A generalisation of the Diffie-Hellman protocol is studied in this dissertation. In the generalisation polynomials are used to reduce the representation size of a public key and linear shift registers for more efficient computations. These changes are important for the implementation of the protocol in con- strained environments. The security of the Diffie-Hellman protocol and its generalisation is based on the same computations problems. Lastly three examples of the generalisation and their implementation are discussed. For two of the protocols, models are given to predict the execution time and it is determined how well these model predictions are. / Dissertation (MSc (Applied Mathematics))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / MSc / unrestricted

In this thesis we study complex harmonic functions of the form f where f is the sum of a nonconstant analytic and a nonconstant anti-analytic function of one variable. The Fundamental Theorem of Algebra does not apply to such functions, so we ask how many zeros a complex harmonic function can have and where those zeros are located. This thesis focuses on the complex harmonic family of polynomials p_c where p_c is the sum of z+(c/2)z^2 and the conjugate of (c/(n-1))z^(n-1)+(1/n)z^n. We first establish properties of the critical curve, which separates orientation preserving and reversing regions. These properties are then used to show the sum of the orders of the zeros of p_c is -n. In turn, we use this to show p_c has n+2 zeros when 04 and n+4 zeros when c>4, n>5. The total number of zeros of p_c changes when zeros interact with the critical curve, so we investigate where zeros occur on the critical curve to understand how the number of zeros of p_c changes for c between 1 and 4.

Please read the abstract in the front of this document. / Thesis (PhD)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

We impose a monotonicity condition with several reversals on the moduli of the coefficients of a polynomial. We then consider three types of polynomials: (1) those satisfying the condition on all of the coefficients, (2) those satisfying the condition on the even indexed and odd indexed coefficients separately, and (3) polynomials of the form P(z) = a0+ Σnj=µ ajzj where µ ≥ 1 with the coefficients aµ; aµ+1;…; an satisfying the condition. For each type of polynomial, we give a result which puts a bound on the number of zeros in a disk (in the complex plane) centered at the origin. For each type, we give an example showing the results are best possible.

We put restrictions on the coefficients of polynomials and give bounds concerning the number of zeros in a specific region. The restrictions involve a monotonicity-type condition on the coefficients of the even powers of the variable and on the coefficients of the odd powers of the variable (treated separately). We present results by imposing the restrictions on the moduli of the coefficients, the real and imaginary parts of the coefficients, and the real parts (only) of the coefficients.

A well known result due to Ankeny and Rivlin [1] states that if p(z) = ∑v=0n avzv is a polynomial of degree n satisfying p(z) ≠ 0 for |z| < 1 then for R > 1 max |z|=R|p(Z)| ≤Rn+1/2 max|z|=1|p(z)|. It was proposed by late Professor R.P. Boas, Jr. to obtain an inequality analogous to this inequality for polynomials having no zeros in |z| < K. K > 0. In this paper, we obtain some results in this direction, by considering polynomials of the form p(z) = a0 + ∑v=tn a vzv1 ≤ t ≤ n which have no zeros in |z| < K, K ≥1.

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We present some results about chains of monoids S₀ → S₁ → S.₂. and their associated representation rings, with particular emphasis to behavior as the index n (viz. S_n) varies. A rich supply of such chains of monoids can be found via specializations of diagrammatic algebras or variations of diagrammatic algebras, where the inclusions involve the addition of loose strands. This thesis comprises of original results along three themes associated with the above: (1) Identifying a certain polynomial property featuring operators on representation rings, and a characterization of chains of groups G₀ →G₁ → G₂ .. which satisfy this polynomial property. (2) Understanding the induced action on homology from topological actions of the chain of Temperley-Lieb monoids TL₁ → TL₂ → TL₃ ... . Making the analogy to classical representation stability. (3) Identifying chains of diagrammatic monoids S₀ → S₁ → S₂ .. on which cryptographic protocols resist linear attacks. Explicitly computes lower bounds on the dimensions of all representations of various truncations of diagrammatic monoids.

Indiana University-Purdue University Indianapolis (IUPUI) / We consider orthogonal polynomials P_n satisfying orthogonality relations where the measure of orthogonality is, in general, a complex-valued Borel measure supported on subsets of the complex plane. In our consideration we will focus on measures of the form d\mu(z) = \rho(z) dz where the function \rho may depend on other auxiliary parameters. Much of the asymptotic analysis is done via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method, and relies heavily on notions from logarithmic potential theory.

In this paper, a relationship between the zeros and critical points of a polynomial p(z) is established. The relationship is used to prove Sendov's conjecture in some special cases.