A New Quantum Claw-finding Algorithm for Three Functions

In this note, we give a quantum algorithm that finds collisions in arbitrary τ-to-one functions after only O ( 3 √ N /τ) expected evaluations of the function. Assuming the function is given by a black box, this is more efficient than the best possible classical algorithm, even allowing probabilism. We also give a similar algorithm for finding claws in pairs of functions. Furthermore, we exhibit a space-time tradeoff for our technique. Our approach uses Grover's quantum searching algorithm in a novel way.In this note, we give a quantum algorithm that finds collisions in arbitrary τ-to-one functions after only O ( 3 √ N /τ) expected evaluations of the function. Assuming the function is given by a black box, this is more efficient than the best possible classical algorithm, even allowing probabilism. We also give a similar algorithm for finding claws in pairs of functions. Furthermore, we exhibit a space-time tradeoff for our technique. Our approach uses Grover's quantum searching algorithm in a novel way.

ACM SIGACT NewsJun 1, 1997132 citations

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

Xingyu Yan, Licheng Wang, Lize Gu +2 more

\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 $^1_κ$ where $κ$ is a polynomial integer. Then, we present an efficient construction of NTCF $^1_κ$ 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 $^1_κ$ ), our work reveals a new quantum reduction path from the LWE problem to the DCP (resp. extrapolated DCP). Finally, we demonstrate the NTCF $^1_κ$ can naturally be reduced to the NTCF $^1_2$ , thereby achieving the same application for proving the quantumness.\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$^1_κ$ where $κ$ is a polynomial integer. Then, we present an efficient construction of NTCF$^1_κ$ 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$^1_κ$), our work reveals a new quantum reduction path from the LWE problem to the DCP (resp. extrapolated DCP). Finally, we demonstrate the NTCF$^1_κ$ can naturally be reduced to the NTCF$^1_2$, thereby achieving the same application for proving the quantumness.

arXiv:2211.16993Nov 30, 20220 citations

^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_κ

where

κ

is a polynomial integer. Then, we present an efficient construction of NTCF

^1_κ

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_κ

), our work reveals a new quantum reduction path from the LWE problem to the DCP (resp. extrapolated DCP). Finally, we demonstrate the NTCF

^1_κ

can naturally be reduced to the NTCF

^1_2

, thereby achieving the same application for proving the quantumness.