This article defines encrypted gate, which is denoted by
$EG[U]:|\alpha\rangle\rightarrow\left((a,b),Enc_{a,b}(U|\alpha\rangle)\right)$.
We present a gate-teleportation-based two-party computation scheme for $EG[U]$,
where one party gives arbitrary quantum state $|\alpha\rangle$ as input and
obtains the encrypted $U$-computing result $Enc_{a,b}(U|\alpha\rangle)$, and
the other party obtains the random bits $a,b$. Based on $EG[P^x](x\in\{0,1\})$,
we propose a method to remove the $P$-error generated in the homomorphic
evaluation of $T/T^\dagger$-gate. Using this method, we design two
non-interactive and perfectly secure QHE schemes named \texttt{GT} and
\texttt{VGT}. Both of them are $\mathcal{F}$-homomorphic and quasi-compact (the
decryption complexity depends on the $T/T^\dagger$-gate complexity).
  Assume $\mathcal{F}$-homomorphism, non-interaction and perfect security are
necessary property, the quasi-compactness is proved to be bounded by $O(M)$,
where $M$ is the total number of $T/T^\dagger$-gates in the evaluated circuit.
\texttt{VGT} is proved to be optimal and has $M$-quasi-compactness. According
to our QHE schemes, the decryption would be inefficient if the evaluated
circuit contains exponential number of $T/T^\dagger$-gates. Thus our schemes
are suitable for homomorphic evaluation of any quantum circuit with low
$T/T^\dagger$-gate complexity, such as any polynomial-size quantum circuit or
any quantum circuit with polynomial number of $T/T^\dagger$-gates.Comment: 32 pages, 11 figure

Liang, Min

English

arXiv

Quantum homomorphic encryption (QHE) is an important cryptographic technology for delegated quantum computation. It enables remote Server performing quantum computation on encrypted quantum data, and the specific algorithm performed by Server is unnecessarily known by Client. Quantum fully homomorphic encryption (QFHE) is a QHE that satisfies both compactness and $\mathcal{F}$-homomorphism, which is homomorphic for any quantum circuits. However, Yu et al.[Phys. Rev. A 90, 050303(2014)] proved a negative result: assume interaction is not allowed, it is impossible to construct perfectly secure QFHE scheme.
So this article focuses on non-interactive and perfectly secure QHE scheme with loosen requirement, specifically quasi-compactness.

This article defines encrypted gate, which is denoted by $EG[U]:|\alpha\rangle\rightarrow\left((a,b),Enc_{a,b}(U|\alpha\rangle)\right)$. We present a gate-teleportation-based two-party computation scheme for $EG[U]$, where one party gives arbitrary quantum state $|\alpha\rangle$ as input and obtains the encrypted $U$-computing result $Enc_{a,b}(U|\alpha\rangle)$, and the other party obtains the random bits $a,b$. Based on $EG[P^x](x\in\{0,1\})$, we propose a method to remove the $P$-error generated in the homomorphic evaluation of $T/T^\dagger$-gate. Using this method, we design two non-interactive and perfectly secure QHE schemes named \texttt{GT} and \texttt{VGT}. Both of them are $\mathcal{F}$-homomorphic and quasi-compact (the decryption complexity depends on the $T/T^\dagger$-gate complexity).

Assume $\mathcal{F}$-homomorphism, non-interaction and perfect security are necessary property, the quasi-compactness is proved to be bounded by $O(M)$, where $M$ is the total number of $T/T^\dagger$-gates in the evaluated circuit. \texttt{VGT} is proved to be optimal and has $M$-quasi-compactness.
According to our QHE schemes, the decryption would be inefficient if the evaluated circuit contains exponential number of $T/T^\dagger$-gates. Thus our schemes are suitable for homomorphic evaluation of any quantum circuit with low $T/T^\dagger$-gate complexity, such as any polynomial-size quantum circuit or any quantum circuit with polynomial number of $T/T^\dagger$-gates

Teleportation-based quantum homomorphic encryption scheme with quasi-compactness and perfect security

Abstract

Similar works

Full text

Available Versions

Cryptology ePrint Archive