  \emph{Noisy trapdoor claw-free function} (NTCF) as a powerful post-quantum
cryptographic tool can efficiently constrain actions of untrusted quantum
devices. However, the original NTCF is essentially \emph{2-to-1} one-way
function (NTCF$^1_2$). In this work, we attempt to further extend the
NTCF$^1_2$ to achieve \emph{many-to-one} trapdoor claw-free functions with
polynomial bounded preimage size. Specifically, we focus on a significant
extrapolation of NTCF$^1_2$ by drawing on extrapolated dihedral cosets, thereby
giving 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 out
that 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 new
quantum reduction path from the LWE problem to the DCP (resp. extrapolated
DCP). Finally, we demonstrate the NTCF$^1_{\kappa}$ can naturally be reduced to
the NTCF$^1_2$, thereby achieving the same application for proving the
quantumness.
