This example protocol implements the functionality of Quantum Encryption with Certified Deletion using single-qubit state preparation and measurement.
Assumptions
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
Notation
- For any string and set denotes the string restricted to the bits indexed by
- For
- denotes the state space of a single qubit,
- denotes the set of density operators on a Hilbert space
- : Security parameter
- : Length, in bits, of the message
- : Total number of qubits sent from encrypting party to decrypting party
- : Length, in bits, of the string used for verification of deletion
- : Length, in bits, of the string used for extracting randomness
- : Length, in bits, of error correction hash
- : Length, in bits, of error syndrome
- : Basis in which the encrypting party prepare her quantum state
- : Threshold error rate for the verification test
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Theta}
: Set of possible bases from which \theta is chosen
- : Universal family of hash functions used in the privacy amplification scheme
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{H}_{ec}}
: Universal family of hash functions used in the error correction scheme
- : Hash function used in the privacy amplification scheme
- : Hash function used in the error correction scheme
- : Function that computes the error syndrome
- : Function that computes the corrected string
Properties
Protocol Description
Circuit 1: Key
The key generation circuit
Input : None
Output: A key state
- Sample
- Sample Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r|_{\tilde{\mathcal{I}}} \gets \{0,1\}^k}
where
- Sample
- Sample
- Sample
- Sample
- Sample
- Output
Circuit 2: Enc
The encryption circuit
Input : A plaintext state and a key state
Output: A ciphertext state
- Sample where
- Compute where
- Compute
- Compute
- Output
Circuit 3: Dec
The decryption circuit
Input : A key state and a ciphertext
Output: A plaintext state and an error flag
- Compute
- Measure in the computational basis. Call the result
- Compute where
- Compute
- If , then set . Else, set
- Compute
- Output
Circuit 4: Del
The deletion circuit
Input : A ciphertext
Output: A certificate string
- Measure in the Hadamard basis. Call the output y.
- Output
Circuit 5: Ver
The verification circuit
Input : A key state and a certificate string
Output: A bit
- Compute where
- Compute
- If , output . Else, output .
Further Information
References