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Thesis (M.S.)--University of Nevada, Reno, 2008. / "May, 2008." Includes bibliographical references (leaves 144-146). Online version available on the World Wide Web.

Diss.--Cologne. / Includes bibliographical references.

Diss.--Cologne. / Includes bibliography.

Herz–Schur multipliers of a locally compact group have a well developed theory coming from a large literature; they have proved very useful in the study of the reduced C∗-algebra of a locally compact group. There is also a rich connection to Schur multipliers,which have been studied since the early twentieth century, and have a large number of applications. We develop a theory of Herz–Schur multipliers of a C∗-dynamical system, extending the classical Herz–Schur multipliers, making Herz–Schur multiplier techniques available to study a much larger class of C∗-algebras. Furthermore, we will also introduce and study generalised Schur multipliers, and derive links between these two notions which extend the classical results describing Herz–Schur multipliers in terms of Schur multipliers. This theory will be developed in as much generality as possible, with reference to the classical motivation. After introducing all the necessary concepts we begin the investigation by defining generalised Schur multipliers. The main result is a dilation type characterisation of these multipliers; we also show how such multipliers can be represented using HilbertC∗-modules. Next we introduce and study generalised Herz–Schur multipliers, first extending a classical result involving the representation theory of SU(2), before studying how such functions are related to our generalised Schur multipliers. We give a characterisation of generalised Herz–Schur multipliers as a certain class of the generalised Schur multipliers, and obtain a description of precisely which Schur multipliers belong to this class. Finally, we consider some ways in which the generalised multipliers can arise; firstly, from the classical multipliers which provide our motivation, secondly, from the Haagerup tensor product of a C∗-algebra with itself, and finally from positivity considerations. We show that our theory behaves well with respect to positivity and give conditions under which our multipliers are automatically positive in a natural sense.

Associated to a split, semisimple linear algebraic group G is a group of invariant quadratic forms, which we denote Q(G). Namely, Q(G) is the group of quadratic forms in characters of a maximal torus which are fixed with respect to the action of the Weyl group of G. We compute Q(G) for various examples of products of the special linear, special orthogonal, and symplectic groups as well as for quotients of those examples by central subgroups. Homomorphisms between these linear algebraic groups induce homomorphisms between their groups of invariant quadratic forms. Since the linear algebraic groups are semisimple, Q(G) is isomorphic to Z^n for some n, and so the induced maps can be described by a set of integers called Rost multipliers. We consider various cases of the Kronecker tensor product map between copies of the special linear, special orthogonal, and symplectic groups. We compute the Rost multipliers of the induced map in these examples, ultimately concluding that the Rost multipliers depend only on the dimensions of the underlying vector spaces.

Binary multiplier has been a staple in the digital circuit design. It is used in microprocessor design, DSP applications etc. Here, we discuss the design of a novel multiplier that employs a modified shift and add logic to multiply two n-bit unsigned binary numbers. In our work, we changed the shift and add algorithm. We used a barrel shifter and a multiplexer to generate the partial products. We also found out a way to reduce the number of partial products so that we would have fewer numbers to add after we generated all of them. An array of Carry Save Adders (CSA) is used to add the partial products. With all our arrangements and setups, we aim to reduce delays and make the design as efficient as possible. As examples, we have shown it to multiply two 16-bit numbers, however, the design can easily be either scaled up or down according to the environment the multiplier is being used.

Clock multipliers are among the most critical elements in high speed digital circuits. Power consumption, area, jitter and wide tuning range are key design metrics in these circuits. To provide a wide range of clock frequencies, Digitally Controlled Ring Oscillators (DCROs), whose frequencies are discretely tuned using a Frequency Code Word (FCW), have been investigated in recent studies. They have several advantages over LC-based Voltage Controlled Oscillators (VCO) including simplicity of design, small die area (i.e. no large inductors), better compatibility with deep submicron CMOS processes,ability to offer multiple output phases, and wider tuning range.A compact differential Injection Locked Clock Multiplier (ILCM) based on an injection locked DCRO is implemented in this thesis. As the transistor features continuously shrink and the supply voltage is reduced, ILCMs are becoming more prone to issues such as increased effect of random mismatch, increased device noise, susceptibility of the design to noise coupling and vulnerability to Process Voltage and Temperature (PVT) variations. Furthermore, ILCMs in recent System on a Chip (SoCs) have stringent design requirements including accurate frequency tuning, fine fractional resolution, high levels of integration and better amenability to technology scaling. In the proposed ILCM, multiple techniques were used to address deep submicron CMOS design challenges, as well as modern applications’ requirements. The design is fully digital, synthesizable and automatically placed and routed. All circuit blocks were implemented using digital design flow and designed using a Hardware Description Language (HDL). This allows the design to be more easily ported to deep submicron processes. Online or offline PVT calibration can be performed using a replica oscillator and high speed digital counters to track frequency drifts with PVT variations. A DCRO based on a matrix structure has been utilized to reduce period variations due to random mismatch. The DCRO is built up from pseudo differential delay cells to enhance design immunity to noise coupling. The key thesis contributions are implementing a new DCRO structure using fully syntheziable differential structure, utilizing a novel PVT calibrator that can compensate for frequency mismatch between the main DCRO and its replica, and using a low complexity fractional ILCM technique that achieves a fine fractional resolution with few number of ring oscillator stages.Designed in a TSMC 65 nm GP CMOS process with no analog or RF enhancements, the proposed ILCM frequency ranges from 1.0 to 1.8 GHz and occupies 124:5 m 170 m of chip area. The ILCM can operate in integer or fractional mode for multiplication ratios up to 9. At 1.7 GHz and 1.1 V, the measured integrated RMS jitter (1 kHz to 30 MHz) for the 3rd and 9th multiplication factors are 197 fs and 381 fs, respectively. The ILCM consumes 13.25 mW of power and has a fraction resolution of fref=32. Furthermore, it achieves a jitter-power FOM of −241 dB, when measured at room temperature and 1.1 V. When tested in the presence of switching noise, it provides up to 7 dB improvement in phase noise when compared to a single ended version of the ILCM. In the presence of voltage variations (from 0.9 V to 1.1 V) and temperature variations (from 30 C to 70 C), the maximum integrated RMS jitter variation observed was 50 fs.

The Galois/Counter Mode of Operation (GCM), recently standardized by NIST, simultaneously authenticates and encrypts data at speeds not previously possible for both software and hardware implementations. In GCM, data integrity is achieved by chaining Galois field multiplication operations while a symmetric key block cipher such as the Advanced Encryption Standard (AES), is used to meet goals of confidentiality. Area optimization in a number of proposed high throughput GCM designs have been approached through implementing efficient composite Sboxes for AES. Not as much work has been done in reducing area requirements of the Galois multiplication operation in the GCM which consists of up to 30% of the overall area using a bruteforce approach. Current pipelined implementations of GCM also have large key change latencies which potentially reduce the average throughput expected under traditional internet traffic conditions. This thesis aims to address these issues by presenting area efficient parallel multiplier designs for the GCM and provide an approach for achieving low latency key changes. The widely known Karatsuba parallel multiplier (KA) and the recently proposed Fan-Hasan multiplier (FH) were designed for the GCM and implemented on ASIC and FPGA architectures. This is the first time these multipliers have been compared with a practical implementation, and the FH multiplier showed note worthy improvements over the KA multiplier in terms of delay with similar area requirements. Using the composite Sbox, ASIC designs of GCM implemented with subquadratic multipliers are shown to have an area savings of up to 18%, without affecting the throughput, against designs using the brute force Mastrovito multiplier. For low delay LUT Sbox designs in GCM, although the subquadratic multipliers are a part of the critical path, implementations with the FH multiplier showed the highest efficiency in terms of area resources and throughput over all other designs. FPGA results similarly showed a significant reduction in the number of slices using subquadratic multipliers, and the highest throughput to date for FPGA implementations of GCM was also achieved. The proposed reduced latency key change design, which supports all key types of AES, showed a 20% improvement in average throughput over other GCM designs that do not use the same techniques. The GCM implementations provided in this thesis provide some of the most area efficient, yet high throughput designs to date.

The Galois/Counter Mode of Operation (GCM), recently standardized by NIST, simultaneously authenticates and encrypts data at speeds not previously possible for both software and hardware implementations. In GCM, data integrity is achieved by chaining Galois field multiplication operations while a symmetric key block cipher such as the Advanced Encryption Standard (AES), is used to meet goals of confidentiality. Area optimization in a number of proposed high throughput GCM designs have been approached through implementing efficient composite Sboxes for AES. Not as much work has been done in reducing area requirements of the Galois multiplication operation in the GCM which consists of up to 30% of the overall area using a bruteforce approach. Current pipelined implementations of GCM also have large key change latencies which potentially reduce the average throughput expected under traditional internet traffic conditions. This thesis aims to address these issues by presenting area efficient parallel multiplier designs for the GCM and provide an approach for achieving low latency key changes. The widely known Karatsuba parallel multiplier (KA) and the recently proposed Fan-Hasan multiplier (FH) were designed for the GCM and implemented on ASIC and FPGA architectures. This is the first time these multipliers have been compared with a practical implementation, and the FH multiplier showed note worthy improvements over the KA multiplier in terms of delay with similar area requirements. Using the composite Sbox, ASIC designs of GCM implemented with subquadratic multipliers are shown to have an area savings of up to 18%, without affecting the throughput, against designs using the brute force Mastrovito multiplier. For low delay LUT Sbox designs in GCM, although the subquadratic multipliers are a part of the critical path, implementations with the FH multiplier showed the highest efficiency in terms of area resources and throughput over all other designs. FPGA results similarly showed a significant reduction in the number of slices using subquadratic multipliers, and the highest throughput to date for FPGA implementations of GCM was also achieved. The proposed reduced latency key change design, which supports all key types of AES, showed a 20% improvement in average throughput over other GCM designs that do not use the same techniques. The GCM implementations provided in this thesis provide some of the most area efficient, yet high throughput designs to date.