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Prepare-and-Measure Certified Deletion: Difference between revisions
Prepare-and-Measure Certified Deletion (edit)
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==Protocol Description== | ==Protocol Description== | ||
===Circuit 1: ''Key''=== | |||
The key generation circuit | |||
'''Input ''': None | |||
'''Output''': A key state <math>\rho \in \mathcal{D}(\mathcal{Q}(k+m+n+\mu+\tau)\otimes\mathfrak{H}_{pa}\otimes\mathfrak{H}_{ec}</math> | |||
# Sample <math>\theta \gets \Theta</math> | |||
# Sample <math> r|_{\tilde{\mathcal{I}}} \gets \{0,1\}^k</math> where <math>\tilde{\mathcal{I}} = \{i \in [m] | \theta_i = 1\}</math> | |||
# Sample <math>u \gets \{0,1\}^n</math> | |||
# Sample <math>d \gets \{0,1\}^\mu</math> | |||
# Sample <math>e \gets \{0,1\}^\tau</math> | |||
# Sample <math>H_{pa} \gets \mathfrak{H}_{pa}</math> | |||
# Sample <math>H_{ec} \gets \mathfrak{H}_{ec}</math> | |||
# Output <math>\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}| </math> | |||
===Circuit 2: ''Enc''=== | |||
The encryption circuit | |||
'''Input :''' A plaintext state <math>|\mathrm{msg}\rangle\langle\mathrm{msg}|</math> and a key state <math>| 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}</math> | |||
'''Output:''' A ciphertext state <math>\rho \in \mathcal{D}(\mathcal{Q}(m+n+\tau+\mu))</math> | |||
# Sample <math>r|_\mathcal{I} \gets \{0,1\}^s</math> where <math>\mathcal{I} = \{i \in [m]| \theta_i = 0 \}</math> | |||
# Compute <math>x = H_{pa}(r|_\mathcal{I})</math> where <math>\mathcal{I} = \{i \in [m]| \theta_i = 0 \}</math> | |||
# Compute <math>p = H_{ec}(r|_\mathcal{I}) \oplus d</math> | |||
# Compute <math>q = \mathrm{synd}(r|_\mathcal{I})\oplus e</math> | |||
# Output <math>\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 |</math> | |||
===Circuit 3: ''Dec''=== | |||
The decryption circuit | |||
'''Input :''' A key state <math>| 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}</math> and a ciphertext <math>\rho \otimes |c,p,q\rangle\langle c,p,q| \in \mathcal{D}(\mathcal{Q}(m + n + \mu + \tau)) </math> | |||
'''Output:''' A plaintext state <math>\sigma \in \mathcal{D}(\mathcal{Q}(n))</math> and an error flag <math>\gamma \in \mathcal{D}(\mathcal{Q})</math> | |||
# Compute <math>\rho^\prime = \mathrm{H}^\theta \rho \mathrm{H}^\theta</math> | |||
# Measure <math>\rho^\prime</math> in the computational basis. Call the result <math>r</math> | |||
# Compute <math>r^\prime = \mathrm{corr}(r|_\mathcal{I},q\oplus e)</math> where <math>\mathcal{I} = \{i \in [m]|\theta_i =0\}</math> | |||
# Compute <math>p^\prime = H_{ec}(r^\prime) \oplus d </math> | |||
# If <math>p \neq p^\prime</math>, then set <math>\gamma = |0\rangle\langle 0|</math>. Else, set <math>\gamma = |1\rangle\langle 1|</math> | |||
# Compute <math>x^\prime = H_{pa}(r^\prime)</math> | |||
# Output <math>\rho \otimes \gamma = |c\oplus x^\prime \oplus u \rangle \langle c\oplus x^\prime \oplus u| \otimes \gamma </math> | |||
===Circuit 4: ''Del''=== | |||
The deletion circuit | |||
'''Input :''' A ciphertext <math>\rho \otimes |c,p,q\rangle\langle c,p,q| \in \mathcal{D}(\mathcal{Q}(m+n+\mu+\tau))</math> | |||
'''Output:''' A certificate string <math>\sigma \in \mathcal{D}(\mathcal{Q}(m))</math> | |||
# Measure <math>\rho</math> in the Hadamard basis. Call the output y. | |||
# Output <math>\sigma = |y\rangle\langle y|</math> | |||
===Circuit 5: ''Ver''=== | |||
The verification circuit | |||
'''Input :''' A key state <math>| 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}</math> and a certificate string <math>|y\rangle\langle y| \in \mathcal{D}(\mathcal{Q}(m))</math> | |||
'''Output:''' A bit | |||
# Compute <math>\hat y^\prime = \hat y|_\mathcal{\tilde{I}}</math> where <math> \mathcal{\tilde{I}} = \{i \in [m] | \theta_i = 1 \}</math> | |||
# Compute <math>q = r|_\tilde{\mathcal{I}}</math> | |||
# If <math>\omega(q\oplus \hat y^\prime) < k\delta</math>, output <math>1</math>. Else, output <math>0</math>. | |||
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