Presemifields, bundles and polynomials over GF (pn)

The content of this thesis is first and foremost about presemifields and the equivalence classes they may be categorized by. This equivalence has been termed “bundle equivalence' by Horadam. Bundle equivalence is inherited from multiplicative orthogonal cocycles, and the final Chapter is devoted entirely to coboundaries and cocycles. In this thesis we provide a complete computational classification of the bundles of presemifields in all presemifield isotopism classes of order pⁿ, provide a formula for the number of bundles in the presemifields isotopism class of GF (p ²)and give a representative of each bundle, for any prime p. We provide computational classification of the bundles of presemifields in the isotopism class of GF (p³)  for the cases p=3,5,7,11 and give representatives, give formulae for two of the three possible size bundles in the presemifield isotopism class of GF (p³)  which we call the minimum and the mid-size bundles. We provide a Conjecture which states the total number of mid-size bundles in the isotopism class of GF (p³) and give a computational classification of the bundles of presemifields in the isotopism class of GF (2⁵) and GF (3⁴). We provide a measurement of the differential uniformity of functions derived from the diagonal map of presemifield multiplications with order pⁿ < 16 and derive bivariate polynomial formulae for cocycles and coboundaries in We produce a basis for the (pⁿ - 1 - n) - dimensional -space of coboundaries. When p = 2 we give a recursive definition of the basis coboundaries. We use the Kronecker product to explain the self-similarity of the binomial coefficients modulo a prime and use the Kronecker product to define recursively the basis coboundaries for p odd, and we demonstrate this holds for the case p = 2. We show that each cocycle has a unique decomposition as a direct sum of a coboundary and a multiplicative cocycle of restricted form when p = 2.  The results of this thesis have been published in the Proceedings of the International Workshop on Coding and Cryptography, Designs, Codes and Cryptography and the Proceedings of IEEE International Symposium on Information Theory and will appear in the Journal of the Australian Mathematical Society.