Decoding High-Order Interleaved Rank-Metric Codes

Abstract

This paper presents an algorithm for decoding homogeneous interleaved codes of high interleaving order in the rank metric. The new decoder is an adaption of the Hamming-metric decoder by Metzner and Kapturowski (1990) and guarantees to correct all rank errors of weight up to d − 2 whose rank over the large base field of the code equals the number of errors, where d is the minimum rank distance of the underlying code. In contrast to previously-known decoding algorithms, the new decoder works for any rank-metric code, not only Gabidulin codes. It is purely based on linear-algebraic computations, and has an explicit and easy-to-handle success condition. Furthermore, a lower bound on the decoding success probability for random errors of a given weight is derived. The relation of the new algorithm to existing interleaved decoders in the special case of Gabidulin codes is given.

On Decoding High-Order Interleaved Sum-Rank-Metric Codes

“…In this section, we propose a Metzner-Kapturowski-like decoder for the sum-rank metric, that is a generalization of the decoders proposed in [14,18,20]. Similar to the Hamming-and the rank-metric case, the proposed decoder works for errors of sum-rank weight t up to d − 2 that satisfy the following conditions:…”

“…Variants of the linear-algebraic Metzner-Kapturowski algorithm were further studied in [5,6,12,13,15,21], often under the name vector-symbol decoding (VSD). Moreover, Puchinger, Renner and Wachter-Zeh adapted the algorithm to the rank-metric case in [18,20].…”

“…Note, that the decoding complexity is independent of the code structure of the constituent code since the proposed algorithm exploits properties of high-order interleaving only. Since the sum-rank metric generalizes both the Hamming and the rank metric, the original Metzner-Kapturowski decoder [14] as well as its rank-metric analog [18,20] can be recovered from our proposal.…”

Interleaving Loidreau’s Rank-Metric Cryptosystem

“…For a small interleaving order ℓ, the currently most efficient algorithm to solve both, the Interleaved Search RSD Problem and the problem given in [29, Definition 7], was presented in [3] and will be analyzed in Section IV. For a high interleaving order ℓ ≥ w, the algorithm proposed in [30] is able to solve the Interleaved Search RSD Problem with high probability in polynomial time. For an interleaving order greater than wk, the algorithm proposed in [31] is able to efficiently solve [29, Definition 7], see [31,Section 6.5].…”

“…Further, this attack has a higher complexity than generic decoding in Loidreau's original system with the same public key if and only if d E > d−1 2λ . 3) (decoding attack): In [30], a polynomial-time decoding algorithm is proposed that works for arbitrary interleaved codes of interleaving degree ℓ ≥ t pub and error matrices of full rank. However in case of ℓ < t pub , one must brute-force through the solution space of a linear system of equations, whose size is exponential in m(t pub − ℓ).…”