Boolean functions and binary sequences are fundamental tools in cryptography. In this work, we introduce a new bijection between the set of Boolean functions and the set of binary sequences whose period is a power of two. This correspondence enables the study of properties of Boolean functions through binary sequences and vice versa. Building on this connection, we propose a novel algebraic description, derived from the algebraic normal form of Boolean functions, which we call the reverse-ANF. Then, we explore how this formulation relates both to existing representations of Boolean functions and to binary sequences. Moreover, several cryptographic properties are examined through this new approach. Finally, we analyse generalized self-shrunken sequences from the perspective of Boolean functions, highlighting several properties that emerge under these different frameworks.

An interleaving sequence is obtained by combining or intertwining elements from two or more sequences. On the other hand, cellular automata are known to be generators for keystream sequences. In this paper we present two families of one-dimensional cellular automata as generators of interleaving sequences. This study aims to close a notable gap within the current body of literature by exploring the capacity of cellular automata to generate interleaving sequences. While previous works have separately examined cellular automata as sequence generators and interleaving sequences, there exists limited literature interconnecting these two topics. Our study seeks to bridge this gap, providing perspectives on the generation of interleaving sequences through the utilisation of cellular automata, thereby fostering a deeper understanding of both disciplines.

We study decoding procedures for a family of MDS array codes previously constructed from the Kronecker product of a superregular matrix and a non-singular matrix over a finite field. By exploiting the particular structure of their parity-check matrices, we develop decoding algorithms for different channel models. For the erasure channel, we provide an algorithm capable of recovering any pattern of up to n-k symbol erasures. For the q-ary symmetric channel, we investigate the decoding of one and two symbol errors and give explicit procedures for determining their locations and values. We also consider the particular case in which the superregular matrix is a Vandermonde matrix, showing how its additional algebraic structure can be exploited in the decoding process. Explicit examples over different finite fields are provided to illustrate the proposed procedures.

The binary binomial sequences correspond to the diagonals of the Pascal's triangle modulo 2. They have interesting properties such as they form a basis of the linear space of all binary sequences with period a power of 2. Other properties of these sequences (period, linear complexity, construction rules or relations among different binomial sequences) have been deeply analysed in detail previously. In this work, we study the binomial p-ary sequences for a prime p, its intrinsic characteristic and formation rules. We also prove that the family of p-ary sequences with period a power of p form a vector space over 𝔽_p and that the family of binomial p-ary sequences is a basis of this space.

The existence, several properties, and constructions of Generalized Weighing-Hadamard (GWH) matrices over finite fields are addressed in this work. We study the subset of invertible GWH matrices and show that it forms a group under matrix multiplication. Besides that, we introduce a strong notion of equivalence between such matrices, defined via orthogonal transformations, and further prove that the corresponding quotient group by the subgroup of orthogonal matrices is abelian. Finally, we discuss some applications of these matrices in coding theory

We investigate structural and enumerative properties of binary single parity-check product codes. For each n≥ 2, SPC(n) denotes the binary single parity-check code of length n, consisting of all binary vectors of length n having even Hamming weight. We determine the generalized Hamming weight hierarchy of the product code 𝒞_m,n=SPC(m)⊗SPC(n), whose codewords can be represented as m× n binary matrices in which every row and every column has even Hamming weight. For the square product 𝒞_n =SPC(n)⊗SPC(n), we also determine the maximum codeword weight and prove that its homogeneous weight enumerator is symmetric if and only if n is even. After characterizing the dual code, we apply the MacWilliams identity in its Walsh–Hadamard formulation to derive an exact closed-form expression for the weight enumerator. By grouping the auxiliary binary vectors according to their Hamming weights, we obtain an explicit formula for each coefficient in terms of binomial coefficients and alternating convolutions. Finally, using Krawtchouk polynomials, we present an exact procedure for computing the full weight distribution without exhaustively enumerating all codewords. Numerical examples illustrate the formulas and verify the resulting computations.

In this paper, we study the relation between the linear subspace of the pseudo-noise (PN)-sequences generated by a primitive polynomial and the simplex code. This family of sequences can be also seen as an Maximum Distance Separable (MDS) [Formula: see text]-linear code over [Formula: see text]. Furthermore, we see how to compute the family of generalized sequences produced by a primitive polynomial by means of a first-order Reed–Muller code.

Nowadays, a wide range of critical services relies on Internet of Things (IoT) devices. Nevertheless, they often lack proper security, becoming the gateway to attack the whole system. IoT security protocols are often based on stream ciphers, where pseudo-random number generators (PRNGs) are an essential part of them. In this work, we introduce a novel algorithm based on Hadamard matrices to evaluate the strength (unpredictability) of binary sequences, a key part of the IoT security stack. A comparative study with other algorithms that compute the same parameter is also presented.

The sequences produced by the cryptographic sequence generator known as the shrinking generator can be modelled as the output sequences of linear elementary cellular automata (CA). These sequences are composed of interleaved m-sequences produced by linear structures based on feedback shifts. This profitable characteristic can be used in the cryptanalysis of this generator. In this work we propose an algorithm that takes advantage of the inherent linearity of these CA and the interleaved m-sequences. Although irregularly decimated generators have been conceived and designed as non-linear ones, in practice they can be easily analysed in terms of simple linear structures.

Binary sequences are algebraic structures currently used as security elements in Internet of Things devices, sensor networks, e-commerce, and cryptography. In this work, a contribution to the evaluation of such sequences is introduced. In fact, we present a novel algorithm to compute a fundamental parameter for this kind of structure: the linear complexity, which is related to the predictability (or non-predictability) of the binary sequences. Our algorithm reduced the computation of the linear complexity to just the addition modulo two (XOR logic operation) of distinct terms of the sequence. The performance of this procedure was better than that of other algorithms found in the literature. In addition, the amount of required sequence to perform this computation was more realistic than in the rest of the algorithms analysed. Tables, figures, and numerical results complete the work.

Keystream sequences should look as random as possible, i.e. should present no logical pattern to be exploited in cryptographic attacks. The generalized self-shrinking generator, a sequence generator based on irregular decimation, produces a family of sequences with good cryptographic properties. In this work, we display a detailed analysis on the randomness of the sequences resulting from the concatenation of elements of this family. We apply the most important batteries of statistical and graphical tests providing powerful results and a new method to construct sequences with good cryptographic properties.

Binary PN-sequences generated by LFSRs exhibit good statistical properties; however, due to their intrinsic linearity, they are not suitable for cryptographic applications. In order to break such a linearity, several approaches can be implemented. For example, one can interleave several PN-sequences to increase the linear complexity. In this work, we present a deep randomness study of the resultant sequences of interleaving binary PN-sequences coming from different characteristic polynomials with the same degree. We analyze the period and the linear complexity, as well as many other important cryptographic properties of such sequences.

Some pseudorandom sequences with good crytographic features can be obtained from the interleaving of other families of sequences with unsuitable properties. PN-sequences obtained from maximum-length Linear Feedback Shift Registers exhibit good statistical aspects, such as balancedness, large period, adequate distribution of 0s and 1s and excellent autocorrelation, although their linearity makes them vulnerable against cryptographic attacks. In this work, we present a preliminary analysis on the random features of the interleaving of shifted versions of a PN-sequence. The application of statistical and graphic tests and their corresponding results complete the work.

The output sequence of the shrinking generator can be considered as an interleaving of determined shifted versions of a single PN -sequence. In this paper, we present a study of the interleaving of a PN-sequence and shifted versions of itself. We analyze some important cryptographic properties as the period and the linear complexity in terms of the shifts. Furthermore, we determine the total number of the interleaving sequences that achieve each possible value of the linear complexity.

A simple algorithm to compute the linear complexity of binary sequences with period a power of 2 has been proposed. The algorithm exploits the fractal structure of the binomial representation in this kind of binary sequences. The application of the general algorithm to a particular family of cryptographic sequences (generalized sequences) improves its performance as decreases the amount of sequence to be processed.

Binary sequences produced by a generator should appear as random as possible, that is, have no logical pattern to be used in cryptographic applications. In this paper, we give a detailed analysis of the randomness of a family of binary sequences obtained from generalized self-shrinking generator, an element in the class of decimation-based sequence generators. We have applied the most important batteries of statistical tests to the sequence resulting from the concatenation of the family of generalized sequences obtained from a PN-sequence. This complete study provides good results and allow us to construct a new binary sequence with good cryptographic properties from a family of generalized self-shrunken sequences.

. In this paper we construct F 2 -linear codes over F b 2 with length n and dimension n − r where n = rb . These codes have good properties, namely cyclicity, low density parity-check matrices and maximum distance separation in some cases. For the construction, we consider an odd prime p , let n = p − 1 and utilize a partition of Z n . Then we apply a Zech logarithm to the elements of these sets and use the results to construct an index array which represents the parity-check matrix of the code. These codes are always cyclic and the density of the parity-check and the generator matrices decreases to 0 as n grows (for a fixed r ). When r = 2 we prove that these codes are always maximum distance separable. For higher r some of them retain this property.

Output sequences of the cryptographic pseudo-random number generator, known as the generalized self-shrinking generator, are obtained self-decimating Pseudo-Noise (PN)-sequences with shifted versions of themselves. In this paper, we present three different representations of this family of sequences. Two of them, the p and G-representations, are based on the parameters p and G corresponding to shifts and binary vectors, respectively, used to compute the shifted versions of the original PN-sequence. In addition, such sequences can be also computed as the binary sum of diagonals of the Sierpinski’s triangle. This is called the B-representation. Characteristics and generalities of the three representations are analyzed in detail. Under such representations, we determine some properties of these cryptographic sequences. Furthermore, these sequences form a family that has a group structure with the bit-wise XOR operation.

The notion of Generalized Hamming weights of block codes has been investigated since the nineties due to its significant role in coding theory and cryptography. In this paper we extend this concept to the context of convolutional codes. In particular, we focus on column distances and introduce the novel notion of generalized column distances (GCD). We first show that the hierarchy of GCD is strictly increasing. We then provide characterizations of such distances in terms of the truncated parity-check matrix of the code, that will allow us to determine their values. Finally, the case in which the parity-check matrix is in systematic form is treated.

Linear complexity is a much used metric of the security of any binary sequence with application in communication systems and cryptography. In this work, we propose a method of computing the linear complexity of a popular family of cryptographic sequences, the so-called generalized sequences. Such a family is generated by means of the irregular decimation of a single Pseudo Noise sequence (PN-sequence). The computation method is based on the comparison of the PN-sequence with shifted versions of itself. The concept of linear recurrence relationship and the rows of the Sierpinski triangle play a leading part in this computation.