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A note on the analysis of Herrmann-May lattices for small exponent RSA.

Abul Kalam1, Sudeshna Karmakar2, Santanu Sarkar3

  • 1Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, Tamil Nadu, India. abulkalam.sunny@gmail.com.

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|August 5, 2025
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Summary

Researchers found an error in a 15-year-old lattice-based cryptanalysis method for small RSA exponents. The corrected bound is significantly lower, and the refined method achieves the Boneh-Durfee bound with reduced lattice dimensions.

Keywords:
Herrmann–May latticeLLLLattice basis reductionLinearizationSmall exponent RSA

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Area of Science:

  • Cryptography
  • Number Theory
  • Computer Science

Background:

  • The Herrmann-May method (PKC 2010) proposed a lattice-based approach for cryptanalysis of RSA with small exponents.
  • This method aimed to achieve a theoretical bound of d < N^(1 - 1/sqrt(2)) using unravelled linearization.

Purpose of the Study:

  • To identify and correct an error in the asymptotic analysis of the Herrmann-May lattice-based method.
  • To refine the Herrmann-May lattice construction for improved cryptanalytic bounds.
  • To reassess the security implications for RSA with small exponents.

Main Methods:

  • Detailed analysis of the asymptotic behavior of the Herrmann-May lattice construction.
  • Revision of the mathematical bounds derived from the lattice-based cryptanalysis.
  • Modification of the lattice construction to improve efficiency and achieve tighter bounds.

Main Results:

  • An error in the original asymptotic analysis of the Herrmann-May method was identified, correcting the bound to d < N^0.292256.
  • This revised bound is strictly lower than the previously established Boneh-Durfee bound.
  • The Herrmann-May lattice construction was refined, successfully achieving the Boneh-Durfee bound with a reduced lattice dimension.

Conclusions:

  • The security analysis of RSA with small exponents needs to account for the corrected, tighter bound.
  • The refined lattice construction offers a more efficient approach to cryptanalysis, matching the Boneh-Durfee bound.
  • This work corrects a long-standing error and advances the understanding of lattice-based cryptanalytic techniques.