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Classical Fully Homomorphic Encryption for Quantum Circuits: Difference between revisions
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Classical Fully Homomorphic Encryption for Quantum Circuits (edit)
Revision as of 16:10, 23 November 2018
, 23 November 2018→Stage 2 Server’s Computation
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####Server entangles above superposition and <math>|\psi\rangle</math> with a third register:<math>\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu,r)}|a,b\rangle|\mu,r\rangle|f_a(r)\rangle</math>, such that </br><math>f_0=\mathrm{AltHE.Enc}_{pk}()</math>;</br><math>f_1(\mu_1,r_1)=f_0 (\mu_0,r_0)\oplus_H \hat{c}=\mathrm{AltHE.Enc}_{pk}(\mu_0,r_0)\oplus_H \mathrm{AltHE.Enc}_{pk}(s)</math> | ####Server entangles above superposition and <math>|\psi\rangle</math> with a third register:<math>\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu,r)}|a,b\rangle|\mu,r\rangle|f_a(r)\rangle</math>, such that </br><math>f_0=\mathrm{AltHE.Enc}_{pk}()</math>;</br><math>f_1(\mu_1,r_1)=f_0 (\mu_0,r_0)\oplus_H \hat{c}=\mathrm{AltHE.Enc}_{pk}(\mu_0,r_0)\oplus_H \mathrm{AltHE.Enc}_{pk}(s)</math> | ||
####Server measures the last register to get <math>y =\mathrm{AltHE.Enc}(\mu_0,r_0)=\mathrm{AltHE.Enc}_{pk}(\mu_1,r_1)\oplus_H AltHE.Enc_{pk}(s)</math>.</br> The resulting superposition state is:<math>\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}|a,b\rangle|\mu_a,r_a\rangle|\mathrm{AltHE.Enc}(\mu_0,r_0)\rangle=\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}|a,b\rangle|\mu_a,r_a\rangle|y\rangle</math> | ####Server measures the last register to get <math>y =\mathrm{AltHE.Enc}(\mu_0,r_0)=\mathrm{AltHE.Enc}_{pk}(\mu_1,r_1)\oplus_H AltHE.Enc_{pk}(s)</math>.</br> The resulting superposition state is:<math>\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}|a,b\rangle|\mu_a,r_a\rangle|\mathrm{AltHE.Enc}(\mu_0,r_0)\rangle=\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}|a,b\rangle|\mu_a,r_a\rangle|y\rangle</math> | ||
***'''Encrypted CNOT operation:''' <math>\sum_{a,b\in\{0,1\}}\alpha_{ab}CNOT_{a,b}^s|a\rangle|b\rangle</br><math>=\alpha_{ab}|a\rangle|b\oplus a\cdot s\rangle</math></br><math>=\alpha_{ab}|a\rangle|b\oplus a\cdot(\mu_0+\mu_1)\rangle</math></br><math>=\alpha_{0b}|0\rangle|b\oplus 0\rangle+\alpha_{1b}|1\rangle|b\oplus \mu_0+\mu_1\rangle</math></br><math>=\alpha_{0b}|0\rangle|b\oplus \mu_0+\mu_1\rangle+\alpha_{1b}|1\rangle|b\oplus \mu_0+\mu_1\rangle</math>, <math>\because q\oplus q=0</math></br><math>=\alpha_{0b}|0\rangle X^{\mu_0}|b\oplus \mu_0\rangle+\alpha_{1b}|1\rangle X^{\mu_0}|b\oplus \mu_1\rangle</math>, <math>\because |q\oplus y\rangle=X^y|q\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}|a\rangle X^{\mu_0}|b\oplus \mu_a\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}(I\otimes X^{\mu_0})|a\rangle |b\oplus \mu_a\rangle</math></br> Thus, Server's superposition state could be written as:</br><math>\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a\rangle |b\oplus \mu_a\rangle\otimes|\mu_a,r_a\rangle</math> | ***'''Encrypted CNOT operation:''' <math>\sum_{a,b\in\{0,1\}}\alpha_{ab}CNOT_{a,b}^s|a\rangle|b\rangle</math></br><math>=\alpha_{ab}|a\rangle|b\oplus a\cdot s\rangle</math></br><math>=\alpha_{ab}|a\rangle|b\oplus a\cdot(\mu_0+\mu_1)\rangle</math></br><math>=\alpha_{0b}|0\rangle|b\oplus 0\rangle+\alpha_{1b}|1\rangle|b\oplus \mu_0+\mu_1\rangle</math></br><math>=\alpha_{0b}|0\rangle|b\oplus \mu_0+\mu_1\rangle+\alpha_{1b}|1\rangle|b\oplus \mu_0+\mu_1\rangle</math>, <math>\because q\oplus q=0</math></br><math>=\alpha_{0b}|0\rangle X^{\mu_0}|b\oplus \mu_0\rangle+\alpha_{1b}|1\rangle X^{\mu_0}|b\oplus \mu_1\rangle</math>, <math>\because |q\oplus y\rangle=X^y|q\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}|a\rangle X^{\mu_0}|b\oplus \mu_a\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}(I\otimes X^{\mu_0})|a\rangle |b\oplus \mu_a\rangle</math></br> Thus, Server's superposition state could be written as:</br><math>\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a\rangle |b\oplus \mu_a\rangle\otimes|\mu_a,r_a\rangle</math> | ||
####Server performs Hadamard on second register. The resulting superposition state is:</br><math>\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a\rangle |b\oplus \mu_a\rangle\otimesH|\mu_a,r_a\rangle</math></br><math>(\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a\rangle |b\oplus \mu_a\rangle\otimes\sum_{d\in\{0,1\}}(-1)^{d\cdot((\sum_{a\in\{0,1\}(\mu_a,r_a)mod 2)) }|d\rangle</math>, <math>\because H|q\rangle=\sum_{d\in\{0,1\}}(-1)^{d\cdot q}|d\rangle</math></br><math>(\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a\rangle |b\oplus \mu_a\rangle\otimes\sum_{d\in\{0,1\}}(-1)^{d\cdot((\mu_0,r_0)\oplus (\mu_1,r_1))) }|d\rangle</math> | ####Server performs Hadamard on second register. The resulting superposition state is:</br><math>\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a\rangle |b\oplus \mu_a\rangle\otimesH|\mu_a,r_a\rangle</math></br><math>(\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a\rangle |b\oplus \mu_a\rangle\otimes\sum_{d\in\{0,1\}}(-1)^{d\cdot((\sum_{a\in\{0,1\}(\mu_a,r_a)mod 2)) }|d\rangle</math>, <math>\because H|q\rangle=\sum_{d\in\{0,1\}}(-1)^{d\cdot q}|d\rangle</math></br><math>(\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a\rangle |b\oplus \mu_a\rangle\otimes\sum_{d\in\{0,1\}}(-1)^{d\cdot((\mu_0,r_0)\oplus (\mu_1,r_1))) }|d\rangle</math> | ||