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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 01:56, 26 November 2018
, 26 November 2018→Stage 2 Server’s Computation
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Final superposition at the end of encrypted CNOT operation is:</br> <math>(Z^{d\cdot ((\mu_0,r_0)\oplus (\mu_1,r_1))}\otimes X^{\mu_0})\mathrm{CNOT}_{1,2}^s|\psi_{12}\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. | Final superposition at the end of encrypted CNOT operation is:</br> <math>(Z^{d\cdot ((\mu_0,r_0)\oplus (\mu_1,r_1))}\otimes X^{\mu_0})\mathrm{CNOT}_{1,2}^s|\psi_{12}\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. | ||
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::::#The server uses <math>pk_{i+1}</math> to recrypt 'c' and encrypt other variables under HE: <math>\mathrm{HE.Enc}_{pk_{i+1}}(c)</math>, <math>\mathrm{HE.Enc}_{pk_{i+1}}(\hat{c},y,d)</math>. | ::::#The server uses <math>pk_{i+1}</math> to recrypt 'c' (previously encrypted using <math>pk_{i}</math>) and encrypt other variables under HE: <math>\mathrm{HE.Enc}_{pk_{i+1}}(c)</math>, <math>\mathrm{HE.Enc}_{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 | ::::#The server computes the encryption of <math>x,z</math> (stored in <math>\tilde{z},\tilde{x}</math>) under <math>pk_{i+1}</math> by performing decryption circuit on <math>\mathrm{HE.Enc}_{pk_{i+1}}(c)</math> using <math>\mathrm{HE.Enc}_{pk_{i+1}}(sk_i)</math> (provided by the evaluation key). Here, c, as stated before was the concatenation of encryptions of x, z under <math>pk_{i}</math>. | ||
::::#The server homomorphically computes <math>(\mu_0,r_0)</math> and <math>(\mu_1,r_1)</math>, using | ::::#The server (homomorphically) computes <math>(\mu_0,r_0)</math> and <math>(\mu_1,r_1)</math>, using <math>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 (<math>\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 (or Quantum one time padded state in case of quantum output) to Client. | 3. Server sends updated encryptions of Pauli corrections <math>\tilde{x},\tilde{z}</math> and the classical outcome after measurement of the output state (or Quantum one time padded state in case of quantum output) to Client. | ||