A method for recovering linear block codes over an arbitrary finite field from sets of distorted code words

Abstract

The article is devoted to one of the practically important problems of information security and cryptanalysis, which consists in recovering an unknown linear block code over an arbitrary field from a set of distorted code words. This is a hard computational problem, and the known problem-solving methods are proposed only for codes over the field of two elements and are based on the algorithms for searching words of small weight in (undistorted) linear block codes.
 The main result of the article is a method for solving the problem posed, which differs in essence from the known ones and consists in recovering the desired code by solving the LPN (Learning Parity with Noise) problem, namely, recovering the solutions of systems of linear equations with distorted right-hand sides and a random equally probable matrix of coefficients over specified field. The LPN problem is well known from the Theory of Computational Algorithms and Cryptanalysis. It is equivalent to the problem of random linear block code decoding, and the security of many modern post-quantum cryptosystems are based on its hardness.
 The proposed method provides an opportunity to apply a wider class of algorithms for recovering linear block codes in comparison with the previously known methods, in particular, algorithms like BKW and also the low weight words search algorithms in co-sets of linear block codes. Moreover, in contrast to previously known ones, the complexity of the proposed method depends linearly on the length of the required code (and increases with increasing of its dimension according to which algorithm for the LPN problem-solving is applied). Thus, the basic parameter determined the complexity of recovering a linear block code is its dimension (not its length), which, in principle, makes it possible to speed up known algorithms for recovering linear block codes from a set of corrupted code words.