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Robust Combiners for Quantum Cryptographic Primitives

Boyang Chen

1 Background

In this project, you will investigate problems related to robust combiners of quantum cryptographic primitives. Robust combiners play an important role in reduction and universal construction of cryptographic primitives.

A robust combiner of cryptographic primitive \(A\) is an algorithm that takes two constructions of the \(A\), \(\mathsf{Gen}_1, \mathsf{Gen}_2\), and outputs a construction of the primitive \(\widetilde{\mathsf{Gen}}\), such that assuming \(\mathsf{Gen}_1, \mathsf{Gen}_2\) is a pair of non-uniform \(A\), then \(\widetilde{\mathsf{Gen}}\) is a secure \(A\). Non-uniform \(A\) means that if for any security parameter \(\lambda\), there exists an index \(i(\lambda) \in \{1,2\}\) satisfying the security requirement of \(A\). See [HKNY23] for more details and examples for definition of robust combiner.

2 Problems

You are asked to work on two problems: the first is relatively straightforward, while the second is research-level.

  1. Can you construct a robust combiner for pseudoentanglment states? Pseudoentanglment states mean two families of pure states \(\ket{\Psi_k}_{AB}\) and \(\ket{\Phi_k}_{AB}\), \(k \in \{0,1\}^{\lambda}\) is the secret key. \(\ket{\Psi_k}\) is efficiently preparable (\(\ket{\Phi_\lambda}\) is not necessarily efficiently computable). Entanglement of \(\ket{\Psi_k}\) is \(O(f(\lambda))\) while entanglement of \(\ket{\Phi_k}\) is \(\omega(f(\lambda))\) for some computable function \(f(\lambda)\). Given polynomial copies, any quantum polynomial-time adversary cannot distinguish whether the state is sampled from \(\ket{\Psi_k}\) or \(\ket{\Phi_k}\) with inverse polynomial probability. See Section 1.2 of [ABF+23] for detailed definitions but notice that we’re considering a variant different from theirs: \(\ket{\Phi_k}\) is not necessarily efficiently preparable and we don’t specify the entropy gap here.

  2. Can you construct a robust combiner for single-copy pseudorandom states? Single-copy pseudorandom states mean a family of \(n(\lambda)\) pure states \(\ket{\phi_k}, k \in \{0,1\}^\lambda\) where \(n(\lambda)\) is a computable function satisfying \(n(\lambda) > \lambda\). For any quantum polynomial time adversary, it cannot distinguish \(\ket{\phi_k}\) from totally mixed state assuming that it takes only a single copy of the state (equivalently, the adversary cannot distinguish \(\mathbb{E}_k\ket{\phi_k}\bra{\phi_k}\) from totally mixed state). See Definition 2.2 of [MY22] for more details.

3 Contacts

Boyang Chen (by-chen24@mails.tsinghua.edu.cn)

4 References

[HKNY23] Hiroka, Taiga, Fuyuki Kitagawa, Ryo Nishimaki, and Takashi Yamakawa. “Robust combiners and universal constructions for quantum cryptography.” In Theory of Cryptography Conference, pp. 126-158. Cham: Springer Nature Switzerland, 2024. (for definition and examples of robust combiner)

[ABF+22] Aaronson, Scott, Adam Bouland, Bill Fefferman, Soumik Ghosh, Umesh Vazirani, Chenyi Zhang, and Zixin Zhou. “Quantum pseudoentanglement.” arXiv preprint arXiv:2211.00747 (2022). (for definition of pseudoentanglement states)

[MY22] Morimae, Tomoyuki, and Takashi Yamakawa. “Quantum commitments and signatures without one-way functions.” In Annual International Cryptology Conference, pp. 269-295. Cham: Springer Nature Switzerland, 2022. (for definition of single-copy pseudorandom states)