We investigate two different probabilistic public key cryptosystems, one proposed by Okamoto and Uchiyama and the other by
Paillier. Both of them are based on the discrete logarithmic function and the messages are calculated from the modular product
of two those functions, one of which has a fixed value depending on a given public key. The improvements are achieved by a
good choice for the public key so that it is possible to get efficient algorithms.
This work is supported by ITRC and COSEF(97-01-00-13-01-5).