# CPA security of a stateless and deterministic encryption system

Why can no stateless and deterministic encryption system be IND-CPA secure?

Is there a formal proof for it?

For CPA security it is sufficient if an adversary can distinguish between encryptions of two messages $m_0$ and $m_1$. That is, an adversary gets to see an encryption $c \gets \textsf{Enc}(pk,m_b)$ for a random bit $b$ together with the public key $pk$. Now in order to figure out if $c$ is an encryption of $m_0$ all the adversary has to do is to recompute $\textsf{Enc}(pk,m_0)$ and check whether or not this value matches the given ciphertext thus breaking CPA security.