Resolution of Linear Algebra for the Discrete Logarithm Problem Using GPU and Multi-core Architectures

In cryptanalysis, solving the discrete logarithm problem (DLP) is key to assessing the security of many public-key cryptosystems. The index-calculus methods, that attack the DLP in multiplicative subgroups of finite fields, require solving large sparse systems of linear equations modulo large primes. This article deals with how we can run this computation on GPU- and multi-core-based clusters, featuring InfiniBand networking. More specifically, we present the sparse linear algebra algorithms that are proposed in the literature, in particular the block Wiedemann algorithm. We discuss the parallelization of the central matrix--vector product operation from both algorithmic and practical points of view, and illustrate how our approach has contributed to the recent record-sized DLP computation in GF($2^{809}$).Comment: Euro-Par 2014 Parallel Processing, Aug 2014, Porto, Portugal. \<http://europar2014.dcc.fc.up.pt/\&gt

A fast method for solving a block tridiagonal quasi-Toeplitz linear system

This paper addresses the problem of solving block tridiagonal quasi-Toeplitz linear systems. Inspired by Du, we propose a more general algorithm for such systems. The algorithm is based on a block decomposition for block tridiagonal quasi-Toeplitz matrices and the Sherman–Morrison–Woodbury inversion formula. We also compare the proposed approach to the standard block LU decomposition method and the Gauss algorithm. A theoretical error analysis is also presented. All algorithms have been implemented in Matlab. Numerical experiments performed on a wide variety of test problems show the e¤ectiveness of our algorithm in terms of efficiency, stability, and robustness.The third author was partially financed by Portuguese Funds through FCT within the Project UID/MAT/00013/2013

Transmit Precoding for MIMO Systems with Partial CSI and Discrete-Constellation Inputs

In this paper, we consider the transmit linear precoding problem for MIMO systems with discrete-constellation inputs. We assume that the receiver has perfect channel state information (CSI) and the transmitter only has partial CSI, namely, the channel covariance information. We first consider MIMO systems over frequency-flat fading channels. We design the optimal linear precoder based on direct maximization of mutual information over the MIMO channels with discrete-constellation inputs. It turns out that the optimal linear precoder is a non-diagonal non-unitary matrix. Then, we consider MIMO systems over frequency selective fading channels via extending our method to MIMO-OFDM systems. To keep reasonable computational complexity of solving the linear precoding matrix, we propose a sub-optimal approach to restrict the precoding matrix as a block-diagonal matrix. This approach has near-optimal performance when we integrate it with a properly chosen interleaver. Numerical examples show that for MIMO systems over frequency flat fading channels, our proposed optimal linear precoder enjoys 6-9 dB gain compared to the same system without linear precoder. For MIMO-OFDM systems, our reduced-complexity sub-optimal linear precoder captures 3-6 dB gain compared to the same system with no precoding. Moreover, for those MIMO systems employing a linear precoder designed based on Gaussian inputs with gap approximation technique for discrete-constellation inputs, significant loss may occur when the signal-to-noise ratio is larger than 0 dB

Inversion of linear dynamical systems under conditions of quasiharmonic signals

Методы обращения динамических систем нашли широкое распространение для решения задач управления механическими и электрическими системами. Инвертирование динамических систем является эффективным способом реализации процессов управления по возмущению, а также в комбинированных системах управления с прогнозирующей моделью. При решении задач обращения возникает ряд трудностей, связанных с высокой чувствительностью результатов по отношению к точности задания параметров математической модели объекта, неустойчивостью при управлении неминимально-фазовыми объектами, нарушении условий физической реализуемости. В работе предлагается приближенный метод решения задачи обращения линейных стационарных динамических систем во многом свободный от указанных недостатков. Рассматриваются математические модели линейных динамических систем в форме "вход-выход", удовлетворяющие требованиям асимптотической устойчивости, а также условию равенства размерностей векторов входа и выхода. В основе метода лежит представление входных и выходных сигналов их приближениями в линейном пространстве квазигармонических функций времени. Особенностью предложенного метода обращения динамических систем является представление многомерных многочленов в виде произведения прямоугольных матриц на вектор степеней времени. Такое представление позволило свести большинство постановок задач обращения к решению линейных систем матричных алгебраических уравнений. Компьютерная реализация, предложенного подхода к обращению линейной системы, разработана для "квадратных" линейных скалярных систем в условиях квазигармонических сигналов и содержит блоки аппроксимации задания по выходу, формирования матриц линейных систем и правых частей линейных алгебраических уравнений, оценку числа обусловленности решения линейной системы и блок сравнения результата обращения с заданием на основе непосредственного интегрирования дифференциальных уравнений математической модели.The methods of inversion of dynamical systems are widely used for solving the problems of controlling mechanical and electrical systems. The inverting of dynamic systems is an effective way of implementing processes of control according to disturbance, as well as in combined control systems with a predictive model. In solving the problems of inversion, there are number of difficulties related to the high sensitivity of the results in relation to accuracy of specifying the parameters of a mathematical model of an object, the instability in the control of non-minimal-phase objects, and the violation of physical feasibility conditions. The paper suggests an approximate method for solving the inversion problem for linear stationary dynamical systems that is largely free of the indicated disadvantages. The method is based on the representation of input and output signals by their approximations in the linear space of quasiharmonic functions of time. Consider mathematical models of linear dynamical systems in the form "input-output". The systems under consideration must satisfy the requirements of asymptotic stability, and also the condition of equality of dimension of the input and output vectors. A feature of the proposed method of inversion of dynamical systems is the representation of multidimensional polynomials, approximating input and output signals, in the form of a product of rectangular matrices and a vector of power of time. Such a representation allowed reducing most of the statements of inversion problems to the solution of linear systems of matrix algebraic equations. The computer implementation of the proposed approach to the inverting of the linear system is developed for "square" linear scalar systems in conditions of polynomial signal and contains blocks of approximation of the output assignment; the formation of matrices of linear systems and right-hand sides of linear algebraic equations; estimation of the condition number of the solution of the linear system and the block of comparison the result of the inversion with the assignment based on the direct integration of the differential equations of the mathematical model