Classical Fully Homomorphic Encryption for Quantum Circuits: Difference between revisions

Further, the computation stage incorporates,

* '''Circuit Evaluation:''' Server starts with the classical one time padded states from the Client and generates the required quantum states. For each gate of the circuit that Server applies, he updates the encrypted Pauli encryption according to rules given in Pseudo code below. In case of Toffoli gate operation, an additional step is incorporated where he corrects the extra Clifford gate error performing encrypted CNOT operation and then Hadamard operation on the target qubit. This step uses evaluation key and can be explained as follows.</br>

#'''TCF:''' This operation uses Trapdoor Claw Free function pairs which have the same image (output) for different pre-images(inputs) called 'random claw pair'. Given the image, it is rendered a hard problem to find this corresponding random claw without its trapdoor information (example, a piece of information required to invert the function). For this protocol, the HE Encryption function (HE.Enc()) is taken as one of the functions. A second function whose distribution is shifted from the previous function by a natural (homomorphic) XOR operation (a requirement for the [[Glossary#Quantum Capable Homomorphic Encryption|classical HE]] scheme used) of encrypted key bit used for that encryption function. This means, the functions have a common range such that for every image (output), the pre-images (input) for each of the functions stated above would also differ by a XOR operation of actual (not encrypted) key bit. Thus, any element in the said range set would have one pre-image in the domain set of each function, together called random claw pair. If one performs a XOR operation on the pair, the result is pad key bit. This is implied from the properties of homomorphic XOR. In simple words, the above paragraph implies that if two functions are separated by encrypted pad key via a homomorphic XOR operation, their inputs for a common output (random claw pair) would be separated by the (not encrypted) pad key bit. Thus, any pre-image pair (random claw) thus, obtained, hides the pad key (to be used later for Encrypted CNOT operation).

#'''Server's preparation''' Thus, Server creates a superposition of inputs for the functions over some distribution. Next, he creates a superposition of quantum states generated from Client's input. After applying the gates on qubits, for correction of CNOT errors, Server creates three registers. First has the superposition of quantum states generated from Client's input, second has the superposition on a distribution chosen for inputs of the function while third register has the output of one of the two functions illustrated above, where the function (one of the two) is chosen according to the first qubit of the first register and its quantum input is taken from the second register. Hence, these registers are entangled. Server, now measures the third register which reduces second register to a random claw pair as discussed before, hiding the pad key. It is still hidden from the Server as he does not know trapdoor information to be able to know the random claw pair and he cannot compute it from the measured output as it is a hard problem.