Skew differential Goppa codes and their application to Mceliece cryptosystem

Research funded by grant (PID2019-110525GB-I00 / AEI / 10.13039/501100011033) and by the IMAG-Maria de Maeztu grant (CEX2020-001105-M / AEI / 10.13039/501100011033).A class of linear codes that extends classical Goppa codes to a non-commutative context is defined. An efficient decoding algorithm, based on the solution of a non-commutative key equation, is designed. We show how the parameters of these codes, when the alphabet is a finite field, may be adjusted to propose a McEliece-type cryptosystem.IMAG-Maria de Maeztu grant PID2019-110525GB-I00 / AEI / 10.13039/501100011033CEX2020-001105-M / AEI / 10.13039/501100011033Universidad de Granada/CBU

Bounds and Genericity of Sum-Rank-Metric Codes

We derive simplified sphere-packing and Gilbert-Varshamov bounds for codes in the sum-rank metric, which can be computed more efficently than previous ones.They give rise to asymptotic bounds that cover the asymptotic setting that has not yet been considered in the literature: families of sum-rank-metric codes whose block size grows in the code length. We also provide two genericity results: we show that random linear codes achieve almost the sum-rank-metric Gilbert-Varshamov bound with high probability. Furthermore, we derive bounds on the probability that a random linear code attains the sum-rank-metric Singleton bound, showing that for large enough extension field, almost all linear codes achieve it

Interpolation-Based Decoding of Folded Variants of Linearized and Skew Reed-Solomon Codes

The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed-Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm.Comment: 32 pages, 3 figures, accepted at Designs, Codes and Cryptograph

Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants

We construct s-interleaved linearized Reed-Solomon (ILRS) codes and variants and propose efficient decoding schemes that can correct errors beyond the unique decoding radius in the sum-rank, sum-subspace and skew metric. The proposed interpolation-based scheme for ILRS codes can be used as a list decoder or as a probabilistic unique decoder that corrects errors of sum-rank up to $t\leq\frac{s}{s+1}(n-k)$, where s is the interleaving order, n the length and k the dimension of the code. Upper bounds on the list size and the decoding failure probability are given where the latter is based on a novel Loidreau-Overbeck-like decoder for ILRS codes. The results are extended to decoding of lifted interleaved linearized Reed-Solomon (LILRS) codes in the sum-subspace metric and interleaved skew Reed-Solomon (ISRS) codes in the skew metric. We generalize fast minimal approximant basis interpolation techniques to obtain efficient decoding schemes for ILRS codes (and variants) with subquadratic complexity in the code length. Up to our knowledge, the presented decoding schemes are the first being able to correct errors beyond the unique decoding region in the sum-rank, sum-subspace and skew metric. The results for the proposed decoding schemes are validated via Monte Carlo simulations.Comment: submitted to IEEE Transactions on Information Theory, 57 pages, 10 figure

Residues of skew rational functions and linearized Goppa codes

This paper constitutes a first attempt to do analysis with skew polynomials. Precisely, our main objective is to develop a theory of residues for skew rational functions (which are, by definition, the quotients of two skew polynomials). We prove in particular a skew analogue of the residue formula and a skew analogue of the classical formula of change of variables for residues. We then use our theory to define and study a linearized version of Goppa codes. We show that these codes meet the Singleton bound (for the sum-rank metric) and are the duals of the linearized Reed-Solomon codes defined recently by Martínez-Peñas. We also design efficient encoding and decoding algorithms for them