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arXiv:2609.25819

On the Construction of Trapdoor Claw-Free Functions with Certifiable Key

Charles Lim, Yao Ma

cs.CRquant-phSubmitted Sep 22, 2026• Announced Sep 23, 2026• 0 citations
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Abstract

Trapdoor claw-free functions (TCFs) underpin much of classical-quantum cryptographic interaction, yet every TCF-based protocol states its guarantees relative to an honestly generated key. We give a family-agnostic abstraction of key certification for (noisy) TCF constructions, built on two notions: a certifiable key relation, an NP relation capturing a family's honest keys with witnesses recoverable from the trapdoor; and certified key generation, which emits with each key a certificate of membership satisfying completeness, certificate soundness with extractability, and key privacy. We instantiate certifiable key relations for different constructions, each met generically by a zero-knowledge argument of knowledge for the relation. As our main application, a generic compiler turns any TCF-based proof of quantumness into a zero-knowledge one, with each security property following from its counterpart in the certification scheme. Finally, we delimit the primitive's reach: for protocols resting on injective invariance, an accepting certificate is itself a family distinguisher, leaking exactly the bit such protocols must hide.

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Post-Quantum κκκ-to-1 Trapdoor Claw-free Functions from Extrapolated Dihedral Cosets

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\emph{Noisy trapdoor claw-free function} (NTCF) as a powerful post-quantum cryptographic tool can efficiently constrain actions of untrusted quantum devices. However, the original NTCF is essentially \emph{2-to-1} one-way function (NTCF$^1_2$). In this work, we attempt to further extend the NTCF$^1_2$ to achieve \emph{many-to-one} trapdoor claw-free functions with polynomial bounded preimage size. Specifically, we focus on a significant extrapolation of NTCF$^1_2$ by drawing on extrapolated dihedral cosets, thereby giving a model of NTCF$^1_κ$ where $κ$ is a polynomial integer. Then, we present an efficient construction of NTCF$^1_κ$ assuming \emph{quantum hardness of the learning with errors (LWE)} problem. We point out that NTCF can be used to bridge the LWE and the dihedral coset problem (DCP). By leveraging NTCF$^1_2$ (resp. NTCF$^1_κ$), our work reveals a new quantum reduction path from the LWE problem to the DCP (resp. extrapolated DCP). Finally, we demonstrate the NTCF$^1_κ$ can naturally be reduced to the NTCF$^1_2$, thereby achieving the same application for proving the quantumness.

arXiv:2211.16993•Nov 30, 2022•0 citations
Trapdoor claw-free functions (TCFs) underpin much of classical-quantum cryptographic interaction, yet every TCF-based protocol states its guarantees relative to an honestly generated key. We give a family-agnostic abstraction of key certification for (noisy) TCF constructions, built on two notions: a certifiable key relation, an NP relation capturing a family's honest keys with witnesses recoverable from the trapdoor; and certified key generation, which emits with each key a certificate of membership satisfying completeness, certificate soundness with extractability, and key privacy. We instantiate certifiable key relations for different constructions, each met generically by a zero-knowledge argument of knowledge for the relation. As our main application, a generic compiler turns any TCF-based proof of quantumness into a zero-knowledge one, with each security property following from its counterpart in the certification scheme. Finally, we delimit the primitive's reach: for protocols resting on injective invariance, an accepting certificate is itself a family distinguisher, leaking exactly the bit such protocols must hide.
Randomness certification aims to guarantee that the output of an untrusted quantum device contains genuine randomness that remains unpredictable even given quantum side information held by an adversary. For a classical verifier interacting with a single quantum device, such a guarantee cannot rely on the physical separation underlying standard Bell tests and must instead follow from computational restrictions on the prover. Building on the framework of Computational Bell Inequalities, we distinguish classical and quantum virtual-input hiding, showing that the former suffices for proofs of quantumness, whereas the latter enables randomness certification against quantum side information. Our main technical contribution is to establish quantum virtual-input hiding for a commitment-based protocol derived from the proof of quantumness from one-way-functions by Morimae and Yamakawa. We show that statistical hiding together with ordinary computational binding against quantum polynomial-time senders suffices, without requiring stronger quantum binding notions such as sum-binding or collapse-binding. This yields a randomness-certification protocol from post-quantum statistically hiding bit commitments. In particular, existing constructions of such commitments from post-quantum one-way permutations imply that post-quantum one-way permutations suffice for randomness certification. We further instantiate our construction using quantum-secure trapdoor claw-free functions, obtaining an efficient classical verifier, and in the quantum random oracle model, obtaining randomness certification without additional computational assumptions but with an inefficient verifier. These instantiations separate the cryptographic properties sufficient for randomness certification from the additional structure required for efficient verification within our commitment-based framework.
Trapdoor claw-free functions (TCFs) are immensely valuable in cryptographic interactions between a classical client and a quantum server. Typically, a protocol has the quantum server prepare a superposition of two-bit strings of a claw and then measure it using Pauli- X or Z measurements. In this paper, we demonstrate a new technique that uses the entire range of qubit measurements from the XY -plane. We show the advantage of this approach in two applications. First, building on (Brakerski et al. 2018, Kalai et al. 2022), we show an optimized two-round proof of quantumness whose security can be expressed directly in terms of the hardness of the LWE (learning with errors) problem. Second, we construct a one-round protocol for blind remote preparation of an arbitrary state on the XY -plane up to a Pauli- Z correction.
Properties of quantum mechanics such as entanglement and the no-cloning theorem enable new forms of cryptography that are unattainable in classical models of computation. Quantum information, however, is harder to process, store, and transmit than classical information. As such, it is desirable to construct semi-quantum cryptographic protocols where only a subset of the parties have quantum capabilities while the rest operate classically. In this dissertation, we analyze the security of semi-quantum protocols based on two approaches: (i) nonlocality-based techniques and (ii) the hardness of the Learning with Errors (LWE) problem. In the first part of the dissertation, we leverage a form of Bell test to construct a quantum position verification (QPV) protocol. Initial QPV protocols required verifiers to prepare and send quantum states to an honest prover to achieve security against adversaries with bounded memory or entanglement. Later work constructed QPV protocols with classical verifiers but required additional assumptions such as the hardness of LWE or of random circuit sampling. We construct a variant of QPV where classical verifiers verify the spatial locations of two provers. We use the nonlocal correlations produced by the two provers to show security against adversaries with bounded quantum memory without making any computational intractability assumptions. In the second part of the dissertation, we use trapdoor claw-free functions (TCFs) to construct semi-quantum cryptographic protocols. TCFs, which are constructible from the hardness of LWE, have been shown to be immensely valuable in the construction of classically verifiable quantum cryptographic protocols. As opposed to previous work where the server is asked to prepare a claw state and perform Pauli-X and Z measurements, we use a technique that leverages the entire range of qubit measurements on the XY-plane. We use this technique to construct three protocols: (i) a one-round blind remote state preparation protocol that delegates the preparation of an arbitrary state on the XY-plane up to a Pauli-X correction; (ii) a two-round proof of quantumness whose security is directly in terms of the hardness of LWE and only requires the prover to prepare and measure a single claw state; and (iii) a 1-of-2 Puzzle with a single qubit auxiliary state.
We consider a new model for the testing of untrusted quantum devices, consisting of a single polynomial time bounded quantum device interacting with a classical polynomial time verifier. In this model, we propose solutions to two tasks—a protocol for efficient classical verification that the untrusted device is “truly quantum” and a protocol for producing certifiable randomness from a single untrusted quantum device. Our solution relies on the existence of a new cryptographic primitive for constraining the power of an untrusted quantum device: post-quantum secure trapdoor claw-free functions that must satisfy an adaptive hardcore bit property. We show how to construct this primitive based on the hardness of the learning with errors (LWE) problem.
Remote state preparation (RSP) is a central primitive in quantum cryptography, enabling classical parties to remotely construct quantum states using only classical communication. As a result, RSP serves as a key building block in numerous protocols involving classical clients and quantum servers, allowing classical parties to leverage the advantages offered by powerful quantum computers. All known constructions of RSP rely on strong cryptographic assumptions, typically variants of trapdoor claw-free functions (TCFs). In this work, we initiate the study of a weaker form of remote state preparation, which we call eavesdropper-blind remote state preparation (EB-RSP). Informally, EB-RSP requires blindness only against external observers who see the transcript of the honest protocol, rather than against the quantum server itself. Despite this relaxed adversarial model, the resulting notion remains sufficient for useful cryptographic applications. In particular, we show that two-message EB-RSP already suffices to construct quantum public-key encryption with classical public keys and quantum ciphertexts. We then construct two-message EB-RSP protocols from specific one-way group actions, yielding a first step toward RSP-type primitives based on assumptions that do not rely on trapdoors. Finally, we observe that existing RSP constructions are likely naturally adaptable to the two-message EB-RSP notion; we demonstrate this explicitly for a concrete TCF-based RSP construction.
\emph{Noisy trapdoor claw-free function} (NTCF) as a powerful post-quantum cryptographic tool can efficiently constrain actions of untrusted quantum devices. However, the original NTCF is essentially \emph{2-to-1} one-way function (NTCF21^1_221​). In this work, we attempt to further extend the NTCF21^1_221​ to achieve \emph{many-to-one} trapdoor claw-free functions with polynomial bounded preimage size. Specifically, we focus on a significant extrapolation of NTCF21^1_221​ by drawing on extrapolated dihedral cosets, thereby giving a model of NTCFκ1^1_κκ1​ where κκκ is a polynomial integer. Then, we present an efficient construction of NTCFκ1^1_κκ1​ assuming \emph{quantum hardness of the learning with errors (LWE)} problem. We point out that NTCF can be used to bridge the LWE and the dihedral coset problem (DCP). By leveraging NTCF21^1_221​ (resp. NTCFκ1^1_κκ1​), our work reveals a new quantum reduction path from the LWE problem to the DCP (resp. extrapolated DCP). Finally, we demonstrate the NTCFκ1^1_κκ1​ can naturally be reduced to the NTCF21^1_221​, thereby achieving the same application for proving the quantumness.