Editing Prepare-and-Measure Certified Deletion

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This [https://arxiv.org/abs/1910.03551 example protocol] implements the functionality of Quantum Encryption with Certified Deletion using single-qubit state preparation and measurement. This scheme is limited to the single-use, private-key setting.
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==Requirements==
* '''Network Stage: ''' [[:Category:Prepare and Measure Network Stage| Prepare and Measure]]

==Outline==
The scheme consists of 5 circuits-
* ''Key'': This circuit generates the key used in later stages
* ''Enc'': This circuit encrypts the message using the key
* ''Dec'': This circuit decrypts the ciphertext using the key and generates an error flag bit
* ''Del'': This circuit deletes the ciphertext state and generates a deletion certificate
* ''Ver'': This circuit verifies the validity of the deletion certificate using the key  
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==Notation==
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* For any string <math>x \in \{0,1\}^n</math> and set <math>\mathcal{I} \subseteq [n], x|_\mathcal{I}</math> denotes the string <math>x</math> restricted to the bits indexed by <math>\mathcal{I}</math>
* For <math>x,\theta \in \{0,1\}^n, |x^\theta\rangle = H^\theta|x\rangle = H^{\theta_1}|x_1\rangle \otimes H^{\theta_2}|x_2\rangle \otimes ... \otimes H^{\theta_n}|x_n\rangle</math>
* <math>\mathcal{Q} := \mathbb{C}^2</math> denotes the state space of a single qubit,<math>\mathcal{Q}(n) := \mathcal{Q}^{\otimes n}</math>
* <math>\mathcal{D(H)}</math> denotes the set of density operators on a Hilbert space <math>\mathcal{H}</math>
* <math>\lambda</math>: Security parameter
* <math>n</math>: Length, in bits, of the message
* <math>\omega : \{0,1\} \rightarrow \mathbb{N}</math> : Hamming weight function
* <math>m = \kappa(\lambda)</math>: Total number of qubits sent from encrypting party to decrypting party
* <math>k</math>: Length, in bits, of the string used for verification of deletion
* <math>s = m - k</math>: Length, in bits, of the string used for extracting randomness
* <math>\tau = \tau(\lambda)</math>: Length, in bits, of error correction hash
* <math>\mu = \mu(\lambda)</math>: Length, in bits, of error syndrome
* <math>\theta</math>: Basis in which the encrypting party prepare her quantum state
* <math>\delta</math>: Threshold error rate for the verification test
* <math>\Theta</math>: Set of possible bases from which \theta is chosen
* <math>\mathfrak{H}_{pa}</math>: Universal<math>_2</math> family of hash functions used in the privacy amplification scheme
* <math>\mathfrak{H}_{ec}</math>: Universal<math>_2</math> family of hash functions used in the error correction scheme
* <math>H_{pa}</math>: Hash function used in the privacy amplification scheme
* <math>H_{ec}</math>: Hash function used in the error correction scheme
* <math>synd</math>: Function that computes the error syndrome
* <math>corr</math>: Function that computes the corrected string
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==Protocol Description==
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==Properties==
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This scheme has the following properties:
*'''Correctness''': The scheme includes syndrome and correction functions and is thus robust against a certain amount of noise, i.e. below a certain noise threshold, the decryption circuit outputs the original message with high probability.
*'''Ciphertext Indistinguishability''': This notion implies that an adversary, given a ciphertext, cannot discern whether the original plaintext was a known message or a dummy plaintext <math>0^n</math>
*'''Certified Deletion Security''': After producing a valid deletion certificate, the adversary cannot obtain the original message, even if the key is leaked (after deletion).
==References==
* The scheme along with its formal security definitions and their proofs can be found in [https://arxiv.org/abs/1910.03551 Broadbent & Islam (2019)]

<div style='text-align: right;'>''*contributed by Chirag Wadhwa''</div>