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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 16:45, 22 November 2018
, 22 November 2018→Stage 2 Server’s Computation
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###The Toffoli gate is applied to the Pauli one time padded state and the state is reduced to combination of Clifford C and Pauli P corrections as follows:</br><math>TZ^{z^{[l]}}X^{x^{[l]}}Z^{z^{[n]}}X^{x^{[n]}}Z^{z^{[o]}}X^{x^{[o]}}|\psi\rangle</math></br><math>=TZ^{z^{[l]}}X^{x^{[l]}}Z^{z^{[n]}}X^{x^{[n]}}Z^{z^{[o]}}X^{x^{[o]}}T\dagger T|\psi\rangle</math></br><math>=CNOT_{l,o}^{x^{[n]}}CNOT_{n,o}^{x^{[l]}}CZ_{l,n}^{z^{[o]}}Z^{z^{[l]}}X^{x^{[l]}}T|\psi\rangle</math></br><math>=CNOT_{l,o}^{x^{[n]}}CNOT_{n,o}^{x^{[l]}}H_nCNOT_{l,n}^{z^{[o]}}H_{n}Z^{z^{[l]}}X^{x^{[l]}}T|\psi\rangle</math></br><math>=C_{zx}P_{zx}T|\psi\rangle</math>, where <math>C\epsilon \{\text{CNOT,H}\}</math> and <math>P\epsilon\{X,Z\}</math> | ###The Toffoli gate is applied to the Pauli one time padded state and the state is reduced to combination of Clifford C and Pauli P corrections as follows:</br><math>TZ^{z^{[l]}}X^{x^{[l]}}Z^{z^{[n]}}X^{x^{[n]}}Z^{z^{[o]}}X^{x^{[o]}}|\psi\rangle</math></br><math>=TZ^{z^{[l]}}X^{x^{[l]}}Z^{z^{[n]}}X^{x^{[n]}}Z^{z^{[o]}}X^{x^{[o]}}T\dagger T|\psi\rangle</math></br><math>=CNOT_{l,o}^{x^{[n]}}CNOT_{n,o}^{x^{[l]}}CZ_{l,n}^{z^{[o]}}Z^{z^{[l]}}X^{x^{[l]}}T|\psi\rangle</math></br><math>=CNOT_{l,o}^{x^{[n]}}CNOT_{n,o}^{x^{[l]}}H_nCNOT_{l,n}^{z^{[o]}}H_{n}Z^{z^{[l]}}X^{x^{[l]}}T|\psi\rangle</math></br><math>=C_{zx}P_{zx}T|\psi\rangle</math>, where <math>C\epsilon \{\text{CNOT,H}\}</math> and <math>P\epsilon\{X,Z\}</math> | ||
###The Pauli key encryptions are homomorphically updated according to <math>P_{zx}</math>. | ###The Pauli key encryptions are homomorphically updated according to <math>P_{zx}</math>. | ||
### Three encrypted CNOTs are used to correct <math>C^{zx}</math> as follows under <math>\mathrm{AltHE | ### Three encrypted CNOTs are used to correct <math>C^{zx}</math> as follows under <math>\mathrm{AltHE}</math>. | ||
####Server converts <math>\hat{c} = </math>\mathrm{HE.Convert | ####Server converts <math>\hat{c} = </math>\mathrm{HE.Convert(c)}</math>. | ||
####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 entangles above superposition and <math>\psi</math> 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>, 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 <math>\mathrm{AltHE.Enc}_{pk}(s)</math> | #### Servers entangles above superposition and <math>\psi</math> 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>, 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 <math>\mathrm{AltHE.Enc}_{pk}(s)</math> | ||