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arXiv:2610.04674

Indistinguishability Lifting for Keyed Oracles, Compressed Ideal Cipher, and More Applications

Ritam Bhaumik, Yu-Hsuan Huang

cs.CRquant-phSubmitted Oct 3, 2026• Announced Oct 6, 2026• 0 citations
arXivHTMLPDF

Abstract

Cryptographic security proofs often involve an adversary interacting with a larger, keyed oracle that consists of (potentially exponentially) many independent instances of a smaller, base oracle. However, showing quantum indistinguishability between two such keyed oracles can be tricky, since a single query made by an adversary may involve a superposition that covers all instances of the base oracles simultaneously. In this paper, we establish a generic indistinguishability lifting theorem of the following form: if the two base oracles are indistinguishable under quantum queries, then their corresponding keyed oracles are too, up to an O(q^2) multiplicative loss in distinguishing advantage, where q is the number of queries made by the adversary. Our lifting theorem applies to both statistical and computational settings, and to oracles that are stateful as well. It is also optimal in that it matches the obvious Grover search attack for a certain (contrived) choice of oracles. As an immediate application, we extend Carolan's compressed permutation oracle to an efficiently implementable compressed ideal cipher, and use it to prove preimage resistance of the Davies-Meyer compression function in the quantum ideal cipher model. Thanks to our lifting theorem, the soundness of our compressed ideal cipher reduces to that of Carolan's oracle, and any further improvement on the latter would automatically carry over to the former. As our second application, we give a modular construction that doubles the message length of any quantum-secure strong pseudorandom permutation. Along the way, we show that an existing two-round tweakable Feistel construction is indistinguishable from a random permutation under quantum bidirectional queries. This is done via a dedicated polynomial-method argument, which may be of independent interest.

Related papers

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We propose a generalization of Zhandry's compressed oracle method to random permutations, where an algorithm can query both the permutation and its inverse. We show how to use the resulting oracle simulation to bound the success probability of an algorithm for any predicate on input-output pairs, a key feature of Zhandry's technique that had hitherto resisted attempts at generalization to random permutations. One key technical ingredient is to use strictly monotone factorizations to represent the permutation in the oracle's database. As an application of our framework, we show that the one-round sponge construction is unconditionally preimage resistant in the random permutation model. This proves a conjecture by Unruh.

arXiv:2407.09655•ACM Symposium on Theory of Computing (STOC)•Jun 15, 2025•6 citations
Cryptographic security proofs often involve an adversary interacting with a larger, keyed oracle that consists of (potentially exponentially) many independent instances of a smaller, base oracle. However, showing quantum indistinguishability between two such keyed oracles can be tricky, since a single query made by an adversary may involve a superposition that covers all instances of the base oracles simultaneously.
In this paper, we establish a generic indistinguishability lifting theorem of the following form: if the two base oracles are indistinguishable under quantum queries, then their corresponding keyed oracles are too, up to an O(q^2) multiplicative loss in distinguishing advantage, where q is the number of queries made by the adversary. Our lifting theorem applies to both statistical and computational settings, and to oracles that are stateful as well. It is also optimal in that it matches the obvious Grover search attack for a certain (contrived) choice of oracles.
As an immediate application, we extend Carolan's compressed permutation oracle to an efficiently implementable compressed ideal cipher, and use it to prove preimage resistance of the Davies-Meyer compression function in the quantum ideal cipher model. Thanks to our lifting theorem, the soundness of our compressed ideal cipher reduces to that of Carolan's oracle, and any further improvement on the latter would automatically carry over to the former.
As our second application, we give a modular construction that doubles the message length of any quantum-secure strong pseudorandom permutation. Along the way, we show that an existing two-round tweakable Feistel construction is indistinguishable from a random permutation under quantum bidirectional queries. This is done via a dedicated polynomial-method argument, which may be of independent interest.
The compressed permutation oracle has been used to analyze the quantum security of a number of cryptographic constructions which resisted prior techniques. However, these analyses were fundamentally limited by the poor soundness of the method: the technique was proven sound only up to O(N1/12)O(N^{1/12})O(N1/12) queries to permutations on NNN elements. We revisit this analysis, improving the soundness bound to a tight Ω(N1/2)Ω(N^{1/2})Ω(N1/2). In addition to being tighter, our proof is conceptually simpler and more direct, and gives the same bound in the ideal cipher model. The main technical idea is to construct the compression isometry from a simple POVM on the naive purification, a technique which may find wider applications. As immediate applications, our results yield tight, concrete collision and pre-image lower bounds for the sponge hash construction underlying SHA3 and the Davies--Meyer compression function used in SHA1 and SHA2. More broadly, the improved soundness theorem provides a general-purpose tool for analyzing quantum security in settings where random permutations or ideal ciphers serve as the underlying primitive.
We study the (quantum) security of pseudorandom generators (PRGs) constructed from random oracles. We prove a "lifting theorem" showing, roughly, that if such a PRG is unconditionally secure against classical adversaries making polynomially many queries to the random oracle, then it is also (unconditionally) secure against quantum adversaries in the same sense. As a result of independent interest, we also show that any pseudo-deterministic quantum-oracle algorithm (i.e., a quantum algorithm that with high probability returns the same value on repeated executions) can be simulated by a computationally unbounded but query bounded classical-oracle algorithm with only a polynomial blowup in the number of queries. This implies as a corollary that our lifting theorem holds even for PRGs that themselves make quantum queries to the random oracle.
We prove near-optimal lower bounds for preprocessing attacks on quantum cryptography in the random oracle model. Specifically, we show that a TTT-query adversary with SSS qubits of non-uniform advice can recover a random key kkk from the nnn-qubit binary phase state ∣ψk⟩∝∑xR(k,x)∣x⟩|ψ_k\rangle \propto \sum_{x} R(k,x) |x\rangle∣ψk​⟩∝∑x​R(k,x)∣x⟩ with probability at most O(T2+STN)O(\frac{T^2 + \sqrt{ST}}{N})O(NT2+ST​​) for N=2nN=2^nN=2n. In contrast, the best known bound for post-quantum one-way functions is O(T2+STN)O(\frac{T^2 + ST}{N})O(NT2+ST​), with a trivial attack at S=NS = NS=N. This demonstrates a new advantage of quantum cryptography over classical cryptography: nnn qubits of communication suffice for security against preprocessing attacks with space up to N2N^2N2 rather than NNN. Our methodology is simple: express the optimal preprocessing attack as the operator norm of a random matrix, and bound this value in expectation over the random oracle via the trace-moment method. These trace moments have a natural interpretation using compressed oracles [Zhandry, Crypto 2019], which we then analyze. This can be viewed as a simplification and generalization of the approach of Liu [Eurocrypt 2023] for proving the security of post-quantum cryptography against preprocessing attacks. We also prove the following results: (1) We tighten Liu's analysis of post-quantum PRGs in QROM, achieving a distinguishing advantage bound of O(T2N+STN)O(\frac{T^2}N + \sqrt{\frac{ST}N})O(NT2​+NST​​). (2) For unitary synthesis, we extend the one-query lower bound of Lombardi-Ma-Wright [STOC 2024] to hold against adversaries that can make one arbitrary function query along with polynomially many (adaptive) queries to the random oracle, either before or after the function query. This also interprets the original LMW24 result in terms of compressed oracles. (3) Finally, we prove a tight O(SN)O(\frac{\sqrt{S}}N)O(NS​​) bound for the pseudorandomness of random binary phase states against space SSS distinguishers.
The analysis of quantum algorithms which query random, invertible permutations has been a long-standing challenge in cryptography. Many techniques which apply to random oracles fail, or are not known to generalize to this setting. As a result, foundational cryptographic constructions involving permutations often lack quantum security proofs. With the aim of closing this gap, we develop and prove soundness of a compressed permutation oracle. Our construction shares many of the attractive features of Zhandry's original compressed function oracle: the purification is a small list of input-output pairs which meaningfully reflect an algorithm's knowledge of the oracle.
Recent results on quantum cryptanalysis show that some symmetric key schemes can be broken in polynomial time even if they are proven to be secure in the classical setting. Liskov, Rivest, and Wagner showed that secure tweakable block ciphers can be constructed from secure block ciphers in the classical setting. However, Kaplan et al. showed that their scheme can be broken by polynomial time quantum superposition attacks, even if underlying block ciphers are quantum-secure. Since then, it remains open if there exists a mode of block ciphers to build quantum-secure tweakable block ciphers. This paper settles the problem in the reduction-based provable security paradigm. We show the first design of quantum-secure tweakable block ciphers based on quantum-secure block ciphers, and present a provable security bound. Our construction is simple, and when instantiated with a quantum-secure n-bit block cipher, it is secure against attacks that query arbitrary quantum superpositions of plaintexts and tweaks up to O(2n/6) quantum queries. Our security proofs use the compressed oracle technique introduced by Zhandry. More precisely, we use an alternative formalization of the technique introduced by Hosoyamada and Iwata.
We propose a generalization of Zhandry's compressed oracle method to random permutations, where an algorithm can query both the permutation and its inverse. We show how to use the resulting oracle simulation to bound the success probability of an algorithm for any predicate on input-output pairs, a key feature of Zhandry's technique that had hitherto resisted attempts at generalization to random permutations. One key technical ingredient is to use strictly monotone factorizations to represent the permutation in the oracle's database. As an application of our framework, we show that the one-round sponge construction is unconditionally preimage resistant in the random permutation model. This proves a conjecture by Unruh.