The purpose of this master\u27s project is to study different probabilistic cryptography schemes. The older probabilistic schemes, Goldwasser-Micali and Blum-Goldwasser, will only be covered briefly for a historical perspective. Several new and promising schemes have appeared in the last 7 years, generating interest. I will be examining the Paillier and Damgard-Jurik schemes in depth. This report explains the mathematics behind the schemes along with their inherent benefits, while also suggesting some potential uses. Details are given on how I optimized the algorithms, with special emphasis on using the Chinese Remainder Theorem (CRT) in the Damgard-Jurik algorithm as well as the other algorithms. One of the main benefits these schemes posses is the additively homomorphic property. I explain the homomorphic properties in the description of the schemes and give an overview of these properties in Appendix A. I create software based in the Java Cryptography Extension (JCE) that is used to do a comparative study. This includes a simple message passing program for encrypted text. I create my own implementations of Paillier, Damgard-Jurik, and a variation of Paillier\u27s scheme as a Provider using the JCE. These implementations use the CRT along with other methods to increase performance and create optimized algorithms. The implementations are plugged into the message passing program with an implementation of RSA from another Provider. A comparative study of the timings of these three schemes is done to show which one performs better in different circumstances. Conclusions are drawn based on the results of the tests and my final opinions are stated
