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\emph{Noisy trapdoor claw-free function} (NTCF) as a powerful post-quantumcryptographic tool can efficiently constrain actions of untrusted quantumdevices. However, the original NTCF is essentially \emph{2-to-1} one-wayfunction (NTCF$^1_2$). In this work, we attempt to further extend theNTCF$^1_2$ to achieve \emph{many-to-one} trapdoor claw-free functions withpolynomial bounded preimage size. Specifically, we focus on a significantextrapolation of NTCF$^1_2$ by drawing on extrapolated dihedral cosets, therebygiving a model of NTCF$^1_{\kappa}$ where $\kappa$ is a polynomial integer.Then, we present an efficient construction of NTCF$^1_{\kappa}$ assuming\emph{quantum hardness of the learning with errors (LWE)} problem. We point outthat NTCF can be used to bridge the LWE and the dihedral coset problem (DCP).By leveraging NTCF$^1_2$ (resp. NTCF$^1_{\kappa}$), our work reveals a newquantum reduction path from the LWE problem to the DCP (resp. extrapolatedDCP). Finally, we demonstrate the NTCF$^1_{\kappa}$ can naturally be reduced tothe NTCF$^1_2$, thereby achieving the same application for proving thequantumness.

Post-Quantum $\kappa$-to-1 Trapdoor Claw-free Functions from Extrapolated Dihedral Cosets