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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 01:02, 26 November 2018
, 26 November 2018→Stage 2 Server’s Computation
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*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>) | *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> | *Output: Updated encryption of pad key <math>\tilde{z},\tilde{x}</math> and final classical outcome after performing the circuit. | ||
**'''Circuit Evaluation (FHE.Eval())''' | **'''Circuit Evaluation (FHE.Eval())''' | ||
#Server creates a quantum superposition state for encrypted input <math>l</math>: <math>Z^zX^x|\psi\rangle</math>, where </br><math>|\psi\rangle=\sum_{a,b\epsilon\{0,1\}}\alpha_{ab}|a,b\rangle</math> represents the two qubits superposition state for classical message m,</br> <math>Z^zX^x</math> represents quantum one time pad. </br> | #Server creates a quantum superposition state for encrypted input <math>l</math>: <math>Z^zX^x|\psi\rangle</math>, where </br><math>|\psi\rangle=\sum_{a,b\epsilon\{0,1\}}\alpha_{ab}|a,b\rangle</math> represents the two qubits superposition state for classical message m,</br> <math>Z^zX^x</math> represents quantum one time pad. </br> | ||
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::::#The server (homomorphically) computes <math>(\mu_0,r_0)</math> and <math>(\mu_1,r_1)</math>, using <math>\mathrm{HE.Enc}_{pk_{i+1}}(t_{sk_i},sk_i)</math>, provided by the evaluation key <math>\mathrm{evk}_i</math> encrypted under <math>pk_{i+1}</math>, and <math>\mathrm{HE.Enc}_{pk_{i+1}}(\hat{c},y,d)</math>, from the previous step. | ::::#The server (homomorphically) computes <math>(\mu_0,r_0)</math> and <math>(\mu_1,r_1)</math>, using <math>\mathrm{HE.Enc}_{pk_{i+1}}(t_{sk_i},sk_i)</math>, provided by the evaluation key <math>\mathrm{evk}_i</math> encrypted under <math>pk_{i+1}</math>, and <math>\mathrm{HE.Enc}_{pk_{i+1}}(\hat{c},y,d)</math>, from the previous step. | ||
::::#The server then uses this results of the last three steps, to (homomorphically) update Pauli encryptions for encrypted <math>CNOT^s_{l,n}</math>: </br>(<math>\tilde {x}^{[l]},\tilde{z}^{[l]};\tilde {x}^{[n]},\tilde{z}^{[n]})\rightarrow (\tilde {x}^{[l]},\tilde{z}^{[l]}+d\cdot ((\mu_0,r_0)\oplus (\mu_1,r_1);\tilde {x}^{[n]}+\mu_0,\tilde{z}^{[n]})</math></br> | ::::#The server then uses this results of the last three steps, to (homomorphically) update Pauli encryptions for encrypted <math>CNOT^s_{l,n}</math>: </br>(<math>\tilde {x}^{[l]},\tilde{z}^{[l]};\tilde {x}^{[n]},\tilde{z}^{[n]})\rightarrow (\tilde {x}^{[l]},\tilde{z}^{[l]}+d\cdot ((\mu_0,r_0)\oplus (\mu_1,r_1);\tilde {x}^{[n]}+\mu_0,\tilde{z}^{[n]})</math></br> | ||
3. Server sends updated encryptions of Pauli corrections <math>\tilde{x},\tilde{z}</math> and the classical outcome after measurement of the output state | 3. Server sends updated encryptions of Pauli corrections <math>\tilde{x},\tilde{z}</math> and the classical outcome after measurement of the output state to Client. | ||
=== '''Stage 3''' Client’s Output Correction === | === '''Stage 3''' Client’s Output Correction === | ||