Doctor of Philosophy
School of Information Technology and Computer Science
Chaudhry, Ghulam Rasool, Combinatorial designs with applications in cryptography, Doctor of Philosophy thesis, School of Information Technology and Computer Science, University of Wollongong, 2001. https://ro.uow.edu.au/theses/2007
Since the evolution of human culture, small groups of trustworthy individuals have always played an important role in making crucial decisions in all areas of life. In information based systems, cryptography supports group activity by offering a wide range of cryptographic operations which can only be successfully executed if a well-defined group of people agrees to cooperate. Most of the stronger modern cryptographic systems have been designed and constructed using mathematical functions.
This dissertation looks at the minimal structures of combinatorial designs and their possible uses in cryptographic schemes. These minimal structures can be used to reconstruct a combinatorial design that can be set as a secret. The first objective of this thesis is to study minimal structures of combinatorial designs, especially of Room squares and latin squares. Uniquely completable and critical sets in Room squares are studied. General constructions for uniquely completable sets of Room squares are given but results on minimal and maximal critical sets are empirical only because of the structure of Room squares. General constructions of uniquely completable and critical sets for five different types of back-circulant latin squares are also given. The terms nest, power, influence and strong box in critical sets of Room squares and back-circulant latin squares are introduced. Latin interchanges are used to construct critical sets in modified-two back-circulant latin squares containing odd order subsquares. Critical sets for modified-two back-circulant latin squares for all odd n ≤ 25 are given and a conjecture is made for the general result. This is to be noted that, in this thesis, only strongly uniquely completable and strong critical sets of Room squares and latin squares are studied.
The second objective is to use the minimal structures of combinatorial designs in cryptographic applications, particularly secret sharing schemes. This thesis proposes generalised, hierarchical, key management and perfect secret sharing schemes based on critical sets of Room squares. It shows how cheating in secret sharing schemes can be detected and prevented using critical sets of Room squares.
The third objective is to study another combinatorial structure, Bhaskar Rao designs (BRDs), which can also be used in cryptographic functions, particularly perfect hashing functions. Some open problems of BRDs for block size 4 are solved. This thesis solves most of the cases of BRD(υ,5,λ) for λ = 4,10,20 and some other values of λ. A few cases of BRD(υ,6,λ) are also solved.