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Information Security pp 185–196Cite as

Factoring a Multiprime Modulus N with Random Bits

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 7807))

Abstract

In 2009, Heninger and Shacham presented an algorithm using the Hensel’s lemma for reconstructing the prime factors of the modulus \(N = r_1r_2\). This algorithm computes the prime factors of N in polynomial time, with high probability, assuming that a fraction greater than or equal to 59 % random bits of its primes \(r_1\) and \(r_2\) is given. In this paper, we present the analysis of Hensel’s lemma for a multiprime modulus \(N = \prod ^u_{i=1}r_i\) (for \(u\ge 2\)) and we generalise the Heninger and Shacham’s algorithm to determine the minimum fraction of random bits of its prime factors that is sufficient to factor N in polynomial time with high probability.

R.C. Villena—Supported by CAPES, Brazil.

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Notes

  1. 1.

    Basic RSA is when the modulus N is product of two primes.

  2. 2.

    Multi-prime RSA is a generalization of the Basic RSA where the modulus N is the product of two or more primes.

  3. 3.

    N is the product of u primes with the same bit length, as in the Basic RSA.

  4. 4.

    It is an algorithm to factor an integer N with a very good performance.

  5. 5.

    It is an algorithm to compute a non-trivial factor of N.

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Acknowledgment

We thank anonymous referees who pointed out the work by Kogure et al. [9].

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Authors and Affiliations

  1. Department of Computer Science, University of São Paulo, São Paulo, Brazil

    Routo Terada & Reynaldo Cáceres Villena

Authors
  1. Routo Terada
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  2. Reynaldo Cáceres Villena
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Corresponding author

Correspondence to Routo Terada .

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Editors and Affiliations

  1. University of Texas at Dallas, Richardson, Texas, USA

    Yvo Desmedt

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Terada, R., Villena, R.C. (2015). Factoring a Multiprime Modulus N with Random Bits. In: Desmedt, Y. (eds) Information Security. Lecture Notes in Computer Science(), vol 7807. Springer, Cham. https://doi.org/10.1007/978-3-319-27659-5_13

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  • DOI: https://doi.org/10.1007/978-3-319-27659-5_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-27658-8

  • Online ISBN: 978-3-319-27659-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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