Prepare-and-Measure Certified Deletion

This 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.

Requirements

• 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

Notation

• For any string $x\in \{0,1\}^{n}$ and set 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 \mathcal{I} \subseteq [n], x|_\mathcal{I}} denotes the string $x$ restricted to the bits indexed by 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 \mathcal{I}}
• For $x,\theta \in \{0,1\}^{n},|x^{\theta }\rangle =H^{\theta }|x\rangle =H^{\theta _{1}}|x_{1}\rangle \otimes H^{\theta _{2}}|x_{2}\rangle \otimes ...\otimes H^{\theta _{n}}|x_{n}\rangle$
• 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 \mathcal{Q} := \mathbb{C}^2} denotes the state space of a single qubit,${\mathcal {Q}}(n):={\mathcal {Q}}^{\otimes n}$
• ${\mathcal {D(H)}}$ denotes the set of density operators on a Hilbert space ${\mathcal {H}}$
• 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 \lambda} : Security parameter
• $n$: Length, in bits, of the message
• $m=\kappa (\lambda )$: Total number of qubits sent from encrypting party to decrypting party
• 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 k} : Length, in bits, of the string used for verification of deletion
• $s=m-k$: Length, in bits, of the string used for extracting randomness
• $\tau =\tau (\lambda )$: Length, in bits, of error correction hash
• 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 \mu = \mu(\lambda)} : Length, in bits, of error syndrome
• 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} : Basis in which the encrypting party prepare her quantum state
• $\delta$: Threshold error rate for the verification test
• $\Theta$: Set of possible bases from which \theta is chosen
• 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}_{pa}} : Universal$_{2}$ family of hash functions used in the privacy amplification scheme
• ${\mathfrak {H}}_{ec}$: Universal$_{2}$ family of hash functions used in the error correction scheme
• $H_{pa}$: Hash function 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 H_{ec}} : Hash function used in the error correction scheme
• $synd$: Function that computes the error syndrome
• $corr$: Function that computes the corrected string

Protocol Description

Circuit 1: Key

The key generation circuit

Input : None

Output: A key state $\rho \in {\mathcal {D}}({\mathcal {Q}}(k+m+n+\mu +\tau )\otimes {\mathfrak {H}}_{pa}\otimes {\mathfrak {H}}_{ec}$

1. Sample $\theta \gets \Theta$
2. Sample $r|_{\tilde {\mathcal {I}}}\gets \{0,1\}^{k}$ where ${\tilde {\mathcal {I}}}=\{i\in [m]|\theta _{i}=1\}$
3. Sample $u\gets \{0,1\}^{n}$
4. 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 d \gets \{0,1\}^\mu}
5. Sample $e\gets \{0,1\}^{\tau }$
6. Sample $H_{pa}\gets {\mathfrak {H}}_{pa}$
7. 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 H_{ec} \gets \mathfrak{H}_{ec}}
8. Output $\rho =|r|_{\tilde {\mathcal {I}}},\theta ,u,d,e,H_{pa},H_{ec}\rangle \langle r|_{\tilde {\mathcal {I}}},\theta ,u,d,e,H_{pa},H_{ec}|$

Circuit 2: Enc

The encryption circuit

Input : A plaintext state $|\mathrm {msg} \rangle \langle \mathrm {msg} |$ and 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}}

Output: A ciphertext 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 \rho \in \mathcal{D}(\mathcal{Q}(m+n+\tau+\mu))}

1. Sample $r|_{\mathcal {I}}\gets \{0,1\}^{s}$ where ${\mathcal {I}}=\{i\in [m]|\theta _{i}=0\}$
2. Compute $x=H_{pa}(r|_{\mathcal {I}})$ where ${\mathcal {I}}=\{i\in [m]|\theta _{i}=0\}$
3. Compute $p=H_{ec}(r|_{\mathcal {I}})\oplus d$
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 $\rho =|r^{\theta }\rangle \langle r^{\theta }|\otimes |\mathrm {msg} \oplus x\oplus u,p,q\rangle \langle \mathrm {msg} \oplus x\oplus u,p,q|$

Circuit 3: Dec

The decryption circuit

Input : A key state 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 ciphertext 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 \otimes |c,p,q\rangle\langle c,p,q| \in \mathcal{D}(\mathcal{Q}(m + n + \mu + \tau)) }

Output: A plaintext state $\sigma \in {\mathcal {D}}({\mathcal {Q}}(n))$ and an error flag $\gamma \in {\mathcal {D}}({\mathcal {Q}})$

1. Compute $\rho ^{\prime }=\mathrm {H} ^{\theta }\rho \mathrm {H} ^{\theta }$
2. Measure $\rho ^{\prime }$ in the computational basis. Call the result $r$
3. Compute $r^{\prime }=\mathrm {corr} (r|_{\mathcal {I}},q\oplus e)$ where ${\mathcal {I}}=\{i\in [m]|\theta _{i}=0\}$
4. Compute $p^{\prime }=H_{ec}(r^{\prime })\oplus d$
5. If $p\neq p^{\prime }$, then set $\gamma =|0\rangle \langle 0|$. Else, set $\gamma =|1\rangle \langle 1|$
6. Compute $x^{\prime }=H_{pa}(r^{\prime })$
7. Output $\rho \otimes \gamma =|c\oplus x^{\prime }\oplus u\rangle \langle c\oplus x^{\prime }\oplus u|\otimes \gamma$

Circuit 4: Del

The deletion circuit

Input : A ciphertext $\rho \otimes |c,p,q\rangle \langle c,p,q|\in {\mathcal {D}}({\mathcal {Q}}(m+n+\mu +\tau ))$

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

1. Measure $\rho$ in the Hadamard basis. Call the output y.
2. Output $\sigma =|y\rangle \langle y|$

Circuit 5: Ver

The verification circuit

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

Output: A bit

1. Compute ${\hat {y}}^{\prime }={\hat {y}}|_{\mathcal {\tilde {I}}}$ where ${\mathcal {\tilde {I}}}=\{i\in [m]|\theta _{i}=1\}$
2. Compute $q=r|_{\tilde {\mathcal {I}}}$
3. If $\omega (q\oplus {\hat {y}}^{\prime })<k\delta$, output $1$. Else, output $0$.

Properties

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 $0^{n}$
• 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 Broadbent & Islam (2019)

*contributed by Chirag Wadhwa