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

TL;DR

This paper extends noisy trapdoor claw-free functions to many-to-one variants using extrapolated dihedral cosets, enabling new quantum reductions and applications in post-quantum cryptography.

Contribution

It introduces a polynomial-sized many-to-one NTCF$^1_{ ext{kappa}}$ based on extrapolated dihedral cosets, expanding the cryptographic toolkit for quantum security.

Findings

Constructed NTCF$^1_{ ext{kappa}}$ assuming LWE hardness

Established a quantum reduction from LWE to extrapolated DCP

Showed NTCF$^1_{ ext{kappa}}$ reduces to NTCF$^1_2$ for quantumness proofs

Abstract

\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.