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We show the following generic result. Whenever a quantum query algorithm in the quantum random-oracle model outputs a classical value $t$ that is promised to be in some tight relation with $H(x)$ for some $x$, then $x$ can be efficiently extracted with almost certainty. The extraction is by means of a suitable simulation of the random oracle and works online, meaning that it is straightline, i.e., without rewinding, and on-the-fly, i.e., during the protocol execution and without disturbing it. The technical core of our result is a new commutator bound that bounds the operator norm of the commutator of the unitary operator that describes the evolution of the compressed oracle (which is used to simulate the random oracle above) and of the measurement that extracts $x$. We show two applications of our generic online extractability result. We show tight online extractability of commit-and-open $Sigma$-protocols in the quantum setting, and we offer the first non-asymptotic post-quantum security proof of the textbook Fujisaki-Okamoto transformation, i.e, without adjustments to facilitate the proof.
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The famous Fiat-Shamir transformation turns any public-coin three-round interactive proof, i.e., any so-called sigma-protocol, into a non-interactive proof in the random-oracle model. We study this transformation in the setting of a quantum adversary that in particular may query the random oracle in quantum superposition. Our main result is a generic reduction that transforms any quantum dishonest prover attacking the Fiat-Shamir transformation in the quantum random-oracle model into a similarly successful quantum dishonest prover attacking the underlying sigma-protocol (in the standard model). Applied to the standard soundness and proof-of-knowledge definitions, our reduction implies that both these security properties, in both the computational and the statistical variant, are preserved under the Fiat-Shamir transformation even when allowing quantum attacks. Our result improves and completes the partial results that have been known so far, but it also proves wrong certain claims made in the literature. In the context of post-quantum secure signature schemes, our results imply that for any sigma-protocol that is a proof-of-knowledge against quantum dishonest provers (and that satisfies some additional natural properties), the corresponding Fiat-Shamir signature scheme is secure in the quantum random-oracle model. For example, we can conclude that the non-optimized version of Fish, which is the bare Fiat-Shamir variant of the NIST candidate Picnic, is secure in the quantum random-oracle model.
Weakened random oracle models (WROMs) are variants of the random oracle model (ROM). The WROMs have the random oracle and the additional oracle which breaks some property of a hash function. Analyzing the security of cryptographic schemes in WROMs, we can specify the property of a hash function on which the security of cryptographic schemes depends. Liskov (SAC 2006) proposed WROMs and later Numayama et al. (PKC 2008) formalized them as CT-ROM, SPT-ROM, and FPT-ROM. In each model, there is the additional oracle to break collision resistance, second preimage resistance, preimage resistance respectively. Tan and Wong (ACISP 2012) proposed the generalized FPT-ROM (GFPT-ROM) which intended to capture the chosen prefix collision attack suggested by Stevens et al. (EUROCRYPT 2007). In this paper, in order to analyze the security of cryptographic schemes more precisely, we formalize GFPT-ROM and propose additional three WROMs which capture the chosen prefix collision attack and its variants. In particular, we focus on signature schemes such as RSA-FDH, its variants, and DSA, in order to understand essential roles of WROMs in their security proofs.
Solving strategic games with huge action space is a critical yet under-explored topic in economics, operations research and artificial intelligence. This paper proposes new learning algorithms for solving two-player zero-sum normal-form games where the number of pure strategies is prohibitively large. Specifically, we combine no-regret analysis from online learning with Double Oracle (DO) methods from game theory. Our method -- emph{Online Double Oracle (ODO)} -- is provably convergent to a Nash equilibrium (NE). Most importantly, unlike normal DO methods, ODO is emph{rationale} in the sense that each agent in ODO can exploit strategic adversary with a regret bound of $mathcal{O}(sqrt{T k log(k)})$ where $k$ is not the total number of pure strategies, but rather the size of emph{effective strategy set} that is linearly dependent on the support size of the NE. On tens of different real-world games, ODO outperforms DO, PSRO methods, and no-regret algorithms such as Multiplicative Weight Update by a significant margin, both in terms of convergence rate to a NE and average payoff against strategic adversaries.
We study the query complexity of quantum learning problems in which the oracles form a group $G$ of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a $t$-query quantum algorithm in terms of group characters. As an application, we show that $Omega(n)$ queries are required to identify a random permutation in $S_n$. More generally, suppose $H$ is a fixed subgroup of the group $G$ of oracles, and given access to an oracle sampled uniformly from $G$, we want to learn which coset of $H$ the oracle belongs to. We call this problem coset identification and it generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation. We provide character-theoretic formulas for the optimal success probability achieved by a $t$-query algorithm for this problem. One application involves the Heisenberg group and provides a family of problems depending on $n$ which require $n+1$ queries classically and only $1$ query quantumly.
In the universal blind quantum computation problem, a client wants to make use of a single quantum server to evaluate $C|0rangle$ where $C$ is an arbitrary quantum circuit while keeping $C$ secret. The clients goal is to use as few resources as possible. This problem, first raised by Broadbent, Fitzsimons and Kashefi [FOCS09, arXiv:0807.4154], has become fundamental to the study of quantum cryptography, not only because of its own importance, but also because it provides a testbed for new techniques that can be later applied to related problems (for example, quantum computation verification). Known protocols on this problem are mainly either information-theoretically (IT) secure or based on trapdoor assumptions (public key encryptions). In this paper we study how the availability of symmetric-key primitives, modeled by a random oracle, changes the complexity of universal blind quantum computation. We give a new universal blind quantum computation protocol. Similar to previous works on IT-secure protocols (for example, BFK [FOCS09, arXiv:0807.4154]), our protocol can be divided into two phases. In the first phase the client prepares some quantum gadgets with relatively simple quantum gates and sends them to the server, and in the second phase the client is entirely classical -- it does not even need quantum storage. Crucially, the protocols first phase is succinct, that is, its complexity is independent of the circuit size. Given the security parameter $kappa$, its complexity is only a fixed polynomial of $kappa$, and can be used to evaluate any circuit (or several circuits) of size up to a subexponential of $kappa$. In contrast, known schemes either require the client to perform quantum computations that scale with the size of the circuit [FOCS09, arXiv:0807.4154], or require trapdoor assumptions [Mahadev, FOCS18, arXiv:1708.02130].
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