Editing Prepare-and-Send Quantum Fully Homomorphic Encryption (section)

=== Stage 2 Server’s Computation===
'''Circuit's Evaluation (QFHE.Eval())'''
*'''Input:''' public key tuple <math>(pk_i)_{i = 0}^{L}</math>, Evaluation key tuple, Encrypted Pad key (<math>\{\tilde{a}^{[0]}...\tilde{a}^{[n]}</math>, <math>\tilde{b}^{[i]}...\tilde{b}^{[n]}\}</math>), QOTP Input State (<math> X^{a^{[1]}}Z^{b^{[1]}}\otimes.....\otimes X^{a^{[n]}}Z^{b^{[n]}}\rho Z^{b^{[1]}}X^{a^{[1]}}\otimes.....\otimes X^{a^{[n]}}Z^{b^{[n]}}</math>)</br>
*'''Output:''' QOTP Circuit Output State (<math> X^{a'^{[1]}}Z^{b'^{[1]}}\otimes.....\otimes X^{a'^{[k]}}Z^{b'^{[k]}}\rho' Z^{b'^{[1]}}X^{a'^{[1]}}\otimes.....\otimes X^{a'^{[k]}}Z^{b'^{[k]}}</math>), Corresponding Encrypted Pad key (<math>\tilde{a'},\tilde{b'}</math>)=(HE.Eval<math>_{evk_L}^\text{C}(\tilde{a}</math>),HE.Eval<math>_{evk_L}^\text{C}(\tilde{b}</math>))</br></br>
Let the Circuit be denoted by C and the gates be <math>c_i</math>
# For all i, <math>c_i</math> gate is applied on qubit m and the <math>m_{th}</math> bits of pad key <math>(\tilde {a}^{[m]},\tilde{b}^{[m]})</math> are updated to <math>(\tilde {a}'^{[m]},\tilde{b}'^{[m]})</math> as follows. 
## If <math>c_i=\{P,H,CNOT\}</math>, a Clifford gate then <math>c_iX^{a^{[m]}}Z^{b^{[m]}}\psi=X^{a'^{[m]}}Z^{b'^{[m]}}c_i\psi</math>)
### if <math>c_i=</math>H then: <math>(\tilde {a}^{[m]},\tilde{b}^{[m]})\rightarrow (\tilde{b}^{[m]},\tilde{a}^{[m]})</math> (Hadamard tranforms X gate into Z and Z into X)
### if <math>c_i=</math>P then: <math>(\tilde {a}^{[m]},\tilde{b}^{[m]})\rightarrow (\tilde{a}^{[m]},\tilde{a}^{[m]}\oplus\tilde{b}^{[m]})</math> 
### if <math>c_i=</math>CNOT with m as target bit and n as control bit then: <math>(\tilde {a}^{[m]},\tilde{b}^{[m]};\tilde {a}^{[n]},\tilde{b}^{[n]})\rightarrow (\tilde {a}^{[m]},\tilde{b}^{[m]}\oplus \tilde {b}^{[n]};\tilde{a}^{[m]}\oplus \tilde {a}^{[n]},\tilde{b}^{[n]})</math>
## If <math>c_i=T_j</math> gate then: <math> (T_jX^{a^{[m]}}Z^{b^{[m]}}\psi=P^{a^{[m]}}X^{a^{[m]}}Z^{b^{[m]}}T_j\psi)</math>
###'''Generate Measurement''' M<math>\leftarrow</math> QFHE.GenMeasurement(<math>\tilde {a}^{[m]},\Gamma_{pk_{j+1}}(sk_j),evk_j)</math>
###'''Gadget Correction'''<math>(X^{a'^{[m]}}Z^{b'^{[m]}}T_j)\psi\leftarrow</math> QFHE.Measurement(M, <math>P^{a^{[m]}}X^{a^{[m]}}Z^{b^{[m]}}T_j\psi)</math>
### Server gets measurement outcome x',z'
###'''Recryption''' Server recrypts one-pad key using pk<math>_{k+1}</math> (<math>\tilde {a''}^{[m]},\tilde{b''}^{[m]})\leftarrow</math> QFHE.Rec<math>_{pk_{k+1}}(\tilde {a}^{[m]},\tilde{b}^{[m]})</math>
### Server updates the recrypted key using x,z and x',z'. (<math>\tilde {a'}^{[m]},\tilde{b'}^{[m]})\leftarrow (\tilde {a''}^{[m]},\tilde{b''}^{[m]}</math>)
## Server sends the updated encryption and QOTP output state to Client.