# If a different plaintext may produce the same ciphertext, is the system perfectly secure?

Define the injective map $$\phi: \Omega\rightarrow \mathbb{N}$$, such that $$\Omega=\mathcal{A}^n$$ denotes the set of all strings of length $$n\in\mathbb{N}^*$$ from an alphabet $$\mathcal{A}$$ of elements $$a_i$$ and length $$m$$. Define a string (to be encoded) of finite length, $$\mathcal{M}\in\Omega$$. A cipher text $$\mathcal{C}$$, of arbitrary length $$\ell$$, exists and may be generated from the probability distribution for all positive integers $$i\leq m$$ $$\begin{equation} p\left(a_i\right)=E\left(\phi\left(\mathcal{M}\right),\alpha,D\left(\mathcal{C},s,k\right)\right). \end{equation}$$ Let the set of all ciphertexts $$\bf{C}$$ be generated from the set of all probability distributions $$\bf{p}\left(a_i\right)/\sim$$, such that there exists some $$D'\neq D$$ or $$k'\neq k$$ whereby $$\begin{equation*} E\left(\phi\left(\mathcal{M}\right),\alpha,D\left(\mathcal{C},s,k\right)\right)\sim E\left(\phi\left(\mathcal{M}'\right),\alpha,D'\left(\mathcal{C},s,k'\right)\right), \end{equation*}$$ if and only if $$\mathcal{M}\neq\mathcal{M}'$$. The previous relation means that if the message (plaintext) $$\mathcal{M}$$ is changed, then there exists a decryption function $$D'$$ or key $$k'$$ such that the ciphertext is the same.

If such a systems existed what would be its security? Since knowledge of the probability distribution provides no information of $$\mathcal{M}$$, I would expect it to be perfectly secure. If so, could this be shown using mutual information?

By the way, the distribution of ciphertext characters converges to $$p(a_i)$$ as $$\ell\rightarrow\infty$$, so the equivalence is at least asymptotically true. I would be grateful for any help.

Note that the encryption scheme is symmetric.

• Trivially, the one-time pad scheme (which is the most secure cipher per se, not including integrity requirements) has this property, unless I'm misunderstanding you? Are you asking if this property entails security, or if it merely doesn't preclude security? May 10 at 16:53
• Thank you for your reply. I am asking whether the above system is equivalent to the one time pad in terms of security. May 11 at 0:32