No description available.

Let p be an odd prime, and define f(x) as follows: f(x) as the sum from 1 to k of a_i times x raised to the power of (p to the power of (alpha_i+1)) in F_(p to the power of n)[x] where 0 is less than or equal to alpha_1 < alpha_2 < ... < alpha_k where alpha_k is equal to alpha. We consider the exponential sum S(f, n) equal to the sum_(x as x runs over the finite field with (p to the n elements) of zeta_(p to the power of Tr_n (f(x))), where zeta_p equals e to the power of (2i times pi divided by p) and Tr_n is the trace from the finite field with p to the n elements to the finite field with p elements.We provide necessary background from number theory and review the basic facts about quadratic forms over a finite field with p elements through both the multivariable and single variable approach. Our main objective is to compute S(f, n) explicitly. The sum S(f, n) is determined by two quantities: the nullity and the type of the quadratic form Tr_n (f(x)). We give an effective algorithm for the computation of the nullity. Tables of numerical values of the nullity are included. However, the type is more subtle and more difficult to determine. Most of our investigation concerns the type. We obtain "relative formulas" for S(f, mn) in terms of S(f, n) when the p-adic order of m is less than or equal to the minimum p-adic order of the alphas. The formulas are obtained in three separate cases, using different methods: (i) m is q to the s power, where q is a prime different from 2 and p; (ii) m is 2 to the s power; and (iii) m is p. In case (i), we use a congruence relation resulting from a suitable Galios action. For case (ii), in addition to the congruence in case (i), a special partition of the finite field with p to the 2n elements is needed. In case (iii), the congruence method does not work. However, the Artin-Schreier Theorem allows us to compute the trace of the extension from the finite field with p to the pn elements to the fi nite field with p to the n elements rather explicitly.When the 2-adic order of each of the alphas is equal and it is less than the 2-adic order of n, we are able to determine S(f, n) explicitly. As a special case, we have explicit formulas for the sum of the monomial, S(ax to the power of (1+ (p to the power of alpha)).Most of the results of the thesis are new and generalize previous results by Carlitz, Baumert, McEliece, and Hou.

Data security will play a central role in the design of future IT systems. The PC has been a major driver of the digital economy. Recently, there has been a shift towards IT applications realized as embedded systems, because they have proved to be good solutions for many applications, especially those which require data processing in real time. Examples include security for wireless phones, wireless computing, pay-TV, and copy protection schemes for audio/video consumer products and digital cinemas. Most of these embedded applications will be wireless, which makes the communication channel vulnerable. The implementation of cryptographic systems presents several requirements and challenges. For example, the performance of algorithms is often crucial, and guaranteeing security is a formidable challenge. One needs encryption algorithms to run at the transmission rates of the communication links at speeds that are achieved through custom hardware devices. Public-key cryptosystems such as RSA, DSA and DSS have traditionally been used to accomplish secure communication via insecure channels. Elliptic curves are the basis for a relatively new class of public-key schemes. It is predicted that elliptic curve cryptosystems (ECCs) will replace many existing schemes in the near future. The main reason for the attractiveness of ECC is the fact that significantly smaller parameters can be used in ECC than in other competitive system, but with equivalent levels of security. The benefits of having smaller key size include faster computations, and reduction in processing power, storage space and bandwidth. This makes ECC ideal for constrained environments where resources such as power, processing time and memory are limited. The implementation of ECC requires several choices, such as the type of the underlying finite field, algorithms for implementing the finite field arithmetic, the type of the elliptic curve, algorithms for implementing the elliptic curve group operation, and elliptic curve protocols. Many of these selections may have a major impact on overall performance. In this dissertation a finite field from a special class called the Optimal Extension Field (OEF) is chosen as the underlying finite field of implementing ECC. OEFs utilize the fast integer arithmetic available on modern microcontrollers to produce very efficient results without resorting to multiprecision operations or arithmetic using polynomials of large degree. This dissertation discusses the theoretical and implementation issues associated with the development of this finite field in a low end embedded system. It also presents various improvement techniques for OEF arithmetic. The main objectives of this dissertation are to --Implement the functions required to perform the finite field arithmetic operations. -- Implement the functions required to generate an elliptic curve and to embed data on that elliptic curve. -- Implement the functions required to perform the elliptic curve group operation. All of these functions constitute a library that could be used to implement any elliptic curve cryptosystem. In this dissertation this library is implemented in an 8-bit AVR Atmel microcontroller. / Dissertation (MEng (Computer Engineering))--University of Pretoria, 2006. / Electrical, Electronic and Computer Engineering / unrestricted