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

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

Revision as of 08:34, 11 July 2019

450 bytes added , 11 July 2019

→‎Stage 1 Client’s Preparation

Shraddha

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 '''Encryption(QFHE.Enc())'''
 *Input: Quantum Input state density matrix (<math>\rho</math>) (say composed of n single qubit states, <math>\sigma</math>)
-*Output: Encrypted pad keys:<math>\{\tilde{a}^{[0]}...\tilde{a}^{[n]}</math>, <math>\tilde{b}^{[i]}...\tilde{b}^{[n]}\}</math>; 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]}}</math>
+*Output: Encrypted pad keys:<math>\{\tilde{a}^{[0]}...\tilde{a}^{[n]}</math>,<math>\tilde{b}^{[i]}...\tilde{b}^{[n]}\}</math>; 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]}}</math>
 #	For i=1 to n
-##	Client chooses pad keys a,b
+## Client chooses pad keys a,b <math>\epsilon_R\{0,1\}</math>
-##	She quantum one time pads the single qubit by applying XaZb on the single qubit state. XaZbσZbXa ← σ
+## She quantum one time pads the single qubit by applying $X^aZ^b$ on the single qubit state. <math>X^aZ^b\sigma Z^bX^a\leftarrow\sigma</math>
-##	She encrypts the pad keys using one bit for each of a and b from the public key string pk0 using HE.Enc. (˜  )=(HE.Enc ),HE.Enc
+## She encrypts the pad keys using one bit for each of a and b from the public key string <math>pk_0</math> using HE.Enc. (<math>\tilde{a}^{[i]},\tilde{b}^{[i]}</math>)=(HE.Enc<math>_{pk_0^{[i]}}(a^{[i]}</math>),HE.Enc<math>_{pk_0^{[i]}}(b^{[i]}))\leftarrow</math> (a,b)
-##	She apprehends encrypted pad keys to the one time padded quantum state to obtain CQ state,
+## She apprehends encrypted pad keys to the one time padded quantum state to obtain CQ state,
-#	Client sends encryptions (˜a[i],˜b[i]) and the quantum one time padded (QOTP) state Xa[1]Zb[1] ⊗ ..... ⊗ Xa[n]Zb[n]ρZb[1]Xa[1] ⊗ ..... ⊗ Xa[n]Zb[n], for all i to the Server with the evaluation keys and public keys.
+<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>
- '''Gadget Construction (QFHE.GenGadgetpki+1(ski))'''
+#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))'''
 #	Generate 4m EPR pairs (
 #	Choose 2m pairs  using sk