Prepare-and-Measure Certified Deletion

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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

Properties

Protocol Description

Circuit 1: Key

The key generation circuit

Input : None

Output: A key state

  1. 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 \theta \gets \Theta}
  2. Sample where 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 \tilde{\mathcal{I}} = \{i \in [m] | \theta_i = 1\}}
  3. Sample
  4. Sample
  5. Sample
  6. Sample
  7. Sample
  8. Output

Circuit 2: Enc

The encryption circuit

Input : A plaintext state and a key state

Output: A ciphertext state

  1. Sample where
  2. Compute where
  3. Compute
  4. Compute 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 q = \mathrm{synd}(r|_\mathcal{I})\oplus e}
  5. Output 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 \rho = |r^\theta\rangle\langle r^\theta |\otimes|\mathrm{msg}\oplus x \oplus u,p,q\rangle\langle \mathrm{msg}\oplus x \oplus u,p,q |}

Circuit 3: Dec

The decryption circuit

Input : A key state and a ciphertext

Output: A plaintext state and an error flag

  1. Compute
  2. Measure 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 \rho^\prime} in the computational basis. Call the result
  3. Compute where
  4. Compute 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 p^\prime = H_{ec}(r^\prime) \oplus d }
  5. If 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 p \neq p^\prime} , then set . Else, set
  6. Compute 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 x^\prime = H_{pa}(r^\prime)}
  7. Output

Circuit 4: Del

The deletion circuit

Input : A ciphertext

Output: A certificate string 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 \sigma \in \mathcal{D}(\mathcal{Q}(m))}

  1. Measure in the Hadamard basis. Call the output y.
  2. Output

Circuit 5: Ver

The verification circuit

Input : A key state 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}},\theta,u,d,e,H_{pa},H_{ec}\rangle \langle r|_\tilde{\mathcal{I}},\theta,u,d,e,H_{pa},H_{ec}| \in \mathcal{D}(\mathcal{Q}(k+m+n+\mu+\tau)\otimes\mathfrak{H}_{pa}\otimes\mathfrak{H}_{ec}} and a certificate string 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 |y\rangle\langle y| \in \mathcal{D}(\mathcal{Q}(m))}

Output: A bit

  1. Compute where
  2. Compute
  3. If 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 \omega(q\oplus \hat y^\prime) < k\delta} , output . Else, output 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 0} .

Further Information

References