A theory of randomness for public key cryptosystems: The ElGamal cryptosystem case

There are many public key cryptosystems that require random inputs to encrypt messages and their security is always discussed assuming that random objects are ideally generated. Since cryptosystems run on computers, it is quite natural that these random objects are computationally generated. One theoretical solution is the use of pseudorandom generators in the Yao's sense [16]. Informally saying, the pseudorandom generators are polynomial-time algorithms whose outputs are computationally indistinguishable from the uniform distribution. Since if we use the Yao's generators then it takes much more time to generate pseudorandom objects than to encrypt messages in public key cryptosystems, we relax the conditions of pseudorandom generators to fit public key cryptosystems and give a minimal requirement for pseudorandom generators within public key cryptosystems. As an example, we discuss the security of the ElGamal cryptosystem [7] with some well-known generators (e.g., the linear congruential generator). We also propose a new pseudorandom number generator, for random inputs to the ElGamal cryptosystem, that satisfies the minimal requirement. The newly proposed generator is based on the linear congruential generator. We show some evidence that the ElGamal cryptosystem with the proposed generator is secure.