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DOI: 10.4230/LIPIcs.ICALP.2016.16
URN: urn:nbn:de:0030-drops-62985
URL: http://drops.dagstuhl.de/opus/volltexte/2016/6298/
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Childs, Andrew M. ; van Dam, Wim ; Hung, Shih-Han ; Shparlinski, Igor E.

Optimal Quantum Algorithm for Polynomial Interpolation

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LIPIcs-ICALP-2016-16.pdf (0.5 MB)


Abstract

We consider the number of quantum queries required to determine the coefficients of a degree-d polynomial over F_q. A lower bound shown independently by Kane and Kutin and by Meyer and Pommersheim shows that d/2 + 1/2 quantum queries are needed to solve this problem with bounded error, whereas an algorithm of Boneh and Zhandry shows that d quantum queries are sufficient. We show that the lower bound is achievable: d/2 + 1/2 quantum queries suffice to determine the polynomial with bounded error. Furthermore, we show that d/2 + 1 queries suffice to achieve probability approaching 1 for large q. These upper bounds improve results of Boneh and Zhandry on the insecurity of cryptographic protocols against quantum attacks. We also show that our algorithmís success probability as a function of the number of queries is precisely optimal. Furthermore, the algorithm can be implemented with gate complexity poly(log(q)) with negligible decrease in the success probability. We end with a conjecture about the quantum query complexity of multivariate polynomial interpolation.

BibTeX - Entry

@InProceedings{childs_et_al:LIPIcs:2016:6298,
  author =	{Andrew M. Childs and Wim van Dam and Shih-Han Hung and Igor E. Shparlinski},
  title =	{{Optimal Quantum Algorithm for Polynomial Interpolation}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{16:1--16:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6298},
  URN =		{urn:nbn:de:0030-drops-62985},
  doi =		{10.4230/LIPIcs.ICALP.2016.16},
  annote =	{Keywords: Quantum algorithms, query complexity, polynomial interpolation, finite fields}
}

Keywords: Quantum algorithms, query complexity, polynomial interpolation, finite fields
Seminar: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 17.08.2016


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