Quantum one-way permutation over the finite field of two elements

TL;DR

This paper proves the security of a quantum one-way permutation based on a specific unitary operator over finite fields, showing it is hard to invert and thus useful for quantum cryptography.

Contribution

It demonstrates that Levin's proposed quantum permutation is provably secure and cannot be inverted efficiently, establishing a link between classical one-way functions and quantum security.

Findings

The permutation's output states are maximally entangled two-qubit states.

The probability of inverting the permutation approaches zero faster than any polynomial inverse.

Existence of classical one-way functions implies a universal quantum one-way permutation.

Abstract

In quantum cryptography, a one-way permutation is a bounded unitary operator $U:\mathcal{H} \to \mathcal{H}$ on a Hilbert space $\mathcal{H}$ that is easy to compute on every input, but hard to invert given the image of a random input. Levin [Probl. Inf. Transm., vol. 39 (1): 92-103 (2003)] has conjectured that the unitary transformation $g(a,x)=(a,f(x)+ax)$, where $f$ is any length-preserving function and $a,x \in GF_{{2}^{\|x\|}}$, is an information-theoretically secure operator within a polynomial factor. Here, we show that Levin's one-way permutation is provably secure because its output values are four maximally entangled two-qubit states, and whose probability of factoring them approaches zero faster than the multiplicative inverse of any positive polynomial $poly(x)$ over the Boolean ring of all subsets of $x$. Our results demonstrate through well-known theorems that existence of classical one-way functions implies existence of a universal quantum one-way permutation that cannot be inverted in subexponential time in the worst ca