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Split-State Non-malleable Codes and Secret Sharing Schemes for Quantum Messages

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Theory of Cryptography (TCC 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15365))

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Abstract

Non-malleable codes are fundamental objects at the intersection of cryptography and coding theory. These codes provide security guarantees even in settings where error correction and detection are impossible, and have found applications to several other cryptographic tasks. One of the strongest and most well-studied adversarial tampering models is 2-split-state tampering. Here, a codeword is split into two parts which are stored in physically distant servers, and the adversary can then independently tamper with each part using arbitrary functions. This model can be naturally extended to the secret sharing setting with several parties by having the adversary independently tamper with each share. Previous works on non-malleable coding and secret sharing in the split-state tampering model only considered the encoding of classical messages. Furthermore, until recent work by Aggarwal, Boddu, and Jain (IEEE Trans. Inf. Theory 2024 & arXiv 2022), adversaries with quantum capabilities and shared entanglement had not been considered, and it is a priori not clear whether previous schemes remain secure in this model.

In this work, we introduce the notions of split-state non-malleable codes and secret sharing schemes for quantum messages secure against quantum adversaries with shared entanglement. Then, we present explicit constructions of such schemes that achieve low-error non-malleability. More precisely, for some constant \(c>0\), we construct efficiently encodable and decodable split-state non-malleable codes and secret sharing schemes for quantum messages preserving entanglement with external systems and achieving security against quantum adversaries having shared entanglement with codeword length n, any message length at most \(n^c\), and error \(\varepsilon =2^{-{n^{c}}}\). In the easier setting of average-case non-malleability, we achieve efficient non-malleable coding with rate close to 1/11.

J. Ribeiro—Work done while at NOVA LINCS and NOVA School of Science and Technology.

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Notes

  1. 1.

    A set \(\varGamma \subseteq 2^{[p]}\) is monotone if \(A\in \varGamma \) and \(A\subseteq B\) imply that \(B\in \varGamma \).

  2. 2.

    Meaning that the codeword is divided into three parts and the adversary tampers each part independently. We note that constructing NMCs in the 3-split-state model is considerably easier than in the 2-split-state model.

  3. 3.

    Tampering maps are assumed to be unitary without any loss of generality. This is because, in the presence of unbounded arbitrary shared entanglement, tampering with unitary maps is equivalent to tampering with CPTP maps. More precisely, consider a tampering adversary that uses two CPTP maps \(\varPhi _1\) and \(\varPhi _2\) acting on registers \(E_1 W_1\) and \(E_2 W_2\), respectively. Then, the action of this adversary is equivalent to another adversary who tampers using Stinespring isometry extensions U and V of \(\varPhi _1\) and \(\varPhi _2\), respectively, which act on \(E_1 W_1 A_1\) and \(E_2 W_2 A_2\), respectively, where \(A_1\) and \(A_2\) are unentangled ancilla registers set to \(|{0}\rangle \) without loss of generality and can be seen as part of the shared entanglement.

  4. 4.

    By this, we mean that \(p_{\mathcal {A}}\) can be computed and the state \(\gamma ^{\mathcal {A}}_{M}\) can be prepared without the knowledge of the input message \(\sigma _{M\hat{M}}\).

  5. 5.

    Clifford-based quantum authentication schemes apply a random (secret) Clifford operator to the message plus several additional “trap registers” initialized to \(|{0}\rangle \). Verifying whether tampering of the authenticated state occurred consists of checking whether the trap registers all return to the \(|{0}\rangle \) state after applying the inverse Clifford operator. If this does not hold, then the verification procedure outputs the special symbol \(\perp \), which we call the “trap flag”.

  6. 6.

    For the experienced reader, Batra, Boddu, and Jain [16] construct an explicit quantum-secure 2-source non-malleable extractor \(\textrm{nmExt}\) with a large output length. We can then sample classical bitstrings X and Y uniformly at random with appropriate lengths and set the classical key R to be \(R=\textrm{nmExt}(X,Y)\); this is the quantum-secure classical NMRE that we use in our optimized coding scheme.

  7. 7.

    “qpa-state” stands for quantum purified adversary state.

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Acknowledgements

We thank Thiago Bergamaschi for insightful discussions about notions of non-malleability in the quantum setting. We also thank Dakshita Khurana for useful discussions in the initial stage of this project.

JR was supported in part by NOVA LINCS (ref. UIDB/04516/2020) with the financial support of FCT - Fundação para a Ciência e a Tecnologia. RJ was supported by the NRF grant NRF2021-QEP2-02-P05 and the Ministry of Education, Singapore, under the Research Centres of Excellence program. This work was done in part while RJ was visiting the Technion-Israel Institute of Technology, Haifa, Israel and the Simons Institute for the Theory of Computing, Berkeley, CA, USA.

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  1. NTT Research, Sunnyvale, CA, USA

    Naresh Goud Boddu & Vipul Goyal

  2. Centre for Quantum Technologies and National University of Singapore, Singapore, Singapore

    Rahul Jain

  3. Instituto de Telecomunicações, Lisbon, Portugal

    João Ribeiro

  4. Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    João Ribeiro

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    Elette Boyle

  2. University of Virginia, Charlottesville, VA, USA

    Mohammad Mahmoody

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Boddu, N.G., Goyal, V., Jain, R., Ribeiro, J. (2025). Split-State Non-malleable Codes and Secret Sharing Schemes for Quantum Messages. In: Boyle, E., Mahmoody, M. (eds) Theory of Cryptography. TCC 2024. Lecture Notes in Computer Science, vol 15365. Springer, Cham. https://doi.org/10.1007/978-3-031-78017-2_3

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