# Integrating Classical Preprocessing into an Optical Encryption Scheme
## 3 Results
### Symmetric-key Encryption Using Mesoscopic Coherent States  
Motivated by AlphaEta, we consider symmetric key encryption, where parts of the ciphertext can be non-classical: it can be a sequence of coherent states. When decrypting a ciphertext, a measurement should be done to obtain a classical bitstring from the coherent states. This together with the classical part forms the _reconstructed ciphertext_. Afterwards, from the reconstructed ciphertext, the plaintext can be recovered:  
**Definition 3**: _A symmetric-key encryption scheme using mesoscopic coherent states is a triple of algorithms as follows:_  
* KeyGen_: Given a key length, outputs a corresponding secret key_ \(k\)_._
* Enc_: Given a plaintext_ \(m\) _and secret key_ \(k\)_, it outputs a ciphertext_ \(c\)_, consisting of a sequence of coherent states and a bitstring:_ \[c=(\ket{\psi_{1}},\dots,\ket{\psi_{j}},c_{1},\dots,c_{\ell})\] (4)
* Dec_: This process consists of two phases. Given a ciphertext_ \(c\) _and a secret key_ \(k\)_, the sequence of coherent states in_ \(c\) _is measured in the first phase. Now_ \(c\) _can be considered a classical bitstring when entering the second phase of the decryption. The final output of the algorithm is the plaintext_ \(m\)_._  
**Remark 1**: _We allow the sequence of coherent states or the classical bitstring to be empty to include both classical and purely quantum symmetric-key encryption schemes, such as AlphaEta._  
Figure 1 illustrates the overall structure of a symmetric-key encryption scheme using mesoscopic coherent states, highlighting the two different communication channels. For such a scheme to be useful, we assume that the intended recipient, knowing the secret key and leaving aside channel imperfections, can recover correctly the plaintext from the ciphertext. This is captured by correctness:  
**Definition 4**: _A symmetric-key encryption scheme using mesoscopic coherent states is \(\delta\)-correct if \((\mathtt{Dec}_{k}\circ\mathtt{Enc}_{k})(m)=m\) except with a probability smaller than \(\delta\), for all \(k\) generated by KeyGen and all plaintexts \(m\)._  
Notice that AlphaEta satisfies Definition 4, as discussed in [10]; the intended recipient can recover a plaintext bit with an error rate below \(10^{-9}\). A user or adversary not knowing the secret information, should obtain as little information about a plaintext bit as possible. The entropy of a plaintext bit given the reconstructed ciphertext should not be very different from the entropy about that plaintext bit. We use this as motivation for  
Figure 1: Overview of a symmetric-key encryption scheme using mesoscopic coherent states. the following security definition. One could consider stronger security notions, in particular when allowing ciphertext expansion, but the following notion seems adequate when aiming at a composition with a classical scheme as discussed in Section 4.1. **Definition 5**. : _Let \((\mathtt{KeyGen},\mathtt{Enc},\mathtt{Dec})\) be a symmetric-key encryption scheme using mesoscopic coherent states. Let \(P_{i}\) denote the random variable taking the value of a plaintext bit and \(C_{i}\) denote the random variables taking the value of classical ciphertext bits. Let \(\tilde{C_{1}},\ldots,\tilde{C_{l}}\) represent the random variables taking the value of the reconstructed ciphertext bits, i.e., \(\tilde{C_{i}}\)s correspond to the bits obtained after the corresponding measurement of the coherent states. We say that \((\mathtt{KeyGen},\mathtt{Enc},\mathtt{Dec})\) is \(\epsilon\)-secure if for \(1\leq i\leq s\):_  
\[|H(P_{i}|\tilde{C_{1}},\ldots,\tilde{C_{\ell}},C_{\ell+1},\ldots,C_{s})-H(P_{ i})|\leq\epsilon \tag{5}\]  
**Remark 2**. : _The AlphaEta protocol satisfies Definition 5, assuming the a priori probability distribution for a bit is the discrete uniform probability._  
_As discussed in [10], the amount of information that the adversary, not knowing the bases, can obtain through eavesdropping can be made sufficiently small by choosing the parameters \(\langle n\rangle\) and \(\beta\) appropriately.