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.

Given the finite field 𝔽_q , for a prime power q, in this paper we present a way of constructing spreads of 𝔽_q^n . Specifically, through the field reduction technique, we will construct the well-known Desarguesian spread of 𝔽_q^n , using for this purpose the action of an Abelian but not cyclic group. First, we construct a family of orbit codes of maximum distance using this group, and then we complete each of these codes to achieve the Desarguesian spread of the whole space, which will thus have a new orbital structure associated with an Abelian and non-cyclic group.

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

The shrinking generator is a pseudorandom bit generator based on the combination of two linear feedback shift registers of maximum period. These registers are synchronized with a common clock and produce binary sequences with good statistical properties. Due to its simplicity and efficient implementation, the shrinking generator is particularly suitable for stream cipher cryptographic schemes and most proposed attacks rely on the properties of the generator. Furthermore, its analysis serves as the foundation for other interleave constructions. In our work, we present a new algorithm which allows to compute the linear complexity for shrunken sequences in an efficient way together with a closed formula for the linear complexity of its output in certain conditions. Additionally, we establish the first bound on its linear complexity profile and a conjecture about the values of the linear complexity of these sequences. Our techniques involve two-dimensional arrays and their interleave structure, which could prove valuable for other pseudorandom bit generators.

Convolutional codes can be viewed as linear systems over a finite field, and therefore can be described by state representations. In this work we focus on the so called inputstate-output (ISO) representations which are different from driving variable state representations commonly found in the literature. Although many fundamental properties have been thoroughly studied within this framework in the last two decades, the crucial notion of minimality has been less investigated. In this paper, we present a novel and relatively simple constructive algorithm to build a minimal input-stateoutput representation for a convolutional code from any of its generator matrices. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar

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.

In this paper we present a concrete algebraic construction of a novel class of convolutional codes. These codes are built upon generalized Vandermonde matrices and therefore can be seen as a natural extension of Reed-Solomon block codes to the context of convolutional codes. For this reason we call them weighted Reed-Solomon (WRS) convolutional codes. We show that under some constraints on the defining parameters these codes are Maximum Distance Profile (MDP), which means that they have the maximal possible growth in their column distance profile. We study the size of the field needed to obtain WRS convolutional codes which are MDP and compare it with the existing general constructions of MDP convolutional codes in the literature, showing that in many cases WRS convolutional codes require significantly smaller fields.

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.

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.

In this paper, we study product convolutional codes described by state-space representations. In particular, we investigate how to derive state-space representations of the product code from the horizontal and vertical convolutional codes. We present a systematic procedure to build such representation with minimal dimension, i.e., reachable and observable.

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.

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.

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.

A constant dimension code consists of a set of k-dimensional subspaces of $\mathbb {F}_{q}^{n}$ , where $\mathbb {F}_{q}$ is a finite field of q elements. Orbit codes are constant dimension codes which are defined as orbits under the action of a subgroup of the general linear group on the set of all k-dimensional subspaces of $\mathbb {F}_{q}^{n}$ . If the acting group is Abelian, we call the corresponding orbit code Abelian orbit code. In this paper we present a construction of an Abelian non-cyclic orbit code for which we compute its cardinality and its minimum subspace distance. Our code is a partial spread and consequently its minimum subspace distance is maximal.

In this paper we study a problem in the area of coding theory. In particular, we focus on a class of error-correcting codes called convolutional codes. We characterize convolutional codes that can correct bursts of erasures with the lowest possible delay. This characterization is given in terms of a block Toeplitz matrix with entries in a finite field that is built upon a given generator matrix of the convolutional code. This result allows us to provide a concrete construction of a generator matrix of a convolutional code with entries being only zeros or ones that can recover bursts of erasures with low delay. This construction admits a very simple decoding algorithm and, therefore, simplifies the existing schemes proposed recently in the literature.

In this paper we study convolutional codes tailor made for fast decoding over burst erasure channels. This class of streaming codes are suitable for multimedia streaming applications where a stream of source packets must be transmitted in strict delay constraints. We show that in the case of dealing with burst erasure channels it is possible to come up with very simply constructions of encoders of convolutional codes that admit the fastest possible decoding delay to correct all bursts of a given length with a fixed rate. An explicit class of such encoders is presented. The last part of the paper is devoted to treat isolated errors. We propose the use of MDP convolutional codes to recover this kind of losses.