite abelian groups. Formal duality in finite abelian groups has been proposed by Cohn, Kumar, Reiher and Schürmann. In this paper we give a precise definition of formally self dual sets and discuss results from the literature in this perspective. Also, we discuss the connection to formally dual codes. We prove that formally self dual sets can be reduced

The transformation algorithm for Nonlinear Feedback Shift Registers (NLFSRs) converts NLFSRs between Fibonacci and Galois configurations. Up to now, three types of Galois NLFSRs namely Type-I, Type-II and Type-III Galois NLFSRs have been discovered to be equivalent to Fibonacci NLFSRs in existing works. However, either no transformation algorithm has been proposed or the proposed algorithm has very

A weighing matrix W = (wi,j) is a square matrix of order n and entries wi,j in {0,± 1} such that WWT = kIn. In his thesis, Strassler gave a table of existence results for circulant weighing matrices with n ≤ 200 and k ≤ 100. In the latest version of Strassler’s table given by Tan, there are 34 open cases remaining. In this paper we give nonexistence proofs for 12 of these cases, report on preliminary

Automatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly predictable despite having a large maximum order complexity. However, recent results show that polynomial subsequences of automatic sequences, such as the Thue–Morse sequence

A 3-phase Golay complementary triad (GCT) is a set of three sequences over the 3-phase alphabet {1, ω, ω2}, where \(\omega =e^{\frac {2\pi \sqrt {-1}}{3}}\) is a 3rd root of unity, whose aperiodic autocorrelations sum up to zero for each out-of-phase non-zero time-shifts. Recent results by Avis and Jedwab proved the non-existence of 3-phase GCTs of length N ≡ 4 (mod 6). In this paper, we introduce

In this paper, we first introduce a class of linear codes which are affine-invariant. Then we obtain infinite families of 2-designs from them and determine the parameters of these 2-designs by considering the weight distribution of the linear codes.

Motivated by the impact of fast algebraic attacks on stream ciphers, and recent constructions using a threshold function as main part of the filtering function, we study the fast algebraic immunity of threshold functions. As a first result, we determine exactly the fast algebraic immunity of all majority functions in more than 8 variables. Then, For all n ≥ 8 and all threshold value between 1 and n

A basic problem about a constant dimension subspace code is to find its maximal possible size Aq(n, d, k). In this paper, we investigate constant dimension codes with parallel linkage construction and multilevel construction and obtain new lower bounds on Aq(18,6,9). By combining the Johnson type bound, we obtain new lower bounds on Aq(17,6,8). These lower bounds are larger than previously best known

We present two new constructions of entanglement-assisted quantum error-correcting codes using some fundamental properties of (classical) linear codes in an effective way. The main ideas include linear complementary dual codes and related concatenation constructions. Numerical examples in modest lengths show that our constructions perform better than known constructions in the literature. We also give

Quantum synchronizable codes (QSCs) are special quantum error-correcting codes that can correct the effects of both quantum noise on qubits and misalignment in block synchronization. In this paper, using the factorizations of cyclotomic polynomials \({{\varPhi }}_{p_{1}p_{2}}(x)\), where p1 and p2 are distinct odd primes, we give a new method to construct QSCs whose synchronization capabilities can

This paper is devoted to studying the symmetric 2-adic complexity of sequences with optimal autocorrelation magnitude and period 8q, where q is a prime satisfying q ≡ 5 (mod 8). These sequences were constructed by interleaving technique from Ding-Helleseth-Martinsen sequences and almost perfect binary sequences. They were presented by Krengel and Ivanov in 2016 and have been proved to have high linear

This article generalizes the simplified Shallue–van de Woestijne–Ulas (SWU) method of a deterministic finite field mapping \(h\!: \mathbb {F}_{q} \to E_{a}(\mathbb {F}_{q})\) to the case of any elliptic \(\mathbb {F}_{q}\)-curve Ea : y2 = x3 − ax of j-invariant 1728. In comparison with the (classical) SWU method the simplified SWU method allows to avoid one quadratic residuosity test in the field \(\mathbb

For any positive integer m > 2 and an odd prime p, let \(\mathbb {F}_{p^{m}}\) be the finite field with pm elements and let \( \text {Tr}^{m}_{e}\) be the trace function from \(\mathbb {F}_{p^{m}}\) onto \(\mathbb {F}_{p^{e}}\) for a divisor e of m. In this paper, for the defining set \(D=\{x\in \mathbb {F}_{p^{m}}:\text {Tr}^{m}_{e}(x)=1\text { and } \text {Tr}^{m}_{e}(x^{2})=0\}=\{d_{1}, d_{2}, \ldots

We propose several constructions for the original multiplication algorithm of D.V. and G.V. Chudnovsky in order to improve its scalar complexity. We highlight the set of generic strategies who underlay the optimization of the scalar complexity, according to parameterizable criteria. As an example, we apply this analysis to the construction of type elliptic Chudnovsky2 multiplication algorithms for

As a subclass of linear codes, cyclic codes have important applications in consumer electronics, data storage systems and communication systems. In this paper, a new family of optimal ternary cyclic codes with minimum distance five are obtained from known perfect nonlinear functions over \(\mathbb {F}_{3^{m}}\). Moreover, we investigate the weight distributions of their duals.

A formula discovered by L. Carlitz in 1935 finds an interesting application in permutation rational functions of finite fields. It allows us to determine all rational functions of degree three that permute the projective line \(\mathbb {P}^{1}(\mathbb {F}_{q})\) over \(\mathbb {F}_{q}\), a result previously obtained by Ferraguti and Micheli through a different method. It also allows us to determine

Asymmetric quantum error-correcting codes (AQECCs) are an extremely efficient coding scheme against the different occurring probabilities of qubit-flip and phase-shift errors in quantum asymmetric channel. It is generally known that good quantum codes play a decisive role in ensuring the reliability and the authenticity of quantum communication. In this paper, our main objective is to obtain good AQECCs

Minimal binary linear codes are a special class of binary codes with important applications in secret sharing and secure two-party computation. These codes are characterized by the property that none of the nonzero codewords is covered by any other codeword. Denoting by \(w_{{\min \limits }}\) and \(w_{{\max \limits }}\) the minimum and maximum weights of the codewords, respectively, such codes are

The problem of finding APN permutations of \({\mathbb {F}}_{2^{n}}\) where n is even and n > 6 has been called the Big APN Problem. Li, Li, Helleseth and Qu recently characterized APN functions defined on \({\mathbb {F}}_{q^{2}}\) of the form f(x) = x3q + a1x2q+ 1 + a2xq+ 2 + a3x3, where q = 2m and m ≥ 4. We will call functions of this form Kim-type functions because they generalize the form of the

Nowadays, ciphers have been widely used in high-end platforms, resource-constrained, and side-channel attacks vulnerable environments. This motivates various S-boxes aimed at providing a good trade-off between security and efficiency. For small S-boxes, the most natural approach of constructing such S-boxes is a comprehensive search in the space of permutations, which inevitably becomes more challenging

In this work, we study a new family of rings, \({\mathscr{B}}_{j,k}\), whose base field is the finite field \({\mathbb {F}}_{p^{r}}\). We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show

Machine learning and deep learning algorithms are increasingly considered as potential candidates to perform black box side-channel security evaluations. Inspired by the literature on machine learning security, we put forward that it is easy to conceive implementations for which such black box security evaluations will incorrectly conclude that recovering the key is difficult, while an informed evaluator

Inner Product Masking (IPM) is a generalization of several masking schemes including the Boolean one to protect cryptographic implementation against side-channel analysis. The core competitiveness of IPM is that it provides higher side-channel resistance than Boolean masking with the same number of shares. In this paper, we follow a coding theoretic approach and categorize all linear codes of IPM with

We show (almost) separation between certain important classes of Boolean functions. The technique that we use is to show that the total influence of functions in one class is less than the total influence of functions in the other class. In particular, we show (almost) separation of several classes of Boolean functions which have been studied in coding theory and cryptography from classes which have

Boolean functions play an important role in coding theory and symmetric cryptography. In this paper, three classes of Boolean functions with six-valued Walsh spectra are presented and their Walsh spectrum distributions are determined. They are derived from three classes of bent functions by complementing the values of the functions at three different points, where the bent functions are the Maiorana-McFarland

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