Coppersmith's lattices and “focus groups”: An attack on small-exponent RSA

Stephen D. Miller, Bhargav Narayanan, Ramarathnam Venkatesan

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


We present a principled technique for reducing the lattice and matrix size in some applications of Coppersmith's lattice method for finding roots of modular polynomial equations. It relies on extrapolating patterns from the actual behavior of Coppersmith's attack for smaller parameter sizes, which can be thought of as “focus group” testing. When applied to the small-exponent RSA problem, our technique reduces lattice dimensions and consequently running times, and hence can be applied to a wider range of exponents. Moreover, in many difficult examples our attack is not only faster but also more successful in recovering the RSA secret key. We include a discussion of subtleties concerning whether or not existing metrics (such as enabling condition bounds) are decisive in predicting the true efficacy of attacks based on Coppersmith's method. Finally, indications are given which suggest certain lattice basis reduction algorithms (such as Nguyen-Stehlé's L2) may be particularly well-suited for Coppersmith's method.

Original languageEnglish (US)
Pages (from-to)376-392
Number of pages17
JournalJournal of Number Theory
StatePublished - May 2021

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


  • Coppersmith's method
  • Factoring
  • Lattice attacks
  • Lattice basis reduction
  • Small exponent RSA


Dive into the research topics of 'Coppersmith's lattices and “focus groups”: An attack on small-exponent RSA'. Together they form a unique fingerprint.

Cite this