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Classical Fully Homomorphic Encryption for Quantum Circuits: Difference between revisions
Classical Fully Homomorphic Encryption for Quantum Circuits (edit)
Revision as of 17:05, 22 November 2018
, 22 November 2018→Stage 2 Server’s Computation
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=== '''Stage 2''' Server’s Computation === | === '''Stage 2''' Server’s Computation === | ||
*Input: <math>\mathrm{evk}_i</math>, | *Input: <math>\mathrm{evk}_i</math>, pad key elements concatenation (<math>s</math>), encryption of s under HE (<math>c=\mathrm{HE.Enc}_{pk}(s)</math>), one time padded message (<math>l</math>) | ||
*Output: Updated encryption of pad key <math>\tilde{z},\tilde{x}</math> (and Quantum One time Padded Output State <math>X^{\tilde {x}}Z^{\tilde{z}}C|\psi\rangle</math> in case of quantum output, where C is the quantum circuit) | *Output: Updated encryption of pad key <math>\tilde{z},\tilde{x}</math> (and Quantum One time Padded Output State <math>X^{\tilde {x}}Z^{\tilde{z}}C|\psi\rangle</math> in case of quantum output, where C is the quantum circuit) | ||
**'''Circuit Evaluation (FHE.Eval())''' | **'''Circuit Evaluation (FHE.Eval())''' | ||
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####Server converts <math>\hat{c} = \mathrm{HE.Convert(c)}</math>. | ####Server converts <math>\hat{c} = \mathrm{HE.Convert(c)}</math>. | ||
####Server generates superposition on distribution D: <math>\sum_{\mu\in\{0,1\},r}\sqrt{D(\mu,r)}|\mu,r\rangle</math> | ####Server generates superposition on distribution D: <math>\sum_{\mu\in\{0,1\},r}\sqrt{D(\mu,r)}|\mu,r\rangle</math> | ||
#### Servers 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>,</br> | #### Servers 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>. The resulting superposition state is: | ####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>, where <math>\beta</math> is the normalization constant. | ||
####Server performs Hadamard on second register and measures it to get a string d. The resulting superposition state is:</br><math>(Z^{d\cdot ((\mu_0,r_0)\oplus (\mu_1,r_1))}\otimes X^{\mu_0})\textrm{CNOT}_{1,2}^s|\psi\rangle</math> </br>where <math>(\mu_0,r_0)=(\mu_1,r_1)\oplus_H s</math>, as <math>\oplus_H</math> is the homomorphic XOR operation. | ####Server performs Hadamard on second register and measures it to get a string d. The resulting superposition state is:</br><math>(Z^{d\cdot ((\mu_0,r_0)\oplus (\mu_1,r_1))}\otimes X^{\mu_0})\textrm{CNOT}_{1,2}^s|\psi\rangle</math> </br>where <math>(\mu_0,r_0)=(\mu_1,r_1)\oplus_H s</math>, as <math>\oplus_H</math> is the homomorphic XOR operation. | ||
####The server uses <math>pk_{i+1}</math> to compute HE.Enc<math>_{pk_{i+1}}(c_{x,z,pk_i})</math> and <math>\mathrm{HE.Enc}_{pk_{i+1}}(\hat{c},y,d)</math>. | ####The server uses <math>pk_{i+1}</math> to compute HE.Enc<math>_{pk_{i+1}}(c_{x,z,pk_i})</math> and <math>\mathrm{HE.Enc}_{pk_{i+1}}(\hat{c},y,d)</math>. | ||