A method of constructing a public key system in Quotient Algebra Partition (QAP)-based homomorphic encryption (HE), by an algebraic structure, QAP, and an arithmetic operation of homomorphic encryption, wherein the method comprises:
- S1:
- encryption:
- a quantum code [n, k, C], which is structurally a QAP and wherein n>k, is chosen by a data receiver at first;
- S11:
- key generation:
- the data receiver generates a public key, Keypub, to encrypt data and a private key, Keypriv, to decrypt data;
- wherein the public key is represented by
- Keypub=(VQen, Genε),
- where Qen is an n-qubit encoding in [n, k, C], V is an n-qubit permutation, Genε is an error generator allowing to randomly provide an error from a modified error set ε composed of a gigantic number of dressed operators Ē=VEV† of errors E in [n, k, C];
- and wherein the private key for decryption is represented by Keypriv=†P†, which is a product of two n-qubit operators, † and P†;
- wherein the public key of encryption, Keypub, is published in public space to transform a plaintext to a ciphertext by anyone;
- and the private key, Keypriv, is retained by the data receiver to decrypt the encrypted ciphertext;
- S12:
- encoding:
- a data provider provides k-qubit plaintext, |x, preparing a blank state |0 and |x to cast into a product state |0⊗|x of n qubits;
- an error generator Genε of Keypub randomly generates an error Ē from ε;
- the data provider encodes the product state |0⊗|x into the n-qubit encoded state |ψen=ĒVQen|0⊗|x, which means when the data provider encrypts a k-qubit basis state sensitive data (which is plaintext), |x, by writing the product state |0⊗|x of n qubits (which is plaintext) from the plaintext |x for the basis state |0 of n−k qubits;
- by a modified encoding VQen provided by the public key Keypub and a modified error Ē generated randomly from Genε of Keypub, acquiring a encoded state ciphertext |ψen by |ψen=ĒVQen|0⊗|x;
- and the data provider sends |ψen to a computation provider;
- S2:
- Computation:
- S21:
- a k-qubit arithmetic operation M is given to be operated on the encrypted state |en;
- the k-qubit arithmetic operation M is written as n-qubit operation =I2n−k⊗M, which is a tensor product of an (n−k)-qubit operation of identity operator I2n−k and a k-qubit operation M;
- the data receiver produces a computational instructions of a homomorphic encryption (HE) operation Uen with a form:
- Uen PQ†enV†=(PW1P1)(P†1W†11W1P1)(P†1P0)(P†0W†12W1P0)(P†0W2),
- where Q†enV†=W1W2 with a qubit permutation W1 and an operator W2 comprising Spinor, CNOTs, Toffolis, SWAPs, Controlled SWAPs, Multi-Control Gate, Pj=0,1 and P are qubit permutations following the nilpotent condition PW1P1=I2n, and the correction operation =12 is written as a product of two operators 1 and 2, which are consisting of Spinor, CNOTs, Toffolis, SWAPs, Controlled SWAPs, Multi-Controlled Gate;
- the data provider sends the encoded computational instructions Uen to the computation provider and the computation provider receives the computational instructions to computes Uen|ψen;
- S3:
- Decryption:
- the computation provider conducts homomorphic encryption computation Uen|ψen and sends the encrypted results to the data receiver;
- the data receiver decrypts the results by applying the private key Keypriv=†P† to the state Uen|ψen, which is written as
- †P†Uen|ψen=|λ⊗M|x;
- with an (n−k)-qubit syndrome state |λ, and then obtains the result M|x.