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Quantum Copy-Protection from Hidden Subspaces.

Scott Aaronson, Jiahui Liu, Ruizhe Zhang

Published Apr 20, 2020• 6 citations
OpenAlex

Abstract

Quantum copy-protection is an innovative idea that uses the no-cloning property of quantum information to copy-protect programs and was first put forward by [Aar09]. The general goal is that a program distributor can distribute a quantum state $|\Psi\rangle$, whose classical description is secret to the users; a user can use this state to run the program P on his own input, but not be able to pirate this program P or create another state with the same functionality. In the copy-protection with oracle setting, the user has access to a public oracle and can use the given quantum state and the oracle to compute on his/her own input for polynomially many times. However, the user is not able to produce an additional program(quantum or classical) that computes the same as P on almost all inputs. We present a first quantum copy protection scheme with a classical oracle for any unlearnable function families. The construction is based on membership oracles for hidden subspaces in $\mathbb{F}_2^n$, an idea derived from the public key quantum money scheme in[Aar12]. We prove the security of the scheme relative to a classical oracle, namely, the subspace membership oracle with the functionality of computing the secret function we want to copy-protect. The security proof builds on the quantum lower bound for the Direct-Product problem ([Aar12],[BDS16]) and the unlearnability of the copy-protected functions. We also show that existence of quantum copy protection and the quantum hardness of Learning-with-Errors (LWE) will imply publicly verifiable quantum money.

Related papers

New Approaches for Quantum Copy-Protection.

Scott Aaronson, Jiahui Liu, Qipeng Liu +2 more

Quantum copy-protection uses the unclonability of quantum states to construct quantum software that provably cannot be pirated. copy-protection would be immensely useful, but unfortunately, little is known about achieving it in general. In this work, we make progress on this goal, by giving the following results: We show how to copy-protect any program that cannot be learned from its input-output behavior relative to a classical oracle. This construction improves on Aaronson (CCC 2009), which achieves the same relative to a quantum oracle. By instantiating the oracle with post-quantum candidate obfuscation schemes, we obtain a heuristic construction of copy-protection. We show, roughly, that any program which can be watermarked can be copy detected, a weaker version of copy-protection that does not prevent copying, but guarantees that any copying can be detected. Our scheme relies on the security of the assumed watermarking, plus the assumed existence of public-key quantum money. Our construction is publicly detectable and applicable to many recent watermarking schemes.

•Jan 1, 2020•1 citation

New Approaches for Quantum Copy-Protection

Scott Aaronson, Jiahui Liu, Qipeng Liu +2 more

Quantum copy protection uses the unclonability of quantum states to construct quantum software that provably cannot be pirated. Copy protection would be immensely useful, but unfortunately little is known about how to achieve it in general. In this work, we make progress on this goal, by giving the following results: - We show how to copy protect any program that cannot be learned from its input/output behavior, relative to a classical oracle. This improves on Aaronson [CCC'09], which achieves the same relative to a quantum oracle. By instantiating the oracle with post-quantum candidate obfuscation schemes, we obtain a heuristic construction of copy protection. -We show, roughly, that any program which can be watermarked can be copy detected, a weaker version of copy protection that does not prevent copying, but guarantees that any copying can be detected. Our scheme relies on the security of the assumed watermarking, plus the assumed existence of public key quantum money. Our construction is general, applicable to many recent watermarking schemes.

arXiv:2004.09674•Apr 20, 2020•2 citations

Quantum copy-protection of compute-and-compare programs in the quantum random oracle model

Andrea Coladangelo, Christian Majenz, Alexander Poremba

Copy-protection allows a software distributor to encode a program in such a way that it can be evaluated on any input, yet it cannot be "pirated" – a notion that is impossible to achieve in a classical setting. Aaronson (CCC 2009) initiated the formal study of quantum copy-protection schemes, and speculated that quantum cryptography could offer a solution to the problem thanks to the quantum no-cloning theorem. In this work, we introduce a quantum copy-protection scheme for a large class of evasive functions known as "compute-and-compare programs" – a more expressive generalization of point functions. A compute-and-compare program CC[f,y] is specified by a function f and a string y within its range: on input x , CC[f,y] outputs 1 , if f(x)=y , and 0 otherwise. We prove that our scheme achieves non-trivial security against fully malicious adversaries in the quantum random oracle model (QROM), which makes it the first copy-protection scheme to enjoy any level of provable security in a standard cryptographic model. As a complementary result, we show that the same scheme fulfils a weaker notion of software protection, called "secure software leasing", introduced very recently by Ananth and La Placa (eprint 2020), with a standard security bound in the QROM, i.e. guaranteeing negligible adversarial advantage. Finally, as a third contribution, we elucidate the relationship between unclonable encryption and copy-protection for multi-bit output point functions.

arXiv:2009.13865•Quantum•May 2, 2024•19 citations

A Note on Copy-Protection from Random Oracles

Prabhanjan Ananth, Fatih Kaleoglu

Quantum copy-protection, introduced by Aaronson (CCC'09), uses the no-cloning principle of quantum mechanics to protect software from being illegally distributed. Constructing copy-protection has been an important problem in quantum cryptography. Since copy-protection is shown to be impossible to achieve in the plain model, we investigate the question of constructing copy-protection for arbitrary classes of unlearnable functions in the random oracle model. We present an impossibility result that rules out a class of copy-protection schemes in the random oracle model assuming the existence of quantum fully homomorphic encryption and quantum hardness of learning with errors. En route, we prove the impossibility of approximately correct copy-protection in the plain model.

arXiv:2208.12884•Aug 26, 2022•1 citation

Quantum State Obfuscation from Classical Oracles

James Bartusek, Zvika Brakerski, Vinod Vaikuntanathan

A major unresolved question in quantum cryptography is whether it is possible to obfuscate arbitrary quantum computation. Indeed, there is much yet to understand about the feasibility of quantum obfuscation even in the classical oracle model, where one is given for free the ability to obfuscate any classical circuit. In this work, we develop a new array of techniques that we use to construct a quantum state obfuscator, a powerful notion formalized recently by Coladangelo and Gunn (arXiv:2311.07794) in their pursuit of better software copy-protection schemes. Quantum state obfuscation refers to the task of compiling a quantum program, consisting of a quantum circuit C with a classical description and an auxiliary quantum state ψ, into a functionally-equivalent obfuscated quantum program that hides as much as possible about C and ψ. We prove the security of our obfuscator when applied to any pseudo-deterministic quantum program, i.e. one that computes a (nearly) deterministic classical input / classical output functionality. Our security proof is with respect to an efficient classical oracle, which may be heuristically instantiated using quantum-secure indistinguishability obfuscation for classical circuits. Our result improves upon the recent work of Bartusek, Kitagawa, Nishimaki and Yamakawa (STOC 2023) who also showed how to obfuscate pseudo-deterministic quantum circuits in the classical oracle model, but only ones with a completely classical description. Furthermore, our result answers a question of Coladangelo and Gunn, who provide a construction of quantum state indistinguishability obfuscation with respect to a quantum oracle, but leave the existence of a concrete real-world candidate as an open problem. Indeed, our quantum state obfuscator together with Coladangelo-Gunn gives the first candidate realization of a “best-possible” copy-protection scheme for all polynomial-time functionalities. Our techniques deviate significantly from previous works on quantum obfuscation. We develop several novel technical tools which we expect to be broadly useful in quantum cryptography. These tools include a publicly-verifiable, linearly-homomorphic quantum authentication scheme with classically-decodable ZX measurements (which we build from coset states), and a method for compiling any quantum circuit into a ”linear + measurement” () quantum program: an alternating sequence of CNOT operations and partial ZX measurements.

arXiv:2401.10200•ACM Symposium on Theory of Computing (STOC)•Jun 10, 2024•19 citations

Obfuscation of Arbitrary Quantum Circuits (via Subspace-Preserving Pseudorandom Unitary)

Miryam Mi-Ying Huang, Er-Cheng Tang

Program obfuscation aims to conceal a computer program's internal structure while preserving its functionality. A central open problem is whether an obfuscation scheme for arbitrary quantum circuits exists. Despite several efforts toward this goal, prior works have succeeded only in obfuscating quantum circuits that implement either pseudo-deterministic functions or unitary transformations. Although unitary transformations already include a broad class of quantum computations, many important quantum tasks, such as state preparation, quantum sampling, and quantum error correction, go beyond unitaries and are described by general completely positive trace-preserving (CPTP) maps. In this work, we construct the first quantum ideal obfuscation scheme for arbitrary quantum circuits that support quantum inputs and outputs in the classical oracle model assuming post-quantum one-way functions, thereby resolving an open problem raised in Bartusek et al. (STOC 2023), Bartusek, Brakerski, and Vaikuntanathan (STOC 2024), and Huang and Tang (FOCS 2025). At the core of our construction lies a novel primitive that we introduce, called the subspace-preserving strong pseudorandom unitary (spsPRU). An spsPRU is a family of efficient unitaries that fix every vector in a given linear subspace $S$, while acting as a Haar random unitary on its orthogonal complement $S^\perp$ under both forward and inverse oracle queries. Furthermore, by instantiating the classical oracle model with the ideal obfuscation scheme for classical circuits proposed by Jain et al. (CRYPTO 2023) and later enhanced by Bartusek et al. (CRYPTO 2026), our obfuscation scheme can also be realized in the quantumly accessible pseudorandom oracle model.

quant-pharXiv:2601.08969•Jan 15, 2026•0 citations
Quantum copy-protection is an innovative idea that uses the no-cloning property of quantum information to copy-protect programs and was first put forward by [Aar09]. The general goal is that a program distributor can distribute a quantum state ∣Ψ⟩|\Psi\rangle∣Ψ⟩, whose classical description is secret to the users; a user can use this state to run the program P on his own input, but not be able to pirate this program P or create another state with the same functionality. In the copy-protection with oracle setting, the user has access to a public oracle and can use the given quantum state and the oracle to compute on his/her own input for polynomially many times. However, the user is not able to produce an additional program(quantum or classical) that computes the same as P on almost all inputs.
We present a first quantum copy protection scheme with a classical oracle for any unlearnable function families. The construction is based on membership oracles for hidden subspaces in F2n\mathbb{F}_2^nF2n​, an idea derived from the public key quantum money scheme in[Aar12]. We prove the security of the scheme relative to a classical oracle, namely, the subspace membership oracle with the functionality of computing the secret function we want to copy-protect. The security proof builds on the quantum lower bound for the Direct-Product problem ([Aar12],[BDS16]) and the unlearnability of the copy-protected functions.
We also show that existence of quantum copy protection and the quantum hardness of Learning-with-Errors (LWE) will imply publicly verifiable quantum money.
Quantum copy-protection uses the unclonability of quantum states to construct quantum software that provably cannot be pirated. copy-protection would be immensely useful, but unfortunately, little is known about achieving it in general. In this work, we make progress on this goal, by giving the following results: We show how to copy-protect any program that cannot be learned from its input-output behavior relative to a classical oracle. This construction improves on Aaronson (CCC 2009), which achieves the same relative to a quantum oracle. By instantiating the oracle with post-quantum candidate obfuscation schemes, we obtain a heuristic construction of copy-protection. We show, roughly, that any program which can be watermarked can be copy detected, a weaker version of copy-protection that does not prevent copying, but guarantees that any copying can be detected. Our scheme relies on the security of the assumed watermarking, plus the assumed existence of public-key quantum money. Our construction is publicly detectable and applicable to many recent watermarking schemes.
Quantum copy protection uses the unclonability of quantum states to construct quantum software that provably cannot be pirated. Copy protection would be immensely useful, but unfortunately little is known about how to achieve it in general. In this work, we make progress on this goal, by giving the following results: - We show how to copy protect any program that cannot be learned from its input/output behavior, relative to a classical oracle. This improves on Aaronson [CCC'09], which achieves the same relative to a quantum oracle. By instantiating the oracle with post-quantum candidate obfuscation schemes, we obtain a heuristic construction of copy protection. -We show, roughly, that any program which can be watermarked can be copy detected, a weaker version of copy protection that does not prevent copying, but guarantees that any copying can be detected. Our scheme relies on the security of the assumed watermarking, plus the assumed existence of public key quantum money. Our construction is general, applicable to many recent watermarking schemes.
Copy-protection allows a software distributor to encode a program in such a way that it can be evaluated on any input, yet it cannot be "pirated" – a notion that is impossible to achieve in a classical setting. Aaronson (CCC 2009) initiated the formal study of quantum copy-protection schemes, and speculated that quantum cryptography could offer a solution to the problem thanks to the quantum no-cloning theorem. In this work, we introduce a quantum copy-protection scheme for a large class of evasive functions known as "compute-and-compare programs" – a more expressive generalization of point functions. A compute-and-compare program CC[f,y] is specified by a function f and a string y within its range: on input x , CC[f,y] outputs 1 , if f(x)=y , and 0 otherwise. We prove that our scheme achieves non-trivial security against fully malicious adversaries in the quantum random oracle model (QROM), which makes it the first copy-protection scheme to enjoy any level of provable security in a standard cryptographic model. As a complementary result, we show that the same scheme fulfils a weaker notion of software protection, called "secure software leasing", introduced very recently by Ananth and La Placa (eprint 2020), with a standard security bound in the QROM, i.e. guaranteeing negligible adversarial advantage. Finally, as a third contribution, we elucidate the relationship between unclonable encryption and copy-protection for multi-bit output point functions.
Quantum copy-protection, introduced by Aaronson (CCC'09), uses the no-cloning principle of quantum mechanics to protect software from being illegally distributed. Constructing copy-protection has been an important problem in quantum cryptography. Since copy-protection is shown to be impossible to achieve in the plain model, we investigate the question of constructing copy-protection for arbitrary classes of unlearnable functions in the random oracle model. We present an impossibility result that rules out a class of copy-protection schemes in the random oracle model assuming the existence of quantum fully homomorphic encryption and quantum hardness of learning with errors. En route, we prove the impossibility of approximately correct copy-protection in the plain model.
A major unresolved question in quantum cryptography is whether it is possible to obfuscate arbitrary quantum computation. Indeed, there is much yet to understand about the feasibility of quantum obfuscation even in the classical oracle model, where one is given for free the ability to obfuscate any classical circuit. In this work, we develop a new array of techniques that we use to construct a quantum state obfuscator, a powerful notion formalized recently by Coladangelo and Gunn (arXiv:2311.07794) in their pursuit of better software copy-protection schemes. Quantum state obfuscation refers to the task of compiling a quantum program, consisting of a quantum circuit C with a classical description and an auxiliary quantum state ψ, into a functionally-equivalent obfuscated quantum program that hides as much as possible about C and ψ. We prove the security of our obfuscator when applied to any pseudo-deterministic quantum program, i.e. one that computes a (nearly) deterministic classical input / classical output functionality. Our security proof is with respect to an efficient classical oracle, which may be heuristically instantiated using quantum-secure indistinguishability obfuscation for classical circuits. Our result improves upon the recent work of Bartusek, Kitagawa, Nishimaki and Yamakawa (STOC 2023) who also showed how to obfuscate pseudo-deterministic quantum circuits in the classical oracle model, but only ones with a completely classical description. Furthermore, our result answers a question of Coladangelo and Gunn, who provide a construction of quantum state indistinguishability obfuscation with respect to a quantum oracle, but leave the existence of a concrete real-world candidate as an open problem. Indeed, our quantum state obfuscator together with Coladangelo-Gunn gives the first candidate realization of a “best-possible” copy-protection scheme for all polynomial-time functionalities. Our techniques deviate significantly from previous works on quantum obfuscation. We develop several novel technical tools which we expect to be broadly useful in quantum cryptography. These tools include a publicly-verifiable, linearly-homomorphic quantum authentication scheme with classically-decodable ZX measurements (which we build from coset states), and a method for compiling any quantum circuit into a ”linear + measurement” () quantum program: an alternating sequence of CNOT operations and partial ZX measurements.
Program obfuscation aims to conceal a computer program's internal structure while preserving its functionality. A central open problem is whether an obfuscation scheme for arbitrary quantum circuits exists. Despite several efforts toward this goal, prior works have succeeded only in obfuscating quantum circuits that implement either pseudo-deterministic functions or unitary transformations. Although unitary transformations already include a broad class of quantum computations, many important quantum tasks, such as state preparation, quantum sampling, and quantum error correction, go beyond unitaries and are described by general completely positive trace-preserving (CPTP) maps. In this work, we construct the first quantum ideal obfuscation scheme for arbitrary quantum circuits that support quantum inputs and outputs in the classical oracle model assuming post-quantum one-way functions, thereby resolving an open problem raised in Bartusek et al. (STOC 2023), Bartusek, Brakerski, and Vaikuntanathan (STOC 2024), and Huang and Tang (FOCS 2025). At the core of our construction lies a novel primitive that we introduce, called the subspace-preserving strong pseudorandom unitary (spsPRU). An spsPRU is a family of efficient unitaries that fix every vector in a given linear subspace SSS, while acting as a Haar random unitary on its orthogonal complement S⊥S^\perpS⊥ under both forward and inverse oracle queries. Furthermore, by instantiating the classical oracle model with the ideal obfuscation scheme for classical circuits proposed by Jain et al. (CRYPTO 2023) and later enhanced by Bartusek et al. (CRYPTO 2026), our obfuscation scheme can also be realized in the quantumly accessible pseudorandom oracle model.