Collapse binding and collapsing were proposed by Unruh (Eurocrypt '16) as post-quantum strengthenings of computational binding and collision resistance, respectively. These notions have been very successful in facilitating the "lifting" of classical security proofs to the quantum setting. A basic and natural question remains unanswered, however: are they the weakest notions that suffice for such lifting? In this work we answer this question in the affirmative by giving a classical commit-and-open protocol which is post-quantum secure if and only if the commitment scheme (resp. hash function) used is collapse binding (resp. collapsing). We also generalise the definition of collapse binding to quantum commitment schemes, and prove that the equivalence carries over when the sender in this commit-and-open protocol communicates quantum information. As a consequence, we establish that a variety of "weak" binding notions (sum binding, CDMS binding and unequivocality) are in fact equivalent to collapse binding, both for post-quantum and quantum commitments. Finally, we prove a "win-win" result, showing that a post-quantum computationally binding commitment scheme that is not collapse binding can be used to build an equivocal commitment scheme (which can, in turn, be used to build one-shot signatures and other useful quantum primitives). This strengthens a result due to Zhandry (Eurocrypt '19) showing that the same object yields quantum lightning.
Collapse binding and collapsing were proposed by Unruh (Eurocrypt '16) as post-quantum strengthenings of computational binding and collision resistance, respectively. These notions have been very successful in facilitating the "lifting" of classical security proofs to the quantum setting. A basic and natural question remains unanswered, however: are they the weakest notions that suffice for such lifting? In this work we answer this question in the affirmative by giving a classical commit-and-open protocol which is post-quantum secure if and only if the commitment scheme (resp. hash function) used is collapse binding (resp. collapsing). We also generalise the definition of collapse binding to quantum commitment schemes, and prove that the equivalence carries over when the sender in this commit-and-open protocol communicates quantum information. As a consequence, we establish that a variety of "weak" binding notions (sum binding, CDMS binding and unequivocality) are in fact equivalent to collapse binding, both for post-quantum and quantum commitments. Finally, we prove a "win-win" result, showing that a post-quantum computationally binding commitment scheme that is not collapse binding can be used to build an equivocal commitment scheme (which can, in turn, be used to build one-shot signatures and other useful quantum primitives). This strengthens a result due to Zhandry (Eurocrypt '19) showing that the same object yields quantum lightning.
Quantum 2-party cryptography differs from its classical counterpart in at least one important way: Given black-box access to a perfect commitment scheme there exists a secure 1-2 quantum oblivious transfer. This reduction proposed by Crépeau and Kilian was proved secure against any receiver by Yao, in the case where perfect commitments are used. However, quantum commitments would normally be based on computational assumptions. A natural question therefore arises: What happens to the security of the above reduction when computationally secure commitments are used instead of perfect ones? In this paper, we address the security of 1-2 QOT when computationally binding string commitments are available. In particular, we analyse the security of a primitive called Quantum Measurement Commitment when it is constructed from unconditionally concealing but computationally binding commitments. As measuring a quantum state induces an irreversible collapse, we describe a QMC as an instance of ``computational collapse of a quantum state''. In a QMC a state appears to be collapsed to a polynomial time observer who cannot extract full information about the state without breaking a computational assumption. We reduce the security of QMC to a weak binding criteria for the string commitment. We also show that secure QMCs implies QOT using a straightforward variant of the reduction above.
Shor’s quantum factoring algorithm and a few other efficient quantum algorithms break many clas-sical crypto-systems. In response, people proposed post-quantum cryptography based on computational problems that are believed hard even for quantum computers. However, security of these schemes against quantum attacks is elusive. This is because existing security analysis (almost) only deals with classical attackers and arguing security in the presence of quantum adversaries is challenging due to unique quan-tum features such as no-cloning. This work proposes a general framework to study which classical security proofs can be restored in the quantum setting. Basically, we split a security proof into (a sequence of) classical security reductions, and investigate what security reductions are “quantum-friendly”. We characterize sufficient conditions such that a classical reduction can be “lifted ” to the quantum setting. We then apply our lifting theorems to post-quantum signature schemes. We are able to show that the classical generic construction of hash-tree based signatures from one-way functions and and a more efficient variant proposed in [BDH11] carry over to the quantum setting. Namely, assuming existence of (classical) one-way functions that are resistant to efficient quantum inversion algorithms, there exists a quantum-secure signature scheme. We note that the scheme in [BDH11] is a promising (post-quantum) candidate to be implemented in practice and our result further justifies it. Actually, to obtain these results, we formalize a simple criteria, which is motivated by many classical proofs in the literature and is straightforward to check. This makes our lifting theorem easier to apply, and it should be useful elsewhere to prove quantum security of proposed post-quantum cryptographic schemes. Finally we demonstrate the generality of our framework by showing that several existing works (Full-Domain hash in the quantum random-oracle model [Zha12b] and the simple hybrid arguments framework in [HSS11]) can be reformulated under our unified framework. 1
Bit commitment is one of the cornerstones of both classical and quantum cryptography. In the classical world, security analysis is straightforward because both hiding and binding each admit a single natural interpretation. In the quantum world, however, security notions can be defined in different ways that are not necessarily equivalent. Over three decades, several works have introduced numerous binding notions whose precise statements and mu- tual implications remain scattered across the literature. This thesis clarifies that ambiguity. We study quantum binding notions within the framework of canonical purified quantum bit commitment (QBC) schemes, a class that captures both the theoretical generality and the practical mechanisms of recent constructions. In the statistical setting, we examine six binding notions: strong-binding, honest-binding, sum-binding, collapse-binding, extractor- binding, and arbitrary-binding. We prove that all these definitions except strong-binding are equivalent for the class of canonical purified schemes. In the computational setting, we establish that honest-binding, sum-binding, and collapse-binding are equivalent for the same class of schemes. Moreover, we introduce a quantitative framework based on the work of Ref. [26] to quantify the tradeoff between hiding and binding properties, and we use it to provide a geometric proof of the impossibility theorem of Mayers [21] and Lo & Chau [20], for canonical purified schemes. We next revisit the quantum notion of hiding, which is far less ambiguous than binding. Finally, we survey some technical tools that can be used to construct secure QBC schemes.
Classically, it is well-known that several fundamental cryptographic primitives, including one-way functions, pseudorandom number generators, commitments, and signatures, characterize the same cryptographic world, which is known as "Minicrypt". In this paper, we investigate to what extent this picture persists in the quantum-computation classical-communication (QCCC) setting, where parties may perform quantum local computation, but communicate through only classical messages.
We demonstrate that the classical Minicrypt landscape breaks down in the QCCC model by oracle separations. We construct two oracle worlds where efficiently verifiable one-way puzzles (EV-OWPuzz) exist. By a known equivalence, QCCC one-time signatures also exist in both worlds. In the first, one-way functions do not exist, even if we allow quantum pseudodeterministic evaluation. In the second, QCCC bit commitments do not exist. Thus, the classical Minicrypt reductions from signatures to one-way functions and bit commitments have no fully black-box counterparts in the QCCC setting. Our proofs develop techniques for analyzing the random phase states, including a concentration theorem and an LOCC decoupling theorem.
As a separate result, we establish quantum sum-binding for the NOVY protocol, settling the question of its post-quantum security. We also generalize it to a fully black-box construction of a QCCC commitment from EV-OWPuzz with nearly uniform puzzle distribution, complementing our oracle separation results.
Shor's quantum factoring algorithm and a few other efficient quantum algorithms break many classical crypto-systems. In response, people proposed post-quantum cryptography based on computational problems that are believed hard even for quantum computers. However, security of these schemes against \emph{quantum} attacks is elusive. This is because existing security analysis (almost) only deals with classical attackers and arguing security in the presence of quantum adversaries is challenging due to unique quantum features such as no-cloning. This work proposes a general framework to study which classical security proofs can be restored in the quantum setting. Basically, we split a security proof into (a sequence of) classical security reductions, and investigate what security reductions are "quantum-friendly". We characterize sufficient conditions such that a classical reduction can be "lifted" to the quantum setting. We then apply our lifting theorems to post-quantum signature schemes. We are able to show that the classical generic construction of hash-tree based signatures from one-way functions and and a more efficient variant proposed in~\cite{BDH11} carry over to the quantum setting. Namely, assuming existence of (classical) one-way functions that are resistant to efficient quantum inversion algorithms, there exists a quantum-secure signature scheme. We note that the scheme in~\cite{BDH11} is a promising (post-quantum) candidate to be implemented in practice and our result further justifies it. Finally we demonstrate the generality of our framework by showing that several existing works (Full-Domain hash in the quantum random-oracle model~\cite{Zha12ibe} and the simple hybrid arguments framework in~\cite{HSS11}) can be reformulated under our unified framework.
We analyze the situation where computationally binding string commitment schemes are used to force the receiver of a BB84 encoding of a classical bitstring to measure upon reception. Since measuring induces an irreversible collapse to the received quantum state, even given extra information after the measurement does not allow the receiver to evaluate reliably some predicates apply to the classical bits encoded in the state. This fundamental quantum primitive is called quantum measure commitment (QMC) and allows for secure two-party computation of classical functions. An adversary to QMC is one that can both provide valid proof of having measured the received states while still able to evaluate a predicate applied to the classical content of the encoding. We give the first quantum black-box reduction for the security of QMC to the binding property of the string commitment. We characterize a class of quantum adversaries against QMC that can be transformed into adversaries against a weak form for the binding property of the string commitment. Our result provides a construction for 1--2-oblivious transfer that is computationally secure against the receiver and unconditionally secure against the sender from any string commitment scheme satisfying a weak binding property.