Paper 2008/234 Original: eprint.iacr.org/2008/234 ↗

On the CCA1-Security of Elgamal and Damgård's Elgamal

Abstract

It is known that there exists a reduction from the CCA1-security of Damgård's Elgamal (DEG) cryptosystem to what we call the $\DDH^{\DSDH}$ assumption. We show that $\DDH^{\DSDH}$ is unnecessary for DEG-CCA1, while DDH is insufficient for DEG-CCA1. We also show that CCA1-security of the Elgamal cryptosystem is equivalent to another assumption $\DDH^{\CSDH}$, while we show that $\DDH^{\DSDH}$ is insufficient for Elgamal's CCA1-security. Finally, we prove a generic-group model lower bound $\Omega (\sqrt[3]{q})$ for the hardest considered assumption $\DDH^{\CSDH}$, where $q$ is the largest prime factor of the group order.