Group theory

With the publication of Shor's quantum algorithm for solving discrete logarithms in
finite cyclic groups, a need for new cryptographic primitives arose; namely, for more secure
primitives that would prevail in the post-quantum era.
The aim of this dissertation is to exploit some hard problems arising from group theory
for use in cryptography. Over the years, there have been many such proposals. We first look
at two recently proposed schemes based on some form of a generalization of the discrete
logari thm problem (DLP), identify their weaknesses, and cryptanalyze them.
By applying the exper tise gained from the above cryptanalyses, we define our own
generalization of the DLP to arbitrary finite groups. We show that such a definition leads
to the design of signature schemes and pseudo-random number generators with provable
security under a security assumption based on a group theoretic problem. In particular,
our security assumption is based on the hardness of factorizing elements of the projective
special linear group over a finite field in some representations. We construct a one-way
function based on this group theoretic assumption and provide a security proof.

Every transitive permutation representation of a finite group is the representation of the group in its action on the cosets of a particular subgroup of the group. The group has a certain rank for each of these representations. We first find almost all rank-3 and rank-4 transitive representations of the projective special linear group P SL(2, q) where q = pm and p is an odd prime. We also determine the rank of P SL (2, p) in terms of p on the cosets of particular given subgroups. We then investigate the construction of rank-3 transitive and primitive extensions of a simple group, such that the extension group formed is also simple. In the latter context we present a new, group theoretic construction of the famous Hoffman-Singleton graph as a rank-3 graph.

Argument is made for the use of variation permissive methods in the study of social judgment; one such dynamic method which purports to track on-line social evaluation (the mouse paradigm) is then introduced. The methodology of the mouse paradigm, which involves updating 'moment-to-moment' feelings via manipulation of a cursor by computer mouse, permits a wide range of experimental contrivance. Three varieties (SCALE, 1D and 2D), which differ in the amount of virtual (on screen) freedom of movement and psychological constraint, were tested with stereotyped targets (negative, ambivalent and positive) to determine any differences in their absolute distance time series and the extent to which aspects of these time series remained correlated with traditional scale-ratings of positivity and stability in feelings about targets. Results indicated a sharp difference between the two-dimensional (2D) variety and the one-dimensional varieties (SCALE and 1D), a finding which supports contention that the 2D variety possesses an appropriate balance of freedom and constraint.