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DOI: 10.4230/LIPIcs.MFCS.2016.6
URN: urn:nbn:de:0030-drops-64234
URL: http://drops.dagstuhl.de/opus/volltexte/2016/6423/
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Agrawal, Manindra ; Saxena, Nitin ; Srivastava, Shubham Sahai

Integer Factoring Using Small Algebraic Dependencies

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LIPIcs-MFCS-2016-6.pdf (0.5 MB)


Abstract

Integer factoring is a curious number theory problem with wide applications in complexity and cryptography. The best known algorithm to factor a number n takes time, roughly, exp(2*log^{1/3}(n)*log^{2/3}(log(n))) (number field sieve, 1989). One basic idea used is to find two squares, possibly in a number field, that are congruent modulo n. Several variants of this idea have been utilized to get other factoring algorithms in the last century. In this work we intend to explore new ideas towards integer factoring. In particular, we adapt the AKS primality test (2004) ideas for integer factoring. In the motivating case of semiprimes n=pq, i.e. p<q are primes, we exploit the difference in the two Frobenius morphisms (one over F_p and the other over F_q) to factor n in special cases. Specifically, our algorithm is polynomial time (on number theoretic conjectures) if we know a small algebraic dependence between p,q. We discuss families of n where our algorithm is significantly faster than the algorithms based on known techniques.

BibTeX - Entry

@InProceedings{agrawal_et_al:LIPIcs:2016:6423,
  author =	{Manindra Agrawal and Nitin Saxena and Shubham Sahai Srivastava},
  title =	{{Integer Factoring Using Small Algebraic Dependencies}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{6:1--6:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Piotr Faliszewski and Anca Muscholl and Rolf Niedermeier},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6423},
  URN =		{urn:nbn:de:0030-drops-64234},
  doi =		{10.4230/LIPIcs.MFCS.2016.6},
  annote =	{Keywords: integer, factorization, factoring integers, algebraic dependence, dependencies}
}

Keywords: integer, factorization, factoring integers, algebraic dependence, dependencies
Seminar: 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)
Issue Date: 2016
Date of publication: 19.08.2016


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