Mullin, Ronald C.

Recent interest in cryptographic systems has generated many mathematical results involving computations in finite fields. In particular, it is known that the use of optimal normal bases significantly reduces the complexity of computations in certain finite fields. This thesis examines three specific aspects of optimal normal bases. First, the effect of optimal normal bases on computations in finite fields is analyzed. Second, the known constructions of optimal normal bases are presented. Finally, the generators of optimal normal bases are discussed in terms of their order in the field.

Minimum Disclosure Proofs by Protocol allow a prover to convince a verifier that the prover knows some verifiable secret information, without allowing the verifier to learn anything about the secret. Quantum Cryptography makes use of the quantum properties of light to allow the prover and the verifier to exchange secret information or to commit to a bit value. The ability to commit to a bit value will be exploited for a minimum disclosure proof by protocol. This thesis unites the two cryptography fields.

Visual cryptography concerns the problem of "hiding" a monochrome image among sets of transparencies, known as shares. These are created in such a fashion that certain sets of shares when superimposed, will reveal the image; while other subsets yield no information. A standard model is the (k, n) scheme, where any k shares will reveal the image, but any k - 1 or fewer shares reveal no information. In this thesis, we explain the basic mechanism for creating shares. We survey the literature and show how to create (k, k) schemes which exist for all k > 2. Then we introduce perfect hash functions, which can be used to construct (k, n) schemes from (k, k) schemes for all 2 < k < n. We introduce generalizations of (k, n) schemes that we call covert cryptographic schemes, and extend this notion to multilevel visual cryptographic schemes. We give conditions for the existence of such schemes, and we conclude with a survey of generalizations.

A zero knowledge identification protocol is an interactive proof system that allows a person to prove that he knows a secret key associated with his identity without revealing the secret key. This type of protocol is the topic of a fairy tale, by Gustavus Simmons called the King's Dilemma, about a king and the problem he has with thieves impersonating his tax collectors. It describes a zero-knowledge identification protocol that will rid the king of his problem. I present this system, the motivation for this thesis, and the transformations from this protocol, that uses lead weights and containers, to protocols that use mathematical elements. The security of these protocols is determined by the complexity of the underlying mathematical problem, such as the knapsack and discrete logarithm problem, and three properties: completeness, soundness, and zero knowledge.

In this paper, we assume a proper pairwise balanced design on 30 points exists. We then get some results that reveal what the largest line size may be and the maximum number of lines of this size. Using these results, we look at one of two cases. We investigate the structure of the linear spaces in this case having no lines of sizes 2, 3, 6 and 30 and rule out many possibilities in order to get a minimum number of possible solutions.

Today new secure cryptosystems are in great demand. Computers are becoming more powerful and old cryptosystems, such as the Data Encryption Standard (DES), are becoming outdated. This thesis describes a new binary additive strewn cipher (HK cryptosystem) that is based on the logistic map. The logistic map is not random, but works under simple rules to become complex, thus making it ideal for implementation in cryptography. Instead of basing the algorithm on one logistic map, the HK cryptosystem. averages several uncoupled logistic maps. Averaging the maps increases the dimension of such a system, thus providing greater security. This thesis will explore the strengths and weaknesses of the HK cryptosystem and will end by introducing a modified version, called the HK8 cryptosystem that does not have the apparent weakness of the HK system.