This article defines encrypted gate, which is denoted by @math . We present a gate-teleportation-based two-party computation scheme for @math , where one party gives arbitrary quantum state @math as input and obtains the encrypted @math -computing result @math , and the other party obtains the random bits @math . Based on @math , we propose a method to remove the @math -error generated in the homomorphic evaluation of @math -gate. Using this method, we design two non-interactive and perfectly secure QHE schemes named GT and VGT . Both of them are @math -homomorphic and quasi-compact (the decryption complexity depends on the @math -gate complexity). Assume @math -homomorphism, non-interaction and perfect security are necessary property, the quasi-compactness is proved to be bounded by @math , where @math is the total number of @math -gates in the evaluated circuit. VGT is proved to be optimal and has @math -quasi-compactness. According to our QHE schemes, the decryption would be inefficient if the evaluated circuit contains exponential number of @math -gates. Thus our schemes are suitable for homomorphic evaluation of any quantum circuit with low @math -gate complexity, such as any polynomial-size quantum circuit or any quantum circuit with polynomial number of @math -gates.
@cite_15 prove a no-go result: if interaction is not allowed, there does not exist QFHE scheme with perfect security. An enhanced no-go result has been proved independently by Newman and Shi @cite_18 and Lai and Chung @cite_27 : if interaction is not allowed, there does not exist ITS QFHE scheme. This article focuses on the non-interactive and perfectly secure QHE schemes. Our schemes are quasi-compact, so they are not QFHE schemes, and then our result does not contradict with those no-go results. Though our schemes are not QFHE schemes, they can implement any unitary quantum circuit homomorphically (maybe inefficiently).
@cite_27 @cite_15 @cite_18
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We show that any number of parties can coherently exchange any one pure quantum state for another, without communication, given prior shared entanglement. Two applications of this fact to the study of multi-prover quantum interactive proof systems are given. First, we prove that there exists a one-round two-prover quantum interactive proof system for which no finite amount of shared entanglement allows the provers to implement an optimal strategy. More specifically, for every fixed input string, there exists a sequence of strategies for the provers, with each strategy requiring more entanglement than the last, for which the probability for the provers to convince the verifier to accept approaches 1. It is not possible, however, for the provers to convince the verifier to accept with certainty with a finite amount of shared entanglement. The second application is a simple proof that multi-prover quantum interactive proofs can be transformed to have near-perfect completeness by the addition of one round of communication. We show that claims of perfect security for keys produced by quantum key exchange (QKE) are limited to privacy and integrity. Unlike a one-time pad, QKE does not necessarily enable Sender and Receiver to pretend later to have established a different key. This result is puzzling in light of Mayers' No-Go theorem showing the impossibility of quantum bit commitment. But even though a simple and intuitive application of Mayers' protocol transformation appears sufficient to provide deniability (else QBC would be possible), we show several reasons why such conclusions are ill-founded. Mayers' transformation arguments, while sound for QBC, are insufficient to establish deniability in QKE. Having shed light on several unadvertised pitfalls, we then provide a candidate deniable QKE protocol. This itself indicates further shortfalls in current proof techniques, including reductions that preserve privacy but fail to preserve deniability. In sum, purchasing undeniability with an off-the-shelf QKE protocol is significantly more expensive and dangerous than the mere optic fiber for which perfect security is advertised. Standard security proofs of quantum-key-distribution (QKD) protocols often rely on symmetry arguments. In this paper, we prove the security of a three-state protocol that does not possess rotational symmetry. The three-state QKD protocol we consider involves three qubit states, where the first two states @math and @math can contribute to key generation, and the third state @math is for channel estimation. This protocol has been proposed and implemented experimentally in some frequency-based QKD systems where the three states can be prepared easily. Thus, by founding on the security of this three-state protocol, we prove that these QKD schemes are, in fact, unconditionally secure against any attacks allowed by quantum mechanics. The main task in our proof is to upper bound the phase error rate of the qubits given the bit error rates observed. Unconditional security can then be proved not only for the ideal case of a single-photon source and perfect detectors, but also for the realistic case of a phase-randomized weak coherent light source and imperfect threshold detectors. Our result in the phase error rate upper bound is independent of the loss in the channel. Also, we compare the three-state protocol with the Bennett-Brassard 1984 (BB84) protocol. For the single-photon source case, our result proves that the BB84 protocol strictly tolerates a higher quantum bit error rate than the three-state protocol, while for the coherent-source case, the BB84 protocol achieves a higher key generation rate and secure distance than the three-state protocol when a decoy-state method is used. We propose a new cryptographic primitive called Delegatable Homomorphic Encryption (DHE). This allows a Trusted Authority to control delegate the evaluation of circuits over encrypted data to untrusted workers evaluators by issuing tokens. This primitive can be both seen as a public-key counterpart to Verifiable Computation, where input generation and output verification are performed by different entities, or as a generalisation of Fully Homomorphic Encryption enabling control over computations on encrypted data. Our primitive comes with a series of extra features: 1) there is a one-time setup procedure for all circuits; 2) senders do not need to be aware of the functions which will be evaluated on the encrypted data, nor do they need to register keys; 3) tokens are independent of senders and receiver; and 4) receivers are able to verify the correctness of computation given short auxiliary information on the input data and the function, independently of the complexity of the computed circuit. We give a modular construction of such a DHE scheme from three components: Fully Homomorphic Encryption (FHE), Functional Encryption (FE), and a (customised) MAC. As a stepping stone, we first define Verifiable Functional Encryption (VFE), and then show how one can build a secure DHE scheme from a VFE and an FHE scheme. We also show how to build the required VFE from a standard FE together with a MAC scheme. All our results hold in the standard model. Finally, we show how one can build a verifiable computation (VC) scheme generically from a DHE. As a corollary, we get the first VC scheme which remains verifiable even if the attacker can observe verification results.