Generic Decoding in the Sum-Rank Metric

Abstract

We propose the first non-trivial generic decoding algorithm for codes in the sum-rank metric. The new method combines ideas of well-known generic decoders in the Hamming and rank metric. For the same code parameters and number of errors, the new generic decoder has a larger expected complexity than the known generic decoders for the Hamming metric and smaller than the known rank-metric decoders. Furthermore, we give a formal hardness reduction, providing evidence that generic sum-rank decoding is computationally hard. As a by-product of the above, we solve some fundamental coding problems in the sum-rank metric: we give an algorithm to compute the exact size of a sphere of a given sum-rank radius, and also give an upper bound as a closed formula; and we study erasure decoding with respect to two different notions of support.

A Survey on Code-Based Cryptography

“…ISD algorithms were originally proposed for the Hamming metric, but have also been translated to the rank metric, which we cover in Section 5.2.9. Recently, ISD algorithms have also been proposed for other metrics, such as the Lee metric in [125,222,89] and the sum-rank metric [183]. To propose an ISD algorithm in a new metric is always the first step to introduce this new metric to code-based cryptography.…”

Error-Erasure Decoding of Linearized Reed-Solomon Codes in the Sum-Rank Metric

“…The interest in LRS and other sum-rank metric codes keeps increasing as they have multiple widespread applications as e.g. multishot network coding [2], [9], locally repairable codes [10], space-time codes [1] and code-based quantum-resistant cryptography [11]. Recently, it was shown that interleaved [12], [13] and folded [14] variants of LRS codes can be decoded beyond the unique decoding radius.…”

“…The concept of row and column erasures, i.e. the partial knowledge of the column and row space of the error, respectively, was generalized from the rank metric [15] to the sum-rank metric in [11].…”

Duals of linearized Reed-Solomon codes

“…Moreover, Martínez-Peñas gives applications to his codes to multishot network coding in [20]. Since then, sumrank metric codes have received some interest (see for example [20,22,6,21]). In particular, the notion of duality for sum-rank metric codes have been addressed in [19] in which the authors proved that the duals of certain linearized Reed-Solomon remains of the same type (see [19,Theorem 4]).…”