Editing Prepare-and-Send Quantum Fully Homomorphic Encryption
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<math>\sum_{a,b\epsilon\{0,1\}}\frac{1}{4}\rho(HE.Enc_{pk_0^{[i]}}(a^{[i]}),HE.Enc_{pk_0^{[i]}}(b^{[i]}))\otimes X^aZ^b\sigma Z^bX^a</math> | <math>\sum_{a,b\epsilon\{0,1\}}\frac{1}{4}\rho(HE.Enc_{pk_0^{[i]}}(a^{[i]}),HE.Enc_{pk_0^{[i]}}(b^{[i]}))\otimes X^aZ^b\sigma Z^bX^a</math> | ||
#Client sends encryptions (<math>\tilde{a}^{[i]},\tilde{b}^{[i]}</math>) and the quantum one time padded (QOTP) 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]}} \forall i</math>, to the Server with the evaluation keys and public keys. | #Client sends encryptions (<math>\tilde{a}^{[i]},\tilde{b}^{[i]}</math>) and the quantum one time padded (QOTP) 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]}} \forall i</math>, to the Server with the evaluation keys and public keys. | ||
'''Gadget Construction ( | \ '''Gadget Construction (QFHE.GenGadgetpki+1(ski))''' | ||
# Generate | # Generate 4m EPR pairs ( | ||
# Choose | # Choose 2m pairs using sk | ||
## If | ## If (sk = 0) then {(a1,a2),(a2,a3),...,(a4m−1,a4m)} ii. If (sk = 1) then {(a1,a3),(a2,a4),...,(a4m−2,a4m)} | ||
# For j=1 to 2m, | |||
# For j=1 to 2m, | ## Choose p[j] | ||
## Choose p[j] | ## Perform Bell Measurement on jth pair with an extra (P†)p operation, get outcomes (x[j],z[j]) | ||
## Perform Bell Measurement on | ## Thus, new EPR pairs are{missing math}<br/>If (sk = 0) then {(b1,b2),(b2,b3),...,(b4m−1,b4m)}<br/>If (sk = 1) then {(b1,b3),(b2,b4),...,(b4m−2,b4m)}<br/>Denote the 2m entangled pairs be denoted by {(s1,t1),(s2,t2),...,(s2m,t2m)}, such that<br/> | ||
## Thus, new EPR pairs are | ###The classical information of gadget be g(sk)= ({(s1,t1),(s2,t2),...,(s2m,t2m),p,sk}.<br/> | ||
###The quantum state of gadget can be written as {missing math} | |||
# Encrypt (x[j],z[j]), p[j] for all j and sk using pki+1. Resulting Gadget is the classical-quantum (CQ) state,{missing math} | |||
## The classical information of gadget be g(sk) | |||
## The quantum state of gadget can be written as | |||
# Encrypt (x[j],z[j]), p[j] for all j and sk using | |||
=== Stage 2 Server’s Computation=== | === Stage 2 Server’s Computation=== |