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

*Alexandre de Castro*

**Abstract: **In quantum cryptography, a one-way permutation is a bounded unitary operator $U: H \mapsto H$ on a Hilbert space $H$ 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 oneway 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 case.

**Category / Keywords: **foundations / Quantum one-way permutation; CHSH inequality; Controlled NOT gate; Negligible probability; (Pseudo)randomness

**Original Publication**** (in the same form): **Quantum Information Processing (2017) 16:149
**DOI: **10.1007/s11128-017-1599-6

**Date: **received 30 Apr 2017

**Contact author: **alexandre castro at embrapa br

**Available format(s): **PDF | BibTeX Citation

**Note: **Journal reference: Quantum Information Processing (2017). 16:149.
DOI: 10.1007/s11128-017-1599-6

**Version: **20170501:135400 (All versions of this report)

**Short URL: **ia.cr/2017/381

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