Paper 2013/512

Rounding LLL: Finding Faster Small Roots of Univariate Polynomial Congruences

Jingguo Bi and Phong Q. Nguyen


In a seminal work at EUROCRYPT '96, Coppersmith showed how to find all small roots of a univariate polynomial congruence in polynomial time: this has found many applications in public-key cryptanalysis and in a few security proofs. However, the running time of the algorithm is a high-degree polynomial, which limits experiments: the bottleneck is an LLL reduction of a high-dimensional matrix with extra-large coefficients. We present in this paper a polynomial speedup over Coppersmith's algorithm. Our improvement is based on a special property of the matrices used by Coppersmith's algorithm, which allows us to speed up the LLL reduction by rounding. The exact speedup depends on the LLL algorithm used: for instance, the speedup is quadratic in the bit-size of the small-root bound if one uses the Nguyen-Stehlé L^2 algorithm.

Available format(s)
Public-key cryptography
Publication info
Coppersmith's AlgorithmSmall Roots of Polynomial EquationsLLLComplexitySpeedupRSA.
Contact author(s)
pnguyen @ di ens fr
2013-08-17: received
Short URL
Creative Commons Attribution


      author = {Jingguo Bi and Phong Q.  Nguyen},
      title = {Rounding LLL:  Finding Faster Small Roots of Univariate Polynomial Congruences},
      howpublished = {Cryptology ePrint Archive, Paper 2013/512},
      year = {2013},
      note = {\url{}},
      url = {}
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