Modelización lineal de los generadores shrinking a través de las leyes 102 y 60

En este trabajo se presenta la modelización lineal de los generadores shrinking y auto-shrinking a través de autómatas celulares lineales uniformes utilizando la ley 102 (o la 60). La linealidad de estos autómatas se puede utilizar para el criptoanálisis de estos generadores de secuencias.El trabajo del primer autor ha sido financiado por una beca postdoctoral de la Generalitat Valenciana con referencia APOSTD/2013/081 y por el proyecto MTM2011-24858 del Ministerio de Ciencia e Innovación del Gobierno de España. El trabajo del segundo autor ha sido financiado por el Ministerio de Ciencia e Innovación del Gobierno de España bajo el proyecto “TUERI: Technologies for secure and efficient wireless networks within the Internet of Things with applications to transport and logistics”, TIN2011-25452

An approach to the performance of SPC product codes on the erasure channel

Product codes can be used to correct errors or recover erasures. In this work we consider the simplest form of a product code, this is, the single parity check (SPC) product code. This code has a minimum distance of four and is thus guaranteed to recover all single, double, and triple erasure patterns. The code is actually capable of recovering a higher number of erasure patterns. We count the number of uncorrectable erasure patterns of size n×n with t erasures, for t=8, 2n−3, 2n−2 and 2n−1, using the relation between erasure patterns and bipartite graphs.The work of the first author was supported by a grant for postdoctoral students from FAPESP with process 2015/07246-0 and a grant for postdoctoral students from Generalitat Valenciana with reference APOSTD/2013/081

Recovering erasures by using MDS codes over extension alphabets

A new family of Fq-linear codes over Fbq can be obtained replacing the elements in the large field Fqb by elements in Fq[C], where C is the companion matrix of a primitive polynomial of degree b and coefficients in Fq. In this work, we propose a decoding algorithm for this family of Fq-linear codes over the erasure channel, based on solving linear systems over the field Fq.The work of the first author was partially supported by a grant for postdoctoral students from FAPESP with reference 2015/07246-0

Generalized Column Distances

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.This work was supported in part by the Sao Paulo Research Foundation (FAPESP) under Grant 2013/25977-7. The work of Sara D. Cardell was supported in part by the FAPESP, under Grant 2015/07246-0 and in part by the CAPES. The work of Marcelo Firer was supported in part by the CNPq. The work of Diego Napp was supported in part by the Spanish, Generalitat Valenciana, Univesitat d’Alacant, under Grant AICO/2017/128 and Grant VIGROB-287

A construction of MDS array codes

In this paper a new construction of MDS array codes is introduced. In order to obtain a code with this property, the parity-check matrix is constructed just using a superregular matrix of blocks composed of powers of the companion matrix of a primitive polynomial. Also a decoding algorithm for these codes is introduced.The work of the first and the second authors was partially supported by Spanish grant MTM2011-24858 of the Ministerio de Economía y Competitividad of the Gobierno de España. The work of first author was also partially supported by a grant for research students from the Generalitat Valenciana with reference BFPI/2008/138. The work of the third author was partially supported by the research project UMH-Bancaja with reference IPZS01

Computational Analysis of Interleaving PN-Sequences with Different Polynomials

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.This work was supported in part by the Spanish State Research Agency (AEI) of the Ministry of Science and Innovation (MICINN), project P2QProMeTe (PID2020-112586RB-I00/AEI/ 10.13039/501100011033). It was also supported by Comunidad de Madrid (Spain) under project CYNAMON (P2018/TCS-4566), co-funded by FSE and European Union FEDER funds. The work of the second author was partially supported by Spanish grant VIGROB-287 of the University of Alicante

Interleaving Shifted Versions of a PN-Sequence

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.This research is partially supported by Ministerio de Economía, Industria y Competitividad (MINECO), Agencia Estatal de Investigación (AEI), and Fondo Europeo de Desarrollo Regional (FEDER, UE) under project COPCIS, reference TIN2017-84844-C2-1-R. It is also supported by Comunidad de Madrid (Spain) under project CYNAMON (P2018/TCS-4566), co-funded by FSE and European Union FEDER funds. Finally, the third author is partially supported by Spanish grant VIGROB-287 of the Universitat d’Alacant

An Efficient Algorithm to Compute the Linear Complexity of Binary Sequences

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.This work was supported in part by the Spanish State Research Agency (AEI) of the Ministry of Science and Innovation (MICINN), Project P2QProMeTe (PID2020-112586RB-I00/AEI/10.13039/501100011033), co-funded by the European Regional Development Fund (ERDF, EU). It is also supported by Comunidad de Madrid (Spain) under Project CYNAMON (P2018/TCS-4566), co-funded by FSE and European Union FEDER funds. The work of the second author was partially supported by Spanish Grant VIGROB-287 of the University of Alicante

A construction of F2-linear cyclic, MDS codes

In this paper we construct F2-linear codes over Fb2 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 Zn. 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.The first author was supported by CAPES (Brazil). The work of the second author was partially supported by Spanish grants AICO/2017/128 of the Generalitat Valenciana and VIGROB-287 of the Universitat d'Alacant. The third and fourth authors were supported by NSERC (Canada). The first, third and fourth authors acknowledge support from FAPESP SPRINT grant 2016/50476-0