# If DDH is hard then CCA-secure PKES exist?

My cryptography slides describe several relations between cryptographic problems. I don't still have a good justification on the following:

If decisional Diffie-Hellman problem is hard then there exists a CCA-secure public key encryption scheme.

How is this justified?

• By the existence of Cramer-Shoup encryption? – Maeher Feb 25 at 16:58
• @Maeher yes i think that's an answer – Javier Feb 25 at 21:58

The construction assumes a cyclic group $$\mathbb{G}$$ of order $$q$$ with two generators $$g_1,g_2$$ in which the DDH problem is hard, as well as a collision resistant hash function $$H : \mathbb{G}^3 \to \mathbb{Z}_q$$.