Параллельный линейный генератор многозначных псевдослучайных последовательностей с контролем ошибок функционирования

A parallel linear generator of multi-valued pseudorandom sequences, which operates under conditions of generating hardware errors caused by destructive adversary actions is proposed. The main types of modification of the pseudorandom sequence in case of adversary attack are considered. A distinctive feature of the iterative process of ensuring the reliability of computational operations is the "arithmetic" of computational operations by representing a system of generating recurring logical formulas as a system of many-valued logic algebra functions. The subsequent realization of multivalued logic algebra functions by means of arithmetic polynomials allowed us to parallelize the process of generating multivalued pseudorandom sequences and level out the existing complexity (specificity) of cryptographic transformations of logical data types which limit the use of redundant coding methods. As a result, a solution that allows to apply redundant modular codes to control the accuracy of the computational operations performed by the nodes of pseudorandom sequence generation is proposed. Moreover, unlike the known solutions, the proposed method provides obtaining fragments of a pseudorandom sequence on the basis of one recursive arithmetic formula with parallel calculation errors control. The use of modular forms made it possible to transfer computations from the rational numbers field arithmetic to integer arithmetic of a simple field. Among the existing variety of codes correcting errors (maximally spaced codes), a special place is occupied by multivalued Reed-Solomon codes. Reed-Solomon codes usage in the formation of pseudorandom sequences allows the formation of code-like structures that monitor and ensure the reliability of computational operations. The calculated probability of failure-free operation of the parallel linear generator of multivalued pseudorandom sequences with an error control function based on the principle of functioning — sliding redundancy is obtained. The achieved results can find wide application at realization of perspective high-efficiency cryptographic information protection facility.Предложен параллельный линейный генератор многозначных псевдослучайных последовательностей, функционирующий в условиях генерации аппаратных ошибок, обусловленных деструктивными воздействиями злоумышленника. Рассмотрены основные виды модификации псевдослучайной последовательности при атаках злоумышленника. Отличительной особенностью рассматриваемого итеративного процесса обеспечения достоверности вычислительных операций является «арифметизация» вычислительных операций путем представления системы порождающих рекуррентных логических формул как системы многозначных функций алгебры логики. Последующая реализация многозначных функций алгебры логики посредством арифметических полиномов позволила распараллелить процесс генерации многозначных псевдослучайных последовательностей и нивелировать существующую сложность (специфику) криптографических преобразований логических типов данных, ограничивающих применение методов избыточного кодирования. В результате предложено решение, позволяющее применить избыточные модулярные коды для контроля безошибочности производимых вычислительных операций узлами генерации псевдослучайной последовательности. Причем в отличие от известных решений предлагаемый метод обеспечивает получение фрагментов псевдослучайной последовательности на основании одной рекурсивной арифметической формулы с параллельным контролем ошибок вычислений. Применение модулярных форм позволило перенести вычисления из арифметики поля рациональных чисел в целочисленную арифметику простого поля. Среди существующего многообразия кодов, исправляющих ошибки (максимально разнесенных кодов), особое место занимают многозначные коды Рида — Соломона. Применение кодов Рида — Соломона при формировании псевдослучайных последовательностей позволяет формировать кодоподобные структуры, осуществляющие контроль и обеспечение достоверности вычислительных операций. Получены расчетные данные вероятности безотказной работы параллельного линейного генератора многозначных псевдослучайных последовательностей с функцией контроля ошибок по принципу функционирования — скользящее резервирование. Достигнутые результаты могут найти широкое применение при реализации перспективных высокопроизводительных средств криптографической защиты информации

A reduced set of submatrices for a faster evaluation of the MDS property of a circulant matrix with entries that are powers of two

In this paper a reduced set of submatrices for a faster evaluation of the MDS property of a circulant matrix, with entries that are powers of two, is proposed. A proposition is made that under the condition that all entries of a t × t circulant matrix are powers of 2, it is sufficient to check only its 2x2 submatrices in order to evaluate the MDS property in a prime field. Although there is no theoretical proof to support this proposition at this point, the experimental results conducted on a sample of 100 thousand randomly generated matrices indicate that this proposition is true. There are benefits of the proposed MDS test on the efficiency of search methods for the generation of circulant MDS matrices, regardless of the correctness of this proposition. However, if this proposition is correct, its impact on the speed of search methods for circulant MDS matrices will be huge, which will enable generation of MDS matrices of large sizes. Also, a modified version of the make_binary_powers function is presented. Based on this modified function and the proposed MDS test, some examples of efficient 16 x 16 MDS matrices are presented. Also, an examples of efficient 24 x 24 matrices are generated, whose MDS property should be further validated

Design of Small Rate, Close to Ideal, GLDPC-Staircase AL-FEC Codes for the Erasure Channel

International audienceThis work introduces the Generalized Low Density Parity Check (GLDPC)-Staircase codes for the erasure channel, that are constructed by extending LDPC-Staircase codes through Reed Solomon (RS) codes based on "quasi" Hankel matrices. This construction has several key benefits: in addition to the LDPC-Staircase repair symbols, it adds extra-repair symbols that can be produced on demand and in large quantities, which provides small rate capabilities. Additionally, with selecting the best internal parameters of GLDPC graph and under hy- brid Iterative/Reed-Solomon/Maximum Likelihood decoding, the GLDPC-Staircase codes feature a very small decoding overhead and a low error floor. These excellent erasure capabilities, close to that of ideal, MDS codes, are obtained both with large and very small objects, whereas, as a matter of comparison, LDPC codes are known to be asymptotically good. Therefore, these properties make GLDPC-Staircase codes an excellent AL-FEC solution for many situations that require erasure protection such as media streaming

Communications (2012)" COMPLEXITY COMPARISON OF THE USE OF VANDERMONDE VERSUS HANKEL MATRICES TO BUILD SYSTEMATIC MDS REED-SOLOMON CODES

Reed Solomon RS(n, k) codes are Maximum Distance Separable (MDS) ideal codes that can be put into a systematic form, which makes them well suited to many situations. In this work we consider use-cases that rely on a software RS codec and for which the code is not fixed. This means that the application potentially uses a different RS(n, k) code each time, and this code needs to be built dynamically. A lightweight code creation scheme is therefore highly desirable, otherwise this stage would negatively impact the encoding and decoding times. Constructing such an RS code is equivalent to constructing its systematic generator matrix. Using the classic Vandermonde matrix approach to that purpose is feasible but adds significant complexity. In this paper we propose an alternative solution, based on Hankel matrices as the base matrix. We prove theoretically and experimentally that the code construction time and the number of operations performed to build the target RS code are largely in favor of the Hankel approach, which can be between 3.5 to 157 times faster than the Vandermonde approach, depending on the (n, k) parameters

GLDPC-Staircase AL-FEC codes: A Fundamental study and New results

International audienceThis paper provides fundamentals in the design and analysis of Generalized Low Density Parity Check (GLDPC)-Staircase codes over the erasure channel. These codes are constructed by extending an LDPC-Staircase code (base code) using Reed Solomon (RS) codes (outer codes) in order to benefit from more powerful decoders. The GLDPC-Staircase coding scheme adds, in addition to the LDPC-Staircase repair symbols, extra-repair symbols that can be produced on demand and in large quantities, which provides small rate capabilities. Therefore, these codes are extremely flexible as they can be tuned to behave either like predefined rate LDPC-Staircase codes at one extreme, or like a single RS code at another extreme, or like small rate codes. Concerning the code design, we show that RS codes with " quasi " Hankel matrix-based construction fulfill the desired structure properties, and that a hybrid (IT/RS/ML) decoding is feasible that achieves Maximum Likelihood (ML) correction capabilities at a lower complexity. Concerning performance analysis, we detail an asymptotic analysis method based on Density evolution (DE), EXtrinsic Information Transfer (EXIT) and the area theorem. Based on several asymptotic and finite length results, after selecting the optimal internal parameters, we demonstrate that GLDPC-Staircase codes feature excellent erasure recovery capabilities, close to that of ideal codes, both with large and very small objects. From this point of view they outperform LDPC-Staircase and Raptor codes, and achieve correction capabilities close to those of RaptorQ codes. Therefore all these results make GLDPC-Staircase codes a universal Application-Layer FEC (AL-FEC) solution for many situations that require erasure protection such as media streaming or file multicast transmission