Paper 2013/512
Rounding LLL: Finding Faster Small Roots of Univariate Polynomial Congruences
Jingguo Bi and Phong Q. Nguyen
Abstract
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.
Metadata
- Available format(s)
-
PDF
- Category
- Public-key cryptography
- Publication info
- Preprint.
- Keywords
- Coppersmith's AlgorithmSmall Roots of Polynomial EquationsLLLComplexitySpeedupRSA.
- Contact author(s)
- pnguyen @ di ens fr
- History
- 2013-08-17: received
- Short URL
- https://ia.cr/2013/512
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2013/512,
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},
url = {https://eprint.iacr.org/2013/512}
}
