Generator matrix

Abstract: This paper presents several novel theoretical results regarding the recovery of a low-rank matrix from just a few measurements consisting of linear combinations of the matrix entries. We show that properly constrained nuclear-norm minimization stably recovers a low-rank matrix from a constant number of noisy measurements per degree of freedom; this seems to be the first result of this nature. Further, with high probability, the recovery error from noisy data is within a constant of three targets: (1) the minimax risk, (2) an “oracle” error that would be available if the column space of the matrix were known, and (3) a more adaptive “oracle” error which would be available with the knowledge of the column space corresponding to the part of the matrix that stands above the noise. Lastly, the error bounds regarding low-rank matrices are extended to provide an error bound when the matrix has full rank with decaying singular values. The analysis in this paper is based on the restricted isometry property (RIP).

Abstract: A new modification of the McEliece public-key cryptosystem is proposed that employs the so-called maximum-rank-distance (MRD) codes in place of Goppa codes and that hides the generator matrix of the MRD code by addition of a randomly-chosen matrix. A short review of the mathematical background required for the construction of MRD codes is given. The cryptanalytic work function for the modified McEliece system is shown to be much greater than that of the original system. Extensions of the rank metric are also considered.

Abstract: A classical method of constructing a linear code over $ {\mathrm {GF}}(q)$ with a $t$ -design is to use the incidence matrix of the $t$ -design as a generator matrix over $ {\mathrm {GF}}(q)$ of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of 2-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of 2-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterization of highly nonlinear Boolean functions is presented.

Abstract: In this paper, we propose a general framework and practical coding methods for constructing steganographic schemes that minimize the statistical impact of embedding. By associating a cost of an embedding change with every element of the cover, we first derive bounds on the minimum theoretically achievable embedding impact and then propose a framework to achieve it in practice. The method is based on syndrome codes with low-density generator matrices (LDGM). The problem of optimally encoding a message (e.g., with the smallest embedding impact) requires a binary quantizer that performs near the rate-distortion bound. We implement this quantizer using LDGM codes with a survey propagation message-passing algorithm. Since LDGM codes are guaranteed to achieve the rate-distortion bound, the proposed methods are guaranteed to achieve the minimal embedding impact (maximal embedding efficiency). We provide detailed technical description of the method for practitioners and demonstrate its performance on matrix embedding.