## A strategy for recovering roots of bivariate polynomials modulo a

### BibTeX

@MISC{Bustillo_astrategy,

author = {Paula Bustillo and Domingo Gómez and Jaime Gutierrez and Álvar Ibeas},

title = {A strategy for recovering roots of bivariate polynomials modulo a},

year = {}

}

### OpenURL

### Abstract

prime

### Citations

1176 |
Geometric algorithms and combinatorial optimization
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- 1988
(Show Context)
Citation Context ...nomials. 2 Preliminaries 2.1 Closest Vector Problem in Lattices This brief introduction is given in order to keep this article auto-contained. For more details and references, we recommend consulting =-=[10, 14, 18, 19, 20]-=-. Let {b1, . . . , bs} be a set of linearly independent vectors in R r . The set L = {c1b1 + . . . + csbs : c1, . . . , cs ∈ Z} is called an s-dimensional lattice with basis {b1, . . . , bs}. One basi... |

244 |
On Lovász lattice reduction and the nearest lattice point problem
- Babai
- 1986
(Show Context)
Citation Context ...e dimension of L is fixed (see [15], for example). For the slightly weaker task of finding a vector whose distance to the target approximates the smallest possible, we use a result which follows from =-=[1]-=-, and which is based on the celebrated LLL algorithm of Lenstra, Lenstra and Lovász [17]. 2Lemma 1 There exists a polynomial time algorithm which, when given an s-dimensional lattice L and a vector t... |

188 |
Small Solutions to Polynomial Equations, and Low Exponent Vulnerabilities
- Coppersmith
- 1996
(Show Context)
Citation Context ... case of the problem of finding small solutions of multivariate polynomial congruences. For polynomial congruences in one variable, an algorithm for solving this problem has been given by Coppersmith =-=[6]-=-, see also [3, 8, 7, 12, 13]. However, in the general case only heuristic results are known. Here we are able to obtain rigorous results for a big class of irreducible bivariate polynomials modulo a p... |

34 |
Finding small solutions to small degree polynomials
- COPPERSMITH
(Show Context)
Citation Context ...roblem of finding small solutions of multivariate polynomial congruences. For polynomial congruences in one variable, an algorithm for solving this problem has been given by Coppersmith [6], see also =-=[3, 8, 7, 12, 13]-=-. However, in the general case only heuristic results are known. Here we are able to obtain rigorous results for a big class of irreducible bivariate polynomials modulo a prime number. The remainder o... |

32 |
Reconstructing truncated integer variables satisfying linear congruences
- Frieze, Hastad, et al.
- 1988
(Show Context)
Citation Context ...question has applications to, and has been motivated by, the predictability problem for non-linear pseudorandom number generators over Fp and the linear congruential generator on elliptic curves (see =-=[2, 4, 5, 9, 11, 14, 16]-=-). 1The task we solve can be considered as a special case of the problem of finding small solutions of multivariate polynomial congruences. For polynomial congruences in one variable, an algorithm fo... |

26 |
Inferring sequences produced by pseudo-random number generators
- Boyar
- 1989
(Show Context)
Citation Context ...question has applications to, and has been motivated by, the predictability problem for non-linear pseudorandom number generators over Fp and the linear congruential generator on elliptic curves (see =-=[2, 4, 5, 9, 11, 14, 16]-=-). 1The task we solve can be considered as a special case of the problem of finding small solutions of multivariate polynomial congruences. For polynomial congruences in one variable, an algorithm fo... |

17 | Finding Small Roots of Bivariate Integer Polynomial Equations Revisited
- Coron
- 2004
(Show Context)
Citation Context ...roblem of finding small solutions of multivariate polynomial congruences. For polynomial congruences in one variable, an algorithm for solving this problem has been given by Coppersmith [6], see also =-=[3, 8, 7, 12, 13]-=-. However, in the general case only heuristic results are known. Here we are able to obtain rigorous results for a big class of irreducible bivariate polynomials modulo a prime number. The remainder o... |

16 |
A strategy for finding roots of multivariate polynomials with new applications in attacking rsa variants
- Jochemsz, May
- 2006
(Show Context)
Citation Context ...roblem of finding small solutions of multivariate polynomial congruences. For polynomial congruences in one variable, an algorithm for solving this problem has been given by Coppersmith [6], see also =-=[3, 8, 7, 12, 13]-=-. However, in the general case only heuristic results are known. Here we are able to obtain rigorous results for a big class of irreducible bivariate polynomials modulo a prime number. The remainder o... |

12 | A Tool Kit for Finding Small Roots of Bivariate Polynomials over the Integers
- Blömer, May
(Show Context)
Citation Context |

12 | The Modular Inversion Hidden Number Problem
- Boneh, Halevi, et al.
- 2001
(Show Context)
Citation Context ...question has applications to, and has been motivated by, the predictability problem for non-linear pseudorandom number generators over Fp and the linear congruential generator on elliptic curves (see =-=[2, 4, 5, 9, 11, 14, 16]-=-). 1The task we solve can be considered as a special case of the problem of finding small solutions of multivariate polynomial congruences. For polynomial congruences in one variable, an algorithm fo... |

1 |
and Álvar Ibeas. Inferring sequences produced by a linear congruential generator on elliptic curves missing high-order bits. Des
- Gutierrez
- 2007
(Show Context)
Citation Context |