Birthday attack can reduce half of the effective key length of public key cryptosystem based on the finite cyclic group theory and get 63.2% attack success rate. ElGamal cryptograms and elliptic curve cryptography (ECC) based on discrete logarithms can also be expressed in the form of finite cyclic groups, which are vulnerable to birthday attacks. There are two ways to solve this problem: one is to increase the complexity of the addition operation in the group and the other is to increase the order n of the group.

finite cyclic Abelian group, asymmetric encryption, birthday attack.