Systolic architectures are capable of achieving high throughput by maximizing pipelining and by eliminating global data interconnects. Recursive algorithms with regular data flows are suitable for systolization. The computation of multiplicative inversion using algorithms based on EEA (Extended Euclidean Algorithm) are particularly suitable for systolization. Implementations based on EEA present a high degree of parallelism and pipelinability at bit level which can be easily optimized to achieve local data flow and to eliminate the global interconnects which represent most important bottleneck in todays sub-micron design process. The net result is to have high clock rate and performance based on efficient systolic architectures. This thesis examines high performance but also scalable implementations of multiplicative inversion or field division over Galois fields <i>GF</i>(2<i>^m</i>) in the specific case of cryptographic applications where field dimension <i>m</i> may be very large (greater than 400) and either <i>m</i> or defining irreducible polynomial may vary. For this purpose, many inversion schemes with different basis representation are studied and most importantly variants of EEA and binary (Stein's) GCD computation implementations are reviewed. A set of common as well as contrasting characteristics of these variants are discussed. As a result a generalized and optimized variant of EEA is proposed which can compute division, and multiplicative inversion as its subset, with divisor in either <i>polynomial</i> or <i>triangular</i> basis representation. Further results regarding Hankel matrix formation for double-basis inversion is provided. The validity of using the same architecture to compute field division with polynomial or triangular basis representation is proved. Next, a scalable unidirectional bit serial systolic array implementation of this proposed variant of EEA is implemented. Its complexity measures are defined and these are compared against the best known architectures. It is shown that assuming the requirements specified above, this proposed architecture may achieve a higher clock rate performance w. r. t. other designs while being more flexible, reliable and with minimum number of inter-cell interconnects. The main contribution at system level architecture is the substitution of all counter or adder/subtractor elements with a simpler distributed and free of carry propagation delays structure. Further a novel restoring mechanism for result sequences of EEA is proposed using a double delay element implementation. Finally, using this systolic architecture a CMD (Combined Multiplier Divider) datapath is designed which is used as the core of a novel systolic elliptic curve processor. This EC processor uses affine coordinates to compute scalar point multiplication which results in having a very small control unit and negligible with respect to the datapath for all practical values of <i>m</i>. The throughput of this EC based on this bit serial systolic architecture is comparable with designs many times larger than itself reported previously.

In this thesis, we make some contributions at the interface between algebra and graph theory. In Chapter 1, we give an overview of the topics and also the definitions and preliminaries. In Chapter 2, we estimate the number of possible types degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a rather unusual combination of two techniques: a bound on the number of zeros of lacunary polynomials and a bound on the so-called domination number of a graph. In Chapter 3, we deal with the determinant of bipartite graphs. The nullity of a graph G is the multiplicity of 0 in the spectrum of G. Nullity of a (molecular) graph (e.g., a bipartite graph corresponding to an alternant hydrocarbon) has important applications in quantum chemistry and Huckel molecular orbital (HMO) theory. A famous problem, posed by Collatz and Sinogowitz in 1957, asks to characterize all graphs with positive nullity. Clearly, examining the determinant of a graph is a way to attack this problem. In this Chapter, we show that the determinant of a bipartite graph with at least two perfect matchings and with all cycle lengths divisible by four, is zero. In Chapter 4, we first introduce an application of spectral graph theory in proving trigonometric identities. This is a very simple double counting argument that gives very short proofs for some of these identities (and perhaps the only existed proof in some cases!). In the rest of Chapter 4, using some properties of the well-known Chebyshev polynomials, we prove some theorems that allow us to evaluate the number of spanning trees in join of graphs, Cartesian product of graphs, and nearly regular graphs. In the last section of Chapter 4, we obtain the number of spanning trees in an (r,s)-semiregular graph and its line graph. Note that the same results, as in the last section, were proved by I. Sato using zeta functions. But our proofs are much shorter based on some well-known facts from spectral graph theory. Besides, we do not use zeta functions in our arguments. In Chapter 5, we present the conclusion and also some possible projects.

An important advantage of Elliptic Curve Cryptosystem (ECC) is the shorter key length in public key cryptographic systems. It can provide adequate security when the bit length over than 160 bits. Therefore, it has become a popular system in recent years. Scalar multiplication also called point multiplication is the core operation in ECC. In this thesis, we propose the ECC architectures of two different irreducible polynomial versions that are trinomial in GF(2167) and pentanomial in GF(2163). These architectures are based on Montgomery point multiplication with projective coordinate. We use polynomial basis representation for finite field arithmetic. All adopted multiplication, square and add operations over binary field can be completed within one clock cycle, and the critical path lies on multiplication. In addition, we use Itoh-Tsujii algorithm combined with addition chain, to execute binary inversion through using iterative binary square and multiplication. Because the double and add operations in point multiplication need to run many iterations, the execution time in overall design will be decreased if we can improve this partition. We propose two ways to improve the performance of point multiplication. The first way is Minus Cycle Version. In this version, we reschedule the double and add operations according to point multiplication algorithm. When the clock cycle time (i.e., critical path) of multiplication is longer than that of add and square, this method will be useful in improving performance. The second way is Pipeline Version. It speeds up the multiplication operations by executing them in pipeline, leading to shorter clock cycle time. For the hardware implementation, TSMC 0.13um library is employed and all modules are organized in a hierarchy structure. The implementation result shows that the proposed 167-bit Minus Cycle Version requires 156.4K gates, and the execution time of point multiplication is 2.34us and the maximum speed is 591.7Mhz. Moreover, we compare the Area x Time (AT) value of proposed architectures with other relative work. The results exhibit that proposed 167-bit Minus Cycle Version is the best one and it can save up to 38% A T value than traditional one.

Let n and l be positive integers and f (x) be an irreducible polynomial over Fq such that ldeg( f (x)) &lt / 2n - 1, where q is 2 or 3. We obtain an effective upper bound for the multiplication complexity of n-term polynomials modulo f (x)^l. This upper bound allows a better selection of the moduli when Chinese Remainder Theorem is used for polynomial multiplication over Fq. We give improved formulae to multiply polynomials of small degree over Fq. In particular we improve the best known multiplication complexities over Fq in the literature in some cases. Moreover, we present a method for multiplication in finite fields improving finite field multiplication complexity muq(n) for certain values of q and n. We use local expansions, the lengths of which are further parameters that can be used to optimize the bounds on the bilinear complexity, instead of evaluation into residue class field. We show that we obtain improved bounds for multiplication in Fq^n for certain values of q and n where 2 &lt / = n &lt / =18 and q = 2, 3, 4.

The representation of field elements has a great impact on the performance of the finite field arithmetic. In this thesis, we give modified version of redundant representation which works for any finite fields of arbitrary characteristics to design arithmetic circuits with small complexity. Using our modified redundant representation, we improve many of the complexity values. We then propose new representations as an alternative way to represent finite fields of characteristic two by using Charlier and Hermite polynomials. We show that multiplication in these representations can be achieved with subquadratic space complexity. Charlier and Hermite representations enable us to find binomial, trinomial or quadranomial irreducible polynomials which allows us faster modular reduction over binary fields when there is no desirable such low weight irreducible polynomial in other representations. These representations are very interesting for the NIST and SEC recommended binary fields GF(2^{283}) and GF(2^{571}) since there is no optimal normal basis (ONB) for the corresponding extensions. It is also shown that in some cases the proposed representations have better space complexity even if there exists an ONB for the corresponding extension.

Let p be a prime and q = pk. The polynomial gn,q isin Fp[x] defined by the functional equation Sigmaa isin Fq (x+a)n = gn,q(xq- x) gives rise to many permutation polynomials over finite fields. We are interested in triples (n,e;q) for which gn,q is a permutation polynomial of Fqe. In Chapters 2, 3, and 4 of this dissertation, we present many new families of permutation polynomials in the form of gn,q. The permutation behavior of gn,q is becoming increasingly more interesting and challenging. As we further explore the permutation behavior of gn,q, there is a clear indication that gn,q is a plenteous source of permutation polynomials. We also describe a piecewise construction of permutation polynomials over a finite field Fq which uses a subgroup of Fq*, a “selection” function, and several “case” functions. Chapter 5 of this dissertation is devoted to this piecewise construction which generalizes several recently discovered families of permutation polynomials.

In this paper, we provide a generalization of binary quadratic residue codes to the cases of higher power prime residues over the finite field of the same order, which we will call qth power residue codes. We find generating polynomials for such codes, define a new notion corresponding to the binary concept of an idempotent, and use this to find square root lower bound for the codeword weight of the duals of such codes, which leads to a lower bound on the weight of the codewords themselves. In addition, we construct a family of asymptotically bad qth power residue codes. / text

This thesis introduces new efficient hardware implementations for the Advanced Encryption Standard (AES) algorithm. Two main contributions are presented in this thesis, the first one is a high speed 128 bits AES encryptor, and the second one is a new 32 bits AES design. In first contribution a 128 bits loop unrolled sub-pipelined AES encryptor is presented. In this encryptor an efficient merging for the encryption process sub-steps is implemented after relocating them. The second contribution presents a 32 bits AES design. In this design, the S-BOX is implemented with internal pipelining and it is shared between the main round and the key expansion units. Also, the key expansion unit is implemented to work on the fly and in parallel with the main round unit. These designs have achieved higher FPGA (Throughput/Area) efficiency comparing to previous AES designs.

In this thesis, we make some contributions at the interface between algebra and graph theory. In Chapter 1, we give an overview of the topics and also the definitions and preliminaries. In Chapter 2, we estimate the number of possible types degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a rather unusual combination of two techniques: a bound on the number of zeros of lacunary polynomials and a bound on the so-called domination number of a graph. In Chapter 3, we deal with the determinant of bipartite graphs. The nullity of a graph G is the multiplicity of 0 in the spectrum of G. Nullity of a (molecular) graph (e.g., a bipartite graph corresponding to an alternant hydrocarbon) has important applications in quantum chemistry and Huckel molecular orbital (HMO) theory. A famous problem, posed by Collatz and Sinogowitz in 1957, asks to characterize all graphs with positive nullity. Clearly, examining the determinant of a graph is a way to attack this problem. In this Chapter, we show that the determinant of a bipartite graph with at least two perfect matchings and with all cycle lengths divisible by four, is zero. In Chapter 4, we first introduce an application of spectral graph theory in proving trigonometric identities. This is a very simple double counting argument that gives very short proofs for some of these identities (and perhaps the only existed proof in some cases!). In the rest of Chapter 4, using some properties of the well-known Chebyshev polynomials, we prove some theorems that allow us to evaluate the number of spanning trees in join of graphs, Cartesian product of graphs, and nearly regular graphs. In the last section of Chapter 4, we obtain the number of spanning trees in an (r,s)-semiregular graph and its line graph. Note that the same results, as in the last section, were proved by I. Sato using zeta functions. But our proofs are much shorter based on some well-known facts from spectral graph theory. Besides, we do not use zeta functions in our arguments. In Chapter 5, we present the conclusion and also some possible projects.

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