Prepare-and-Send Quantum Fully Homomorphic Encryption: Difference between revisions

Prepare-and-Send Quantum Fully Homomorphic Encryption (edit)

Revision as of 09:49, 11 July 2019

741 bytes added , 11 July 2019

→‎Stage 1 Client’s Preparation

Shraddha

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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>
 #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 (QFHE.GenGadgetpki+1(ski))'''
+'''Gadget Construction (<math>\text{QFHE.GenGadget}_{pk_{i+1}}(sk_i)</math>)'''
-#	Generate 4m EPR pairs (
+# Generate <math>4m</math> EPR pairs (<math>|\phi\rangle=\frac{1}{\sqrt{2}}(00+11))</math>, <math>\{(a_1,b_1),...,(a_{4m},b_{4m})\}</math>
-#	Choose 2m pairs  using sk
+# Choose <math>2m</math> pairs <math>\epsilon \{a_1, a_2,....,a_{4m}\}</math> using sk
-##	If (sk = 0) then {(a1,a2),(a2,a3),...,(a4m−1,a4m)} ii. If (sk = 1) then {(a1,a3),(a2,a4),...,(a4m−2,a4m)}
+## If <math>(sk=0)</math> then <math>\{(a_1,a_2),(a_2,a_3),...,(a_{4m-1},a_{4m})\}</math>
-#	For j=1 to 2m,
+## If <math>(sk=1)</math> then <math>\{(a_1,a_3),(a_2,a_4),...,(a_{4m-2},a_{4m})\}</math>
-##	Choose p[j]
+# For j=1 to 2m,
-##	Perform Bell Measurement on jth pair with an extra (P†)p operation, get outcomes (x[j],z[j])
+## Choose p[j] <math>\epsilon_R \{0,1\}</math>
-##	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/>
+## Perform Bell Measurement on <math>j^{th}</math> pair with an extra <math>(P^\dagger)^p</math> operation, get outcomes (x[j],z[j])
-###The classical information of gadget be g(sk)= ({(s1,t1),(s2,t2),...,(s2m,t2m),p,sk}.<br/>
+## Thus, new EPR pairs are
-###The quantum state of gadget can be written as {missing math}
+### If <math>(sk=0)</math> then <math>\{(b_1,b_2),(b_2,b_3),...,(b_{4m-1},b_{4m})\}</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}
+### If <math>(sk=1)</math> then <math>\{(b_1,b_3),(b_2,b_4),...,(b_{4m-2},b_{4m})\}</math>
+## Denote the <math>2m</math> entangled pairs be denoted by <math>\{(s_1,t_1),(s_2,t_2),...,(s_{2m},t_{2m})\}</math>, such that
+## The classical information of gadget be g(sk)<math>=(\{(s_1,t_1),(s_2,t_2),...,(s_{2m},t_{2m}),p,sk\}</math>.
+## The quantum state of gadget can be written as <math>\gamma_{x,z}(g(sk))=\pi_{j=1}^mX^{x[i]}Z^{z[i]}(P^\dagger){p[i]}|\phi\rangle\langle\phi|_{s_jt_j}(P^\dagger){p[i]}Z^{z[i]}X^{x[i]}</math>
+# Encrypt (x[j],z[j]), p[j] for all j and sk using <math>pk_{i+1}</math>. Resulting Gadget is the classical-quantum  (CQ) state,
+<math>\Gamma_{pk_{i+1}}(sk_i)=\rho(HE.Enc_{pk_{i+1}}(g(sk))\otimes \frac{1}{2^{2m}}\sum_{x,z\epsilon\{0,1\}^m}\rho(HE.Enc_{pk_{i+1}}(x,z)\otimes \gamma_{x,z}(g(sk))</math>
 === Stage 2 Server’s Computation===