We consider the problem of secure leader election and propose two cheat-proof election algorithms : Secure Extrema Finding Algorithm (SEFA) and Secure Preference-based Leader Election Algorithm (SPLEA). Both algorithms assume a synchronous distributed system in which the various rounds of election proceed in a lock-step fashion. SEFA assumes that all elector-nodes share a single common evaluation function that returns the same value at any elector-node when applied to a given candidate-node. When elector-nodes can have different preferences for a candidate-node, the scenario becomes more complicated. Our Secure Preference-based Leader Election Algorithm (SPLEA) deals with this case. Here, individual utility functions at each elector-node determine an elector-node?s preference for a given candidate-node. We relax the assumption of a synchronous distributed system in our Asynchronous Extrema Finding Algorithm (AEFA) and also allow the topology to change during the election process. In AEFA, nodes can start the process of election at different times, but eventually after topological changes stop long enough for the algorithm to terminate, all nodes agree on a unique leader. Our algorithm has been proven to be "weakly" self-stabilizing.