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Solving Linear Equations Modulo Divisors: On Factoring Given Any Bits

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Part of the Lecture Notes in Computer Science book series (LNSC,volume 5350)

Abstract

We study the problem of finding solutions to linear equations modulo an unknown divisor p of a known composite integer N. An important application of this problem is factorization of N with given bits of p. It is well-known that this problem is polynomial-time solvable if at most half of the bits of p are unknown and if the unknown bits are located in one consecutive block. We introduce an heuristic algorithm that extends factoring with known bits to an arbitrary number n of blocks. Surprisingly, we are able to show that ln (2) ≈ 70% of the bits are sufficient for any n in order to find the factorization. The algorithm’s running time is however exponential in the parameter n. Thus, our algorithm is polynomial time only for \(n = {\mathcal O}(\log\log N)\) blocks.

Keywords

  • Lattices
  • small roots
  • factoring with known bits

This research was supported by the German Research Foundation (DFG) as part of the project MA 2536/3-1.

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Herrmann, M., May, A. (2008). Solving Linear Equations Modulo Divisors: On Factoring Given Any Bits. In: Pieprzyk, J. (eds) Advances in Cryptology - ASIACRYPT 2008. ASIACRYPT 2008. Lecture Notes in Computer Science, vol 5350. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89255-7_25

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  • DOI: https://doi.org/10.1007/978-3-540-89255-7_25

  • Publisher Name: Springer, Berlin, Heidelberg

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