A Study of Games Over Finite Groups

There are many fruitful connections between the research areas of game theory and cryptography. For example, game theory was used to solve cryptographic information exchange problems, such as secret sharing and secure multiparty computation (SMPC). Our research involves a study of two-player games defined over several algebraic structures. These structures are used as mathematical platforms for numerous modern cryptosystems. The aim of this research is to identify algebraic structures providing a defense against more recently developed protocol-based attacks while still supporting other security objectives. Our methods include investigation of games defined over various fundamental finite groups, followed by investigation of the effects of mathematical constructions on the strategic features of the games. Our research gives a complete analysis of certain classes of games defined over the finite abelian groups. We also give partial results for nonabelian groups; we conjecture that the problem for arbitrary groups is NP-complete.