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Algebraic Cryptanalysis of a Quantum Money Scheme The Noise-Free Case

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9020)

Abstract

We investigate the Hidden Subspace Problem (\(\mathrm{HSP}_q\)) over \({\mathbb {F}}_q\):

Input : \(p_1,\ldots ,p_m,q_1,\ldots ,q_m\in {\mathbb {F}}_q[x_1,\ldots ,x_n]\) of degree \(d\ge 3\) (and \(n\le m\le 2n\)).

Find : a subspace \(A\subset {{\mathbb {F}}_q}^n\) of dimension \(n/2\) (\(n\) is even) such that
$$\begin{aligned} p_i(A)=0\,\,\forall i\in \{1,\ldots ,m\}\,\,\text {and}\,\, q_j(A^{\perp })=0\,\,\forall j\in \{1,\ldots ,m\}, \end{aligned}$$
where \(A^{\perp }\) denotes the orthogonal complement of \(A\) with respect to the usual scalar product in \({\mathbb {F}}_q\).

This problem underlies the security of the first public-key quantum money scheme that is proved to be cryptographically secure under a non quantum but classic hardness assumption. This scheme was proposed by S. Aaronson and P. Christiano [1] at STOC’12. In particular, it depends upon the hardness of \({\mathrm{HSP}}_2\). More generally, Aaronson and Christiano left as an open problem to study the security of the scheme for a general field \({\mathbb {F}}_q\). We present a randomized polynomial-time algorithm that solves the \({\mathrm{HSP}}_q\) for \(q>d\) with success probability \(\approx 1-1/q\). So, the quantum money scheme extended to \({\mathbb {F}}_q\) is not secure for big \(q\). Finally, based on experimental results and a structural property of the polynomials that we prove, we conjecture that there is also a randomized polynomial-time algorithm solving the \({\mathrm{HSP}}_2\) with high probability. To support our theoretical results we also present several experimental results confirming that our algorithms are very efficient in practice. We emphasize that [1] proposes a non-noisy and a noisy version of the public-key quantum money scheme. The noisy version of the quantum money scheme remains secure.

Keywords

Success Probability Vector Subspace Homogeneous Component Algebraic Attack Noisy Version 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© International Association for Cryptologic Research 2015

Authors and Affiliations

  1. 1.Institute of Physical and Information Technologies (ITEFI) – Spanish National Research Council (CSIC)MadridSpain
  2. 2.Sorbonne Universités, UPMC Univ Paris 06, POLSYS, UMR 7606, LIP6ParisFrance
  3. 3.INRIA, Paris-Rocquencourt Center, POLSYS ProjectParisFrance
  4. 4.CNRS, UMR 7606, LIP6ParisFrance

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