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Construction, decoding and implementation of F_q linear codes. Performanca of SPC product codes and cryptanalysis of the shrinking generators.

Grant number: 15/07246-0
Support Opportunities:Scholarships in Brazil - Post-Doctoral
Effective date (Start): August 01, 2015
Effective date (End): December 31, 2017
Field of knowledge:Engineering - Electrical Engineering - Telecommunications
Principal Investigator:Marcelo Firer
Grantee:Sara Díaz Cardell
Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
Associated research grant:13/25977-7 - Security and reliability of Information: theory and practice, AP.TEM


First, we work on a construction of MDS Fq-linear codes over Fqb based on the isomorphism between _fields Fq[C] and Fqb , where C is the companion matrix of a primitive polynomial of degree b in Fq[x]. If the parameters of one of our codes are [n; k; d], we can recover up to n-k erasures. We propose an algorithm to recover the lost information symbols just solving a linear system with tb unknowns, where t in the number of erased information symbols. We would like to deeply study this algorithm to make it e_cient and compare these codes with other MDS codes. We would also like to find good cryptographic applications of these codes, such as the construction of optimal linear di_usion layers in block ciphers.At the same time, we study the performance of the SPC (single parity-check) simple product codes. These codes have a small minimum distance and, thus, their error correction capability is very limited. However, they are able to recover a higher number of erasures in special cases. We would like to count and analyse these cases in order to study the performance of these codes. Furthermore, SPC product codes obtained with more than two SPC codes have never been studied. Solving this problem can help us to solve graph theory problems, since an erasure pattern representing a codeword of an SPC product code can be also seen as a bipartite graph, where the erasures are the edges.Regarding cryptography, we model some cryptographic non-linear sequence generators, called shrinking generators, using linear cellular automata (CA). The sequences produced by these generators can be obtained as one of the output sequences generated by a family of regular CA. We can take advantage of this linearity and propose an e_cient cryptanalysis of these generators. We would also like study the cryptographic properties of the other sequences generated by the CA and try to model model other generators using CA. Besides, we want to connect CA with Neural Networks (NN) and then study our cryptographic problem from a new perspective never considered before.

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Scientific publications (7)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
CARDELL, SARA D.; CLIMENT, JOAN-JOSEP. A construction of primitive polynomials over finite fields. LINEAR & MULTILINEAR ALGEBRA, v. 65, n. 12, p. 2424-2431, . (15/07246-0)
CARDELL, SARA D.; FUSTER-SABATER, AMPARO. Discrete linear models for the generalized self-shrunken sequences. FINITE FIELDS AND THEIR APPLICATIONS, v. 47, p. 222-241, . (15/07246-0)
CARDELL, SARA D.; CLIMENT, JOAN-JOSEP. AN APPROACH TO THE PERFORMANCE OF SPC PRODUCT CODES ON THE ERASURE CHANNEL. Advances in Mathematics of Communications, v. 10, n. 1, SI, p. 11-28, . (15/07246-0)
CARDELL, SARA D.; FUSTER-SABATER, AMPARO. MODELLING THE SHRINKING GENERATOR IN TERMS OF LINEAR CA. Advances in Mathematics of Communications, v. 10, n. 4, p. 797-809, . (15/07246-0)
CARDELL, SARA D.; FUSTER-SABATER, AMPARO. Linear Models for the Self-Shrinking Generator Based on CA. JOURNAL OF CELLULAR AUTOMATA, v. 11, n. 2-3, p. 195-211, . (15/07246-0)
CARDELL, SARA D.; ARANHA, DIEGO F.; FUSTER-SABATER, AMPARO. Recovering Decimation-Based Cryptographic Sequences by Means of Linear CAs. LOGIC JOURNAL OF THE IGPL, v. 28, n. 4, p. 430-448, . (16/50476-0, 15/07246-0)
CARDELL, SARA D.; FIRER, MARCELO; NAPP, DIEGO. Generalized Column Distances. IEEE TRANSACTIONS ON INFORMATION THEORY, v. 66, n. 11, p. 6863-6871, . (15/07246-0, 13/25977-7)

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