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Classical Fully Homomorphic Encryption for Quantum Circuits: Difference between revisions
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Classical Fully Homomorphic Encryption for Quantum Circuits (edit)
Revision as of 16:31, 19 November 2018
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== Outline== | == Outline== | ||
FHE presents a classical protocol with the help of which a completely classical Client could assign Server a quantum computation for her encrypted (hidden) input/output. Similar to any classical HE this scheme is divided into four steps: Key Generation generates keys for encryption, decryption and evaluation of the circuit; Encryption encodes the input into a secret text using the encryption key generated during Key Generation; Evaluation performs operations (implements the circuit) on the encrypted input using evaluation key generated and Decryption transforms result of the evaluation step hidden in the secret text, to outcome of the circuit for Client's input using decryption key. Following the stages of [[Secure Delegated Quantum Computation]], in preparation stage, Client encrypts her input to hide it from the Server performing [[Supplementary Information#Quantum Cryptography Techniques#Quantum One Time Pad|Quantum One Time Pad]] (QOTP) using her encryption key, who, in the computation stage, performs quantum computation by a completely classical evaluation step. There are two kinds of gates in Quantum Computation (See Heirarchy of Quantum Gates in [[Supplementary Information]]) Clifford Gates, which consists of Hadamard gate, CNOT and Pauli gates (X, Y, Z) and Toffoli gates (any single qubit phase/rotation gate). A universal scheme can perform both these types of gates implying that it can perform any quantum operation. Now, applying [[Supplementary Information#A General Introduction to Quantum Information#Heirarchy of Quantum Gates|Clifford gates]] remains a simple step as it leaves the state with only Pauli corrections (X, Z) which are easy to handle as these gates commute with every quantum gate and hence can be shifted and cancelled out by applying corresponding inverse gate later by the Client, but when applying [[Supplementary Information#A General Introduction to Quantum Information#Heirarchy of Quantum Gates|Toffoli Gates]], it leaves the state with some Pauli corrections and Clifford gate corrections depending on the one pad key used for encryption key used by Client. Decryption key cannot deal with Clifford gate errors as they do not commute with all quantum operations and hence it needs to be corrected by applying corresponding inverse gate before the operation of next gate for computation by the Server. These Clifford gate corrections are a combination of CNOT corrections dependent on encryption key and a Hadamard correction independent of encryption key. Applying Hadamard requires no extra information but CNOT gate errors require revelation of the encryption key. FHE deals with this problem via [[Supplementary Information#Quantum Cryptography Techniques#Encrypted CNOT operation|Encrypted CNOT operation]] using [[Supplementary Information#Quantum Cryptography Techniques#Trapdoor Claw-Free Function|TCF]] | FHE presents a classical protocol with the help of which a completely classical Client could assign Server a quantum computation for her encrypted (hidden) input/output. Similar to any classical HE this scheme is divided into four steps: Key Generation generates keys for encryption, decryption and evaluation of the circuit; Encryption encodes the input into a secret text using the encryption key generated during Key Generation; Evaluation performs operations (implements the circuit) on the encrypted input using evaluation key generated and Decryption transforms result of the evaluation step hidden in the secret text, to outcome of the circuit for Client's input using decryption key. Following the stages of [[Secure Delegated Quantum Computation]], in preparation stage, Client encrypts her input to hide it from the Server performing [[Supplementary Information#Quantum Cryptography Techniques#Quantum One Time Pad|Quantum One Time Pad]] (QOTP) using her encryption key, who, in the computation stage, performs quantum computation by a completely classical evaluation step. There are two kinds of gates in Quantum Computation (See Heirarchy of Quantum Gates in [[Supplementary Information]]) Clifford Gates, which consists of Hadamard gate, CNOT and Pauli gates (X, Y, Z) and Toffoli gates (any single qubit phase/rotation gate). A universal scheme can perform both these types of gates implying that it can perform any quantum operation. Now, applying [[Supplementary Information#A General Introduction to Quantum Information#Heirarchy of Quantum Gates|Clifford gates]] remains a simple step as it leaves the state with only Pauli corrections (X, Z) which are easy to handle as these gates commute with every quantum gate and hence can be shifted and cancelled out by applying corresponding inverse gate later by the Client, but when applying [[Supplementary Information#A General Introduction to Quantum Information#Heirarchy of Quantum Gates|Toffoli Gates]], it leaves the state with some Pauli corrections and Clifford gate corrections depending on the one pad key used for encryption key used by Client. Decryption key cannot deal with Clifford gate errors as they do not commute with all quantum operations and hence it needs to be corrected by applying corresponding inverse gate before the operation of next gate for computation by the Server. These Clifford gate corrections are a combination of CNOT corrections dependent on encryption key and a Hadamard correction independent of encryption key. Applying Hadamard requires no extra information but CNOT gate errors require revelation of the encryption key. FHE deals with this problem via [[Supplementary Information#Quantum Cryptography Techniques#Encrypted CNOT operation|Encrypted CNOT operation]] using [[Supplementary Information#Quantum Cryptography Techniques#Trapdoor Claw-Free Function (TCF)|Trapdoor Claw-Free Function (TCF)]] without revelation of encryption key to the Server. Finally, in the Output Correction stage, Client gets her inputs and updated encryption keys to get the correct final outcome from the secret text using her decryption key. Following is an outline of the steps to illustrate the above mentioned scheme, assuming depth of circuit (see notations used) equal to L. | ||
* '''Key Generation:''' Client generates L+1 classical homomorphic key sets consisting of public key, evaluation key, secret key, trapdoor information using HE.KeyGen (classical HE step). Evaluation key consists of first L pairs of secret key-trapdoor information encrypted with last L public keys such that secret key-trapdoor key pair and public key do not belong to the same key set. Evaluation key also contains this public key used to encrypt the pair. | * '''Key Generation:''' Client generates L+1 classical homomorphic key sets consisting of public key, evaluation key, secret key, trapdoor information using HE.KeyGen (classical HE step). Evaluation key consists of first L pairs of secret key-trapdoor information encrypted with last L public keys such that secret key-trapdoor key pair and public key do not belong to the same key set. Evaluation key also contains this public key used to encrypt the pair. | ||
* '''Encryption Client''' uses one time pad to hide her input and encrypts the pad key using a public key not used to encrypt the trapdoors and secret in the previous step. She then sends the hidden input with encypted pad key and classical evaluation key to the Server. In case of classical input Client uses the public key to encrypt her classical message and send it to the Server over classical channel. | * '''Encryption Client''' uses one time pad to hide her input and encrypts the pad key using a public key not used to encrypt the trapdoors and secret in the previous step. She then sends the hidden input with encypted pad key and classical evaluation key to the Server. In case of classical input Client uses the public key to encrypt her classical message and send it to the Server over classical channel. | ||
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Server repeats the same procedure for each layer and finally sends the updated Pauli encryption and quantum one time padded output of the circuit to Client. Client uses the encrypted pad keys of both qubits related to the CNOT error and converts it to another ciphertext using AltHE. | Server repeats the same procedure for each layer and finally sends the updated Pauli encryption and quantum one time padded output of the circuit to Client. Client uses the encrypted pad keys of both qubits related to the CNOT error and converts it to another ciphertext using AltHE. | ||
* '''Decryption''' The sent Pauli corrections are updated with public key of the last evaluation key used. This is the (L + 1)th public key and hence, Client uses (L + 1)th secret key (which was not included in the evaluation keys) to decrypt the updated encryption of pad key sent by the Server. Thus, she uses the resulting pad key to undo the quantum one time pad on the sent output state. | * '''Decryption''' The sent Pauli corrections are updated with public key of the last evaluation key used. This is the (L + 1)th public key and hence, Client uses (L + 1)th secret key (which was not included in the evaluation keys) to decrypt the updated encryption of pad key sent by the Server. Thus, she uses the resulting pad key to undo the quantum one time pad on the sent output state. | ||
== Notations == | == Notations == | ||
* k, security parameter | * k, security parameter | ||