Nielsen (CRYPTO 2002) demonstrated that constructing a non-committing encryption (NCE) scheme for an unbounded number of messages is impossible in both the standard model and the non-programmable random oracle model, underscoring fundamental barriers to achieving NCE within conventional cryptographic frameworks. However, he circumvented this impossibility in the programmable random oracle model by presenting a construction based on trapdoor functions, exploiting the oracle’s programmability to enable ciphertext equivocation. In this work, we analyze Nielsen’s construction in the quantum random oracle model (QROM), where quantum adversaries can query the oracle in superposition, and examine whether its security remains intact in this significantly stronger setting.
Nielsen (CRYPTO 2002) demonstrated that constructing a non-committing encryption (NCE) scheme for an unbounded number of messages is impossible in both the standard model and the non-programmable random oracle model, underscoring fundamental barriers to achieving NCE within conventional cryptographic frameworks. However, he circumvented this impossibility in the programmable random oracle model by presenting a construction based on trapdoor functions, exploiting the oracle’s programmability to enable ciphertext equivocation. In this work, we analyze Nielsen’s construction in the quantum random oracle model (QROM), where quantum adversaries can query the oracle in superposition, and examine whether its security remains intact in this significantly stronger setting.
Recent developments in quantum computers pose an imminent threat to cryptography since many widely used cryptographic schemes will no longer be secure. This worrisome scenario motivated a growing interest in the field of post-quantum cryptography, which aims to design cryptosystems that are postquantum secure, i.e. secure against quantum-capable adversaries. While finding post-quantum computational hardness assumptions is pivotal to achieve this goal, an issue that is often overlooked is provable security. In fact, to prove that the post-quantum computational hardness necessarily implies post-quantum security of the derived cryptosystem, the reduction argument that establishes this implication also needs to be post-quantum. As such, many new methods have been devised in the last few years to handle this adaptation to the quantum setting. In particular, the quantum analogue of the Random Oracle Model has garnered special interest, as it is the foundation of many cryptographic schemes, from public-key encryption to digital signatures. It is also very problematic and usually the main culprit of the failure of security proofs when transitioning to the quantum setting. This work offers a comprehensive exposition on the most influential techniques developed in the last decade to enable or facilitate the construction of security proofs in the Quantum Random Oracle Model (QROM). Each presented technique offers an unique purpose and a solution to a problem provoked by the QROM, and, as such, as a whole, this works allows for an understanding of the most common strategies to construct these type of security proofs. This is accomplished by presenting and analyzing the security proof of three cryptographic schemes set in this model, which underlines the central role these techniques take in provable quantum security.
In the random oracle model, the parties are given oracle access to a random member of a (typically huge) function family, and are assumed to have unbounded computational power (though they can only make a bounded number of oracle queries). This model provides powerful properties that allow proving the security of many protocols, even such that cannot be proved secure in the standard model (under any hardness assumption). The random oracle model is also used to show that a given cryptographic primitive cannot be used in a black-box way to construct another primitive; in their seminal work, Impagliazzo and Rudich [STOC ’89] showed that in the random function model – when the function family is the set of all functions – it is impossible to construct (secure) key-agreement protocols, yielding that key-agreement cannot be black-box reduced to one-way functions. Their work has a long line of followup works (Simon [EC ’98], Gertner et al. [STOC ’00] and Gennaro et al. [SICOMP ’05], to name a few), showing that given oracle access to a certain type of function family (e.g., the family that “implements ” public-key encryption) is not sufficient for building a given cryptographic primitive (e.g., oblivious transfer). Yet, in the more general sense, the following fundamental question remained open: What is the exact power of the random oracle model, and more specifically, of the random function model? We make progress towards answering the above question, showing that any (no private input) semi-honest two-party functionality that can be securely implemented in the random function model, can be securely implemented information theoretically (where parties are assumed to be all powerful, and no oracle is given). We further generalize the above result to function families that provide some natural combinatorial property. To exhibit the power of our result, we use the recent information theoretic impossibility result of McGregor et al. [FOCS ’10], to show the existence of functionalities (e.g., inner product) that cannot be computed both accurately and in a differentially private manner in the random function model; yielding that protocols for computing these functionalities cannot be black-box reduced to the existence of one-way functions.
One of the key distinctions between classical and quantum information is given by the no-cloning theorem: unlike bits, arbitrary qubits cannot be perfectly copied. This fact has been the inspiration for many quantum cryptographic protocols. In this thesis, we introduce a new cryptographic functionality called uncloneable encryption. This functionality allows the encryption of a classical message such that two collaborating but non-communicating adversaries may not both simultaneously recover the message, even when the encryption key is revealed. We achieve this functionality by using Wiesner’s conjugate coding scheme to encrypt the message. We show that the adversaries cannot both obtain all the necessary information for the correct decryption with high probability. Quantum-secure pseudorandom functions, modelled as random oracles, are then used to ensure that any partial information that the adversaries obtain does not give them an advantage in recovering the message.
In a recent breakthrough, Yamakawa and Zhandry (J. ACM 2024) constructed a proof of quantumness in the quantum random oracle model (QROM) in which the quantum prover samples a codeword preimage of a publicly computable function H. They conjectured that given any H, a successful prover must sample their preimage from a high-entropy distribution over possible answers. If true, this would give a certifiable randomness protocol in the quantum random oracle model. As partial evidence for their conjecture, Yamakawa and Zhandry proved the security of their certifiable randomness protocol assuming the Aaronson-Ambainis conjecture.
We prove the security of the certifiable randomness protocol of Yamakawa-Zhandry unconditionally, without relying on the unproven Aaronson-Ambainis conjecture, against low query-depth quantum adversaries: specifically, adversaries that make up to o(\log λ) adaptive quantum queries to the random oracle.
The Fiat-Shamir transformation is a famous technique to turn identification schemes into signature schemes. The derived scheme is provably secure in the random-oracle model against classical adversaries. Still, the technique has also been suggested to be used in connection with quantum-immune identification schemes, in order to get quantum-immune signature schemes. However, a recent paper by Boneh et al. (Asiacrypt 2011) has raised the issue that results in the random-oracle model may not be immediately applicable to quantum adversaries, because such adversaries should be allowed to query the random oracle in superposition. It has been unclear if the Fiat-Shamir technique is still secure in this quantum oracle model (QROM). Here, we discuss that giving proofs for the Fiat-Shamir transformation in the QROM is presumably hard. We show that there cannot be black-box extractors, as long as the underlying quantum-immune identification scheme is secure against active adversaries and the first message of the prover is independent of its witness. Most schemes are of this type. We then discuss that for some schemes one may be able to resurrect the Fiat-Shamir result in the QROM by modifying the underlying protocol first. We discuss in particular a version of the Lyubashevsky scheme which is provably secure in the QROM.
A signature scheme is a fundamental component in modern digital communication. It allows for authenticated messages, without which it would be nearly impossible to ensure security when using most modern technologies. However, there is a growing threat to this fundamental piece of electronic infrastructure. Universal quantum computers, which were originally envisioned by Richard Feynman, have moved from being a theoretical future technology into one that could realistically be available in a matter of decades. In 1994, Peter Shor devised an algorithm that would run on a quantum computer that could be used to solve mathematical problems that formed the foundation of public-key cryptography. \n \nWhile Shor's algorithm clearly establishes that new mathematical problems must be found and studied that can admit efficient cryptographic protocols, it is equally important that the models in which we consider security are also updated to consider the possibility of a malicious adversary having a quantum computer. \n \nIn the random-oracle model, a hash function is replaced by a truly random function that any relevant party is able to query. This model can enable security reductions where otherwise none are known. However, it has been noted that this model does not properly consider the possibility of a quantum computer. For this, we must instead consider the quantum random-oracle model. \n \nIn this thesis, we explain the basics of quantum physics and quantum computation in order to give a complete motivation for the quantum random-oracle model. We explain many of the difficulties that may be encountered in the quantum random-oracle model, and how some of these problems may be solved. We then show prove three signature schemes secure in the quantum random-oracle model: the LMS hash-based scheme, TESLA, a lattice-based scheme, and the TOO transformation using chameleon hashes. The first two schemes are strong candidates for post-quantum standardization.