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This thesis is devoted to efficient VHDL design of Systolic Array Architecture for Polynomial GF(2^m) multiplication. The hardware implements the Processor Elements(PE) and Systolic Array design for Progressive Product Reduction (PPR) method proposed by Gebali and Atef. The experiment first implements a simpler irreducible polynomials GF(2^5) based on the defined algorithms for PPR in order to confirm the functionality of the design and then tries the bigger value of m for GF(2^133) and GF(2^233), recommended by NIST. The thesis is comparing the three designs based on their power consumption, Maximum Data path delay and device utilization. It also looking in to the different optimization method for the designs and recommends a design optimization based on circuit modification. / Graduate / 0544 / alinia@uvic.ca

RAID-6 is a data encoding scheme used to provide single drive error detection and dual drive error correction for data redundancy on an array of disks. Here we present a thorough study of efficient implementations of RAID-6 on field programmable gate arrays (FPGAs). Since RAID-6 relies heavily on Galois Field Algebra (GFA), an efficient implementation of a GFA FPGA library is also presented. Through rigorous performance analysis, this work shows the most efficient ways to tradeoff FPGA resources and execution time when implementing GFA functions as well as RAID-6 encoding and decoding.

No / A finite quantum system in which the position and momentum take values in the Galois field GF(p¿l) is constructed from a smaller quantum system in which the position and momentum take values in Zp , using field extension. The Galois trace is used in the definition of the Fourier transform. The Heisenberg¿Weyl group of displacements and the Sp(2, GF(p¿l)) group of symplectic transformations are studied. A class of transformations inspired by the Frobenius maps in Galois fields is introduced. The relationship of this 'Galois quantum system' with its subsystems in which the position and momentum take values in subfields of GF(p¿l) is discussed.

Modern public-key cryptography relies extensively on modular multiplication with long operands. We investigate the opportunities to optimize this operation in a heterogeneous multiprocessing platform such as TI OMAP3530. By migrating the long operand modular multiplication from a general-purpose ARM Cortex A8 to a specialized C64x+ VLIW DSP, we are able to exploit the XOR-Multiply instruction and the inherent parallelism of the DSP. The proposed multiplication utilizes Multi-Precision Binary Polynomial Multiplication with Unbalanced Exponent Modular Reduction. The resulting DSP implementation performs a GF(2^233) multiplication in less than 1.31us, which is over a seven times speed up when compared with the ARM implementation on the same chip. We present several strategies for different field sizes and field polynomials, and show that a 360MHz DSP easily outperforms the 500MHz ARM. / Master of Science

La première partie de ce mémoire porte sur l’identification aveugle des codes correcteurs d’erreurs non-binaires, travaillant dans le corps de Galois GF(2m). Une étude sur les propriétés des corps de Galois et des codes non-binaires a été conduite afin d’obtenir les éléments indispensables à la mise en oeuvre des méthodes d’identification aveugle. A partir de la seule connaissance des symboles reçus, nous avons développé des méthodes permettant d’identifier les paramètres des codes non-binaires lors d’une transmission non-bruitée et nous avons mis en évidence la pertinence de cette approche lorsque les paramètres de GF(2m) utilisés à l’émission sont connus à la réception. Nous avons aussi mené une étude théorique approfondie pour justifier l’utilisation du critère du rang par la plupart des méthodes d’identification existantes. Dans le cas d’une transmission bruitée, nous avons développé trois algorithmes dédiés à l’identification en aveugle de la taille des mots de code pour des codes binaires et non-binaires. Pour identifier une base du code dual, nous avons généralisé une technique existante pour les codes binaires, basée sur l’utilisation d’un démodulateur à décision ferme, au cas des codes non-binaires. Puis, nous avons amélioré les performances de détection de cette technique en introduisant un processus itératif basé sur l’utilisation conjointe d’un démodulateur à décision souple et d’un algorithme de décodage à décision souple. Dans la deuxième partie de ce mémoire, nous avons tout d’abord proposé un formalisme mathématique pour étudier l’impact des fonctions de mapping-demapping sur la manipulation des données d’un corps de Galois dans le cas des codes non-binaires. Ensuite, nous avons exploité ce formalisme pour détecter et corriger quelques défauts de transmission. Enfin, nous avons étudié l’impact de certaines fonctions de mapping-demapping sur l’identification aveugle des paramètres des codes non-binaires. / In the first part of this thesis, we have focused on the blind identification of non-binary error correcting codes over the Galois field GF(2m). A study of the properties of Galois fields and non-binarycodes has been presented so as to get the essential elements for a blind identification of non-binary codes parameters. From the knowledge of only the received symbols, methods have been developed to identify the code parameters in the case of a noiseless transmission. The relevance of this approach has been highlighted when the parameters of the used Galois field are known bythe receiver. A theoretical study of rank criterion behaviors has been also presented to justify its use by the most existing identification methods. Then, three blind identification methods of the codeword size for binary and non-binary linear codes have been developped in the framework of a noisy transmission. In order to identify a dual code basis, an existing method for binary codes based on the use of a hard decision demodulation has been generalized to non-binary codes. The detection performance of this technique has been improved when an iterative process based on the joint use of a soft-decision demodulator and a soft-decision iterative decoding is introduced. In the second part of this thesis manuscript, a mathematical formalism is proposed in order to investigate the impact of mapping-demapping functions on linear algebra computations and properties over Galois field in the case of non-binary error correcting codes. Finally, this formalism has been exploited to detect or/and correct some digital transmission problems such as a bad synchronization. Finally, we have studied the impact of some mapping-demapping functions on the blind identification ofnon-binary error correcting codes parameters.

Der Vortrag erläutert das Grundprinzip des Diffie-Hellman-Schlüsseltausches mithilfe des diskreten Logarithmus unter Zuhilfenahme elliptischer Kurven über endlichen Körpern.

Der Vortrag erläutert das Grundprinzip des Diffie-Hellman-Schlüsseltausches mithilfe des diskreten Logarithmus unter Zuhilfenahme elliptischer Kurven über endlichen Körpern.

Today’s communication systems are in need of spectrally efficient and high throughput techniques more than ever because of high data rate applications and the scarcity and expense of bandwidth. To cope with increased data rate demands, more base stations are needed which is not cost and energy efficient in cellular networks. It has been shown that wireless relay networks can provide higher network throughput and increase power efficiency with low complexity and cost. Furthermore, network resources can be utilized more efficiently by using network coding in relay networks. A wireless relay network in which multiple nodes exchange information with the help of relay node(s) is called a multi-way relay channel (MWRC). MWRCs are expected to be an integral part of next generation wireless standards. The main focus of this dissertation is the investigation of transmission schemes in an MWRC to improve the throughput and error performance. An MWRC with full data exchange is assumed in which a half-duplex relay station (RS) is the enabler of communication. One of the challenges with signal demodulation in MWRCs is the existence of ambiguous points in the received constellation. The first part of this dissertation investigates a transmission scheme for full data exchange in MWRC that benefits from these points and improves its throughput by 33% compared to traditional relaying. Then an MWRC is considered where a RS assists multiple nodes to exchange messages. A different approach is taken to avoid ambiguous points in the superposition of user symbols at the relay. This can be achieved by employing complex field network coding (CFNC) which results in full data exchange in two communication phases. CFNC may lead to small Euclidean distances between constellation points, resulting in poor error performance. To improve this performance, the optimal user precoding values are derived such that the power efficiency of the relay constellation is highest when channel state information is available at the users. The error performance of each user is then analyzed and compared with other relaying schemes. Finally, focusing on the uplink of multi-way relay systems, the performance of an MWRC is studied in which users can employ arbitrary modulation schemes and the links between the users and the relay have different gains, e.g. Rayleigh fading. Analytical expressions for the exact average pairwise error probability of these MWRCs are derived. The probability density function (PDF) and the mean of the minimum Euclidean distance of the relay constellation are closely approximated, and a tight upper bound on the symbol error probability is developed. / Graduate