Classical Fully Homomorphic Encryption for Quantum Circuits: Difference between revisions

Classical Fully Homomorphic Encryption for Quantum Circuits (edit)

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Shraddha

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 <div style="background-color: gray; border: solid thin black;title=Functionality Description;">The Toffoli gate application can be deduced 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>
 </div>
-###The Pauli key encryptions are homomorphically updated  according to <math>P_{zx}</math>.</br> (<math>\tilde {x}^{[l]},\tilde{z}^{[l]};\tilde {x}^{[n]},\tilde{z}^{[n]};\tilde {x}^{[o]},\tilde{z}^{[o]})\rightarrow (\tilde {x}^{[l]},\tilde{z}^{[l]};0,0;0,0)</math>
+:::#The Pauli key encryptions are homomorphically updated  according to <math>P_{zx}</math>.</br> (<math>\tilde {x}^{[l]},\tilde{z}^{[l]};\tilde {x}^{[n]},\tilde{z}^{[n]};\tilde {x}^{[o]},\tilde{z}^{[o]})\rightarrow (\tilde {x}^{[l]},\tilde{z}^{[l]};0,0;0,0)</math>
-### Three encrypted CNOTs are used to correct <math>C^{zx}</math> as follows under <math>\mathrm{AltHE}</math>.</br></br>
+:::# Three encrypted CNOTs are used to correct <math>C^{zx}</math> as follows under <math>\mathrm{AltHE}</math>.</br></br>
 ***'''Server's Preparation:'''
-####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>
-####Server 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> </br><math>\therefore \mu_0\oplus\mu_1=s</math>
+::::#Server 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> </br><math>\therefore \mu_0\oplus\mu_1=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>.</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>
+::::#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>
 ***'''Encrypted CNOT operation:'''<div style="background-color: gray; border: solid thin black;title=Functionality Description;"><math>\sum_{a,b\in\{0,1\}}\alpha_{ab}CNOT_{a,b}^s|a,b\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}|a,b\oplus a\cdot s\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}|a,b\oplus a\cdot(\mu_0\oplus\mu_1)\rangle</math></br><math>=\sum_{b\in\{0,1\}}\alpha_{0b}|0,b\oplus \mu_0\oplus\mu_0\rangle+\alpha_{1b}|1,b\oplus \mu_0\oplus\mu_1\rangle</math>,   <math>\because q\oplus q=0</math></br><math>=\sum_{b\in\{0,1\}}\alpha_{0b}|0\rangle\otimes X^{\mu_0}|b\oplus \mu_0\rangle+\alpha_{1b}|1\rangle \otimes X^{\mu_0}|b\oplus \mu_1\rangle</math>, <math>\because |q\oplus y\rangle=X^y|q\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}|a\rangle\otimes X^{\mu_0}|b\oplus \mu_a\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}(I\otimes X^{\mu_0})|a,b\oplus \mu_a\rangle</math></br> </div>
-####Server XORs the second qubit of first register with <math>\mu_a</math> to get:</br><math>\sum_{a,b\in\{0,1\}}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{a,b}^s|a,b\rangle\otimes|\mu_a,r_a\rangle|y\rangle</math>
+::::#Server XORs the second qubit of first register with <math>\mu_a</math> to get:</br><math>\sum_{a,b\in\{0,1\}}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{a,b}^s|a,b\rangle\otimes|\mu_a,r_a\rangle|y\rangle</math>
-####Server performs Hadamard on second register. The resulting superposition state is:</br><math>\sum_{a,b\in\{0,1\}}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a,b\rangle\otimes H^k|\mu_a,r_a\rangle|y\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a,b\rangle\otimes\bigg(\sum_{e\in\{0,1\}^k}(-1)^{e\cdot(\mu_a,r_a) }|e\rangle\bigg)|y\rangle</math>, <math>\because H^k|q\rangle=\sum_{e\in\{0,1\}^k}(-1)^{e\cdot q}|e\rangle</math>, where q has k qubits</br>
+::::#Server performs Hadamard on second register. The resulting superposition state is:</br><math>\sum_{a,b\in\{0,1\}}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a,b\rangle\otimes H^k|\mu_a,r_a\rangle|y\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a,b\rangle\otimes\bigg(\sum_{e\in\{0,1\}^k}(-1)^{e\cdot(\mu_a,r_a) }|e\rangle\bigg)|y\rangle</math>, <math>\because H^k|q\rangle=\sum_{e\in\{0,1\}^k}(-1)^{e\cdot q}|e\rangle</math>, where q has k qubits</br>
-####Server measures the second register to get d. The resulting superposition is:</br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a,b\rangle\otimes(-1)^{d\cdot(\mu_a,r_a)}|d\rangle|y\rangle</math></br>
+::::#Server measures the second register to get d. The resulting superposition is:</br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{ab}^s|a,b\rangle\otimes(-1)^{d\cdot(\mu_a,r_a)}|d\rangle|y\rangle</math></br>
 <div style="background-color: gray; border: solid thin black;title=Functionality Description;">
 The first register could be equivalently written as:</br><math>(-1)^{d\cdot(\mu_0,r_0)}|0,b\rangle+(-1)^{d\cdot(\mu_1,r_1)}|1,b\rangle</math></br><math>=(-1)^{d\cdot (\mu_0,r_0)}((-1)^{d\cdot((\mu_0,r_0)\oplus(\mu_0,r_0))}|0,b\rangle+(-1)^{d\cdot((\mu_0,r_0)\oplus(\mu_1,r_1))}|1,b\rangle)</math></br><math>=(-1)^{d\cdot (\mu_0,r_0)}((-1)^{0\cdot(d\cdot((\mu_0,r_0)\oplus(\mu_1,r_1))})|0,b\rangle+(-1)^{1\cdot(d\cdot((\mu_0,r_0)\oplus(\mu_1,r_1)))}|1,b\rangle)</math></br><math>=(-1)^{d\cdot (\mu_0,r_0)}((-1)^{a\cdot(d\cdot((\mu_0,r_0)\oplus(\mu_1,r_1))})|a,b\rangle</math></br><math>=(-1)^{d\cdot (\mu_0,r_0)}(Z^{d\cdot((\mu_0,r_0)\oplus(\mu_1,r_1))}|a,b\rangle)</math>, <math>\because Z|q\rangle=(-1)^q|q\rangle</math></br>Thus, the resulting state (upto a global phase) is: </br><math>\approx(Z^{d\cdot((\mu_0,r_0)\oplus(\mu_1,r_1))}\otimes X^{\mu_0})CNOT_{12}^s\sum_{a,b\in\{0,1\}}\alpha_{ab}|a,b\rangle</math></br>
 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.
 </div>
-####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>.
-####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 <math>\mathrm{HE.Enc}_{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 <math>\mathrm{HE.Enc}_{pk_{i+1}}(c_{x,z,pk_i})</math>.
-####The server homomorphically computes <math>(\mu_0,r_0)</math> and <math>(\mu_1,r_1)</math>, using the secret texts encrypting <math>t_{sk_i},sk_i,\hat{c},y,d</math> (all encrypted with HE under public key <math>pk_{i+1}</math>). The server then uses this result, along with the secret texts encrypting <math>x,z,d</math>, to homomorphically compute <math>\tilde{z} = z + (d\cdot ((\mu_0,r_0)\oplus (\mu_1,r_1)),0)</math> and <math>\tilde{x} = x + (0,\mu_0)</math>.
+::::#The server homomorphically computes <math>(\mu_0,r_0)</math> and <math>(\mu_1,r_1)</math>, using the secret texts encrypting <math>t_{sk_i},sk_i,\hat{c},y,d</math> (all encrypted with HE under public key <math>pk_{i+1}</math>). The server then uses this result, along with the secret texts encrypting <math>x,z,d</math>, to homomorphically compute <math>\tilde{z} = z + (d\cdot ((\mu_0,r_0)\oplus (\mu_1,r_1)),0)</math> and <math>\tilde{x} = x + (0,\mu_0)</math>.
 #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.