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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 12:49, 20 November 2018
, 20 November 2018→Stage 2 Server’s Computation
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### if <math>c_i=</math>P then <div class="floatright">//Pauli Gate</div></br><math>(\tilde {a}^{[l]},\tilde{b}^{[l]})\rightarrow (\tilde{a}^{[l]},\tilde{a}^{[l]}\oplus\tilde{b}^{[l]})</math> | ### if <math>c_i=</math>P then <div class="floatright">//Pauli Gate</div></br><math>(\tilde {a}^{[l]},\tilde{b}^{[l]})\rightarrow (\tilde{a}^{[l]},\tilde{a}^{[l]}\oplus\tilde{b}^{[l]})</math> | ||
### if <math>c_i=</math>CNOT with m as target bit and n as control bit then <div class="floatright">//CNOT</div></br>(<math>\tilde {a}^{[l]},\tilde{b}^{[l]};\tilde {a}^{[n]},\tilde{b}^{[n]})\rightarrow (\tilde {a}^{[l]},\tilde{b}^{[l]}\oplus \tilde {b}^{[n]};\tilde{a}^{[l]}\oplus \tilde {a}^{[n]},\tilde{b}^{[n]})</math> | ### if <math>c_i=</math>CNOT with m as target bit and n as control bit then <div class="floatright">//CNOT</div></br>(<math>\tilde {a}^{[l]},\tilde{b}^{[l]};\tilde {a}^{[n]},\tilde{b}^{[n]})\rightarrow (\tilde {a}^{[l]},\tilde{b}^{[l]}\oplus \tilde {b}^{[n]};\tilde{a}^{[l]}\oplus \tilde {a}^{[n]},\tilde{b}^{[n]})</math> | ||
## If | ## If <math>c_i=T</math> gate then <div class="floatright">//Toffoli Gate on $l_{th}, n_{th}, o_{th}$ key bits</div> | ||
### 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 | ###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>TX^{a^{[l]}}Z^{b^{[l]}}X^{a^{[n]}}Z^{b^{[n]}}X^{a^{[o]}}Z^{b^{[o]}}|\psi\rangle</math></br><math>=TX^{a^{[l]}}Z^{b^{[l]}}X^{a^{[n]}}Z^{b^{[n]}}X^{a^{[o]}}Z^{b^{[o]}}T\dagger T|\psi\rangle</math></br><math>=CNOT_{l,o}^{a^{[n]}}CNOT_{n,o}^{a^{[l]}}CZ_{l,n}^{b^{[o]}}X^{a^{[l]}}Z^{b^{[l]}}T|\psi\rangle</math></br><math>=CNOT_{l,o}^{a^{[n]}}CNOT_{n,o}^{a^{[l]}}H_nCNOT_{l,n}^{b^{[o]}}H_{n}X^{a^{[l]}}Z^{b^{[l]}}T|\psi\rangle</math></br><math>=C_{ab}P_{ab}T|\psi\rangle</math>, where <math>C\epsilon \{\textnormal{CNOT,H}\}</math> and <math>P\epsilon\{X,Z\}</math> | ||
### The Pauli key encryptions are homomorphically updated according to | ###The Pauli key encryptions are homomorphically updated according to <math>P_{ab}</math>. | ||
### Three encrypted CNOTs are used to correct | ### Three encrypted CNOTs are used to correct <math>C^{ab}</math> as follows. | ||
#### The server applies encrypted CNOT operation to the two qubit state ZzXx |ψi using the ciphertext ˆc =HE.Convert(c). | #### The server applies encrypted CNOT operation to the two qubit state ZzXx |ψi using the ciphertext ˆc =HE.Convert(c). | ||
#### Server generates following superposition sampled over random distribution D for the TCF function pairs (f0 =AltHE.Encpk(),f1) based on the condition f0 ⊕H f1 = cˆ{euqation missing} | #### Server generates following superposition sampled over random distribution D for the TCF function pairs (f0 =AltHE.Encpk(),f1) based on the condition f0 ⊕H f1 = cˆ{euqation missing} | ||