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

Quantum one-way permutation over the finite field of two elements In quantum cryptography, a one-way permutation is a bounded unitary operator $$U:\mathcal {H} \rightarrow \mathcal {H}$$ U : H → H on a Hilbert space $$\mathcal {H}$$ H that is easy to compute on every input, but hard to invert given the image of a random input. Levin (Probl Inf Transm 39(1):92–103, 2003) has conjectured that the unitary transformation $$g(a,x)=(a,f(x)+ax)$$ g ( a , x ) = ( a , f ( x ) + a x ) , where f is any length-preserving function and $$a,x \in \hbox {GF}_{{2}^{\Vert x\Vert }}$$ a , x ∈ 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 case. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

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

, Volume 16 (6) – Apr 27, 2017
18 pages

/lp/springer_journal/quantum-one-way-permutation-over-the-finite-field-of-two-elements-0sLiMm2s8q
Publisher
Springer Journals
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-017-1599-6
Publisher site
See Article on Publisher Site

### Abstract

In quantum cryptography, a one-way permutation is a bounded unitary operator $$U:\mathcal {H} \rightarrow \mathcal {H}$$ U : H → H on a Hilbert space $$\mathcal {H}$$ H that is easy to compute on every input, but hard to invert given the image of a random input. Levin (Probl Inf Transm 39(1):92–103, 2003) has conjectured that the unitary transformation $$g(a,x)=(a,f(x)+ax)$$ g ( a , x ) = ( a , f ( x ) + a x ) , where f is any length-preserving function and $$a,x \in \hbox {GF}_{{2}^{\Vert x\Vert }}$$ a , x ∈ 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 case.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Apr 27, 2017

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