Ivan Damgård, Jakob Funder, Jesper Buus Nielsen, Louis Salvail
Attacks on classical cryptographic protocols are usually modeled by allowing an adversary to ask queries from an oracle. Security is then defined by requiring that as long as the queries satisfy some constraint, there is some problem the adversary cannot solve, such as compute a certain piece of information. In this paper, we introduce a fundamentally new model of quantum attacks on classical cryptographic protocols, where the adversary is allowed to ask several classical queries in quantum superposition. This is a strictly stronger attack than the standard one, and we consider the security of several primitives in this model. We show that a secret-sharing scheme that is secure with threshold $t$ in the standard model is secure against superposition attacks if and only if the threshold is lowered to $t/2$. We use this result to give zero-knowledge proofs for all of NP in the common reference string model. While our protocol is classical, it is sound against a cheating unbounded quantum prover and computational zero-knowledge even if the verifier is allowed a superposition attack. Finally, we consider multiparty computation and show that for the most general type of attack, simulation based security is not possible. However, putting a natural constraint on the adversary, we show a non-trivial example of a protocol that can indeed be simulated.
The SOLID concept is to develop small solid-state hybrid systems with 3-8 qubits capable of performing elementary processing and communication of quantum information. This involves design, fabrication and investigation of combinations of qubits, oscillators, cavities, and transmission lines, creating hybrid devices interfacing different types of qubits for quantum data storage, qubit interconversion, and communication. The agenda is to provide proofs of concept, to identify roadblocks, and to stake out roads that seem particularly promising, including practical applications of quantum technologies.
We motivate, define and construct quantum proofs of knowledge, proofs of knowledge secure against quantum adversaries. Our constructions are based on a new quantum rewinding technique that allows us to extract witnesses in many classical proofs of knowledge. We give criteria under which a classical proof of knowledge is a quantum proof of knowledge. Combining our results with Watrous’ results on quantum zeroknowledge, we show that there are zero-knowledge quantum proofs of knowledge for all languages in NP.
In this paper, we prove classical coin-flipping secure in the presence of quantum adversaries. The proof uses a recent result of Watrous [Wat09] that allows quantum rewinding for protocols of a certain form. We then discuss two applications. First, the combination of coin-flipping with any non-interactive zero-knowledge protocol leads to an easy transformation from non-interactive zero-knowledge to interactive quantum zero-knowledge. Second, we discuss how our protocol can be applied to a recently proposed method for improving the security of quantum protocols [DFL+09], resulting in an implementation without set-up assumptions. Finally, we sketch how to achieve efficient simulation for an extended construction in the common-reference-string model.
In this paper, we prove classical coin-flipping secure in the presence of quantum adversaries. The proof uses a recent result of Watrous [Wat09] that allows quantum rewinding for protocols of a certain form. We then discuss two applications. First, the combination of coin-flipping with any non-interactive zero-knowledge protocol leads to an easy transformation from non-interactive zero-knowledge to interactive quantum zero-knowledge. Second, we discuss how our protocol can be applied to a recently proposed method for improving the security of quantum protocols [DFL+09], resulting in an implementation without set-up assumptions. Finally, we sketch how to achieve efficient simulation for an extended construction in the common-reference-string model.
We define quantum expanders in a natural way. We give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL(2, q) given by Lubotzky et al. (1988). The second construction is combinatorial, and is based on a quantum variant of the Zig-Zag product introduced by Reingold et al. (2000). Both constructions are of constant degree, and the second one is explicit. Using quantum expanders, we characterize the complexity of comparing and estimating quantum entropies. Specifically, we consider the following task: given two mixed states, each given by a quantum circuit generating it, decide which mixed state has more entropy. We show that this problem is QSZK-complete (where QSZK is the class of languages having a zero-knowledge quantum interactive protocol). This problem is very well motivated from a physical point of view. Our proof resembles the classical proof that the entropy difference problem is SZK-complete, but crucially depends on the use of quantum expanders.
Zero-knowledge proof system is an important protocol that can be used as a basic block for construction of other more complex cryptographic protocols. Quantum zero-knowledge protocols have been proposed but, since their implementation requires advanced quantum technology devices, experimental implementation of zero-knowledge protocols have not being reported. In this work, we present a quantum zero-knowledge protocol based on a quantum bit commitment protocol that can be implemented with today technology. Hence, our quantum zero-knowledge protocol can be readily implemented.
Zero-knowledge proof system is an important protocol that can be used as a basic block for construction of other more complex cryptographic protocols. An intrinsic characteristic of a zero-knowledge systems is the assumption that is impossible for the verifier to show to a third part that he has interacted with the prover. However, it has been shown that using quantum correlations the impossibility of transferring proofs can be successfully attacked. In this work we show two new protocols for proof transference, being the first one based on teleportation and the second one without using entangled states.
Rahul Jain, Alexandra Kolla, Gatis Midrijānis, Ben W. Reichardt
Let L be a language decided by a constant-round quantum Arthur-Merlin (QAM)\nprotocol with negligible soundness error and all but possibly the last message\nbeing classical. We prove that if this protocol is zero knowledge with a\nblack-box, quantum simulator S, then L in BQP. Our result also applies to any\nlanguage having a three-round quantum interactive proof (QIP), with all but\npossibly the last message being classical, with negligible soundness error and\na black-box quantum simulator.\n These results in particular make it unlikely that certain protocols can be\ncomposed in parallel in order to reduce soundness error, while maintaining zero\nknowledge with a black-box quantum simulator. They generalize analogous\nclassical results of Goldreich and Krawczyk (1990).\n Our proof goes via a reduction to quantum black-box search. We show that the\nexistence of a black-box quantum simulator for such protocols when L notin BQP\nwould imply an impossibly-good quantum search algorithm.\n
Watrous had presented the first proof of zero-knowledge property of a proof system against a quantum verifier. The key of the proof is the construction of a quantum simulator. In the construction, the 'failure state' is rotated to the 'success' state by a tricky operation which is initially developped for the amplification of QMA proof systems. This manuscript presents a new and simpler construction of a simulator. In the construction, we simply amplify the success probability of a classical simulator using Grover's amplification.
The cut polytope of a graph arises in many fields. Although much is known about facets of the cut polytope of the complete graph, very little is known for general graphs. The study of Bell inequalities in quantum information science requires knowledge of the facets of the cut polytope of the complete bipartite graph or, more generally, the complete k-partite graph. Lifting is a central tool to prove certain inequalities are facet inducing for the cut polytope. In this paper we introduce a lifting operation, named triangular elimination, applicable to the cut polytope of a wide range of graphs. Triangular elimination is a specific combination of zero-lifting and Fourier-Motzkin elimination using the triangle inequality. We prove sufficient conditions for the triangular elimination of facet inducing inequalities to be facet inducing. The proof is based on a variation of the lifting lemma adapted to general graphs. The result can be used to derive facet inducing inequalities of the cut polytope of various graphs from those of the complete graph. We also investigate the symmetry of facet inducing inequalities of the cut polytope of the complete bipartite graph derived by triangular elimination.
This paper proves that several interactive proof systems are zero-knowledge against general quantum attacks. This includes the well-known Goldreich–Micali–Wigderson classical zero-knowledge protocols for graph isomorphism and graph 3-coloring (assuming the existence of quantum computationally concealing commitment schemes in the second case). Also included is a quantum interactive proof system for a complete problem for the complexity class of problems having honest verifier quantum statistical zero-knowledge proofs, which therefore establishes that honest verifier and general quantum statistical zero-knowledge are equal: $\mathrm{QSZK}= \mathrm{QSZK}_{\mathrm{HV}}$. Previously no nontrivial interactive proof systems were known to be zero-knowledge against quantum attacks, except in restricted settings such as the honest verifier and common reference string models. This paper therefore establishes for the first time that true zero-knowledge is indeed possible in the presence of quantum information and computation.
We study two group theoretic problems, GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP, in the setting of black-box groups, where DOUBLE COSET MEMBERSHIP generalizes a set of problems, including GROUP MEMBERSHIP, GROUP FACTORIZATION, and COSET INTERSECTION. No polynomial-time classical algorithms are known for these problems. We show that for solvable groups, there exist efficient quantum algorithms for GROUP INTERSECTION if one of the underlying solvable groups has a smoothly solvable commutator subgroup, and for DOUBLE COSET MEMBERSHIP if one of the underlying solvable groups is smoothly solvable. We also study the decision versions of STABILIZER and ORBIT COSET, which generalizes GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP, respectively. We show that they reduce to ORBIT COSET under certain conditions. Finally, we show that DOUBLE COSET MEMBERSHIP and DOUBLE COSET NONMEMBERSHIP have zero knowledge proof systems.
In this paper we propose a definition for (honest verifier) quantum statistical zero-knowledge interactive proof systems and study the resulting complexity class, which we denote QSZK/sub HV/. We prove several facts regarding this class, including: the following problem is a complete promise problem for QSZKHV: given instructions for preparing two mixed quantum states, are the states close together or far apart in the trace norm metric? This problem is a quantum generalization of the complete promise problem of Sahai and Vadhan (1997) for (classical) statistical zero-knowledge; QSZK/sub HV/ is closed under complement; QSZK/sub HV//spl sube/PSPACE. (At present it is not known if arbitrary quantum interactive proof systems can be simulated in PSPACE even for one-round proof systems); any polynomial-round honest verifier quantum statistical zero-knowledge proof system can be simulated by a two-message (i.e., one-round) honest verifier quantum statistical zero-knowledge proof system. Similarly, any polynomial-round honest verifier quantum statistical zero-knowledge proof system can be simulated by a three-message public-coin honest verifier quantum statistical zero-knowledge proof system. These facts establish close connections between classical statistical zero-knowledge and our definition for quantum statistical zero-knowledge, and give some insight regarding the effect of this zero-knowledge restriction on quantum interactive proof systems. The relationship between our definition and possible definitions of general (i.e., not necessarily honest) quantum statistical zero-knowledge are also discussed.
In this paper we propose a definition for (honest verifier) quantum statistical zero-knowledge interactive proof systems and study the resulting complexity class, which we denote QSZK. We prove several facts regarding this class that establish close connections between classical statistical zero-knowledge and our definition for quantum statistical zero-knowledge, and give some insight regarding the effect of this zero-knowledge restriction on quantum interactive proof systems.
There had been well known claims of unconditionally secure quantum protocols for bit commitment. However, we, and independently Mayers, showed that all proposed quantum bit commitment schemes are, in principle, insecure because the sender, Alice, can almost always cheat successfully by using an Einstein-Podolsky-Rosen (EPR) type of attack and delaying her measurements. One might wonder if secure quantum bit commitment protocols exist at all. We answer this question by showing that the same type of attack by Alice will, in principle, break any bit commitment scheme. The cheating strategy generally requires a quantum computer. We emphasize the generality of this ``no-go theorem'': Unconditionally secure bit commitment schemes based on quantum mechanics---fully quantum, classical or quantum but with measurements---are all ruled out by this result. Since bit commitment is a useful primitive for building up more sophisticated protocols such as zero-knowledge proofs, our results cast very serious doubt on the security of quantum cryptography in the so-called ``post-cold-war'' applications. We also show that ideal quantum coin tossing is impossible because of the EPR attack. This no-go theorem for ideal quantum coin tossing may help to shed some lights on the possibility of non-ideal protocols.