Post-Quantum κ-to-1 Trapdoor Claw-free Functions from Extrapolated Dihedral Cosets
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Noisy Trapdoor Claw-free functions (NTCF) as powerful post-quantum cryptographic tools can efficiently constrain actions of untrusted quantum devices. Recently, Brakerski et al. at FOCS 2018 showed a remarkable use of NTCF for a classically verifiable proof of quantumness and also derived a protocol for cryptographically certifiable quantum randomness generation. However, the original NTCF used in their work is essentially 2-to-1 one-way function, namely NTCF^1_2, which greatly limits the rate of randomness generation. In this work, we attempt to further extend the NTCF^1_2 to achieve a κ-to-1 function with poly-bounded preimage size. Specifically, we focus on a significant extrapolation of NTCF^1_2 by drawing on extrapolated dihedral cosets, giving a model of NTCF^1_κ with κ = poly(n). Then, we present an efficient construction of NTCF^1_κ under the well-known quantum hardness of the Learning with Errors (QLWE) assumption. As a byproduct, our work manifests an interesting connection between the NTCF^1_2 (resp. NTCF^1_κ) and the Dihedral Coset States (resp. Extrapolated Dihedral Coset States). Finally, we give a similar interactive protocol for proving quantumness from the NTCF^1_κ.
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