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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 15:09, 22 November 2018
, 22 November 2018→Stage 2 Server’s Computation
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####Server generates following superposition sampled over random distribution D\sqrt{D(\mu,r)}|\mu,r\rangle</math> | ####Server generates following superposition sampled over random distribution D\sqrt{D(\mu,r)}|\mu,r\rangle</math> | ||
#### Servers then entangles the two superposition states on quantum input and random distribution D with a third register for function output as follows:</br><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> such that <math>f_0=AltHE.Enc_{pk}();f_1(\mu_1,r_1)=f_0 (\mu_0,r_0)\oplus_H \hat{c}=AltHE.Enc_{pk}(\mu_0,r_0)\oplus_H AltHE.Enc_{pk}(s)</math> | #### Servers then entangles the two superposition states on quantum input and random distribution D with a third register for function output as follows:</br><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> such that <math>f_0=AltHE.Enc_{pk}();f_1(\mu_1,r_1)=f_0 (\mu_0,r_0)\oplus_H \hat{c}=AltHE.Enc_{pk}(\mu_0,r_0)\oplus_H AltHE.Enc_{pk}(s)</math> | ||
####Server measures the last register to get a secret text (function output) <math>y = AltHE.Enc_{pk}(\mu_0,r_0)=AltHE.Enc_{pk}(\mu_1,r_1)\oplus_H AltHE.Enc_{pk}(s) | ####Server measures the last register to get a secret text (function output) <math>y = AltHE.Enc_{pk}(\mu_0,r_0)=AltHE.Enc_{pk}(\mu_1,r_1)\oplus_H AltHE.Enc_{pk}(s)</math>. | ||
####Server performs Hadamard on second register and measures it to get a string d such that first register of input quantum state is reduced to the following ideal state:</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> | ####Server performs Hadamard on second register and measures it to get a string d such that first register of input quantum state is reduced to the following ideal state:</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 HE.Enc<math>_{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 HE.Enc<math>_{pk_{i+1}}(\hat{c},y,d)</math>. | ||
####The server computes the encryption of <math>x,z</math> under <math>pk_{i+1}</math> by homomorphically running the decryption circuit on inputs <math>\mathrm{HE.Enc}_{pk_{i+1}}(sk_i)</math> and HE.Enc<math>_{pk_{i+1}}(c_{x,z,pk_i})</math>. | ####The server computes the encryption of <math>x,z</math> under <math>pk_{i+1}</math> by homomorphically running the decryption circuit on inputs <math>\mathrm{HE.Enc}_{pk_{i+1}}(sk_i)</math> and HE.Enc<math>_{pk_{i+1}}(c_{x,z,pk_i})</math>. | ||