### Abstract

Lattice-based signature schemes following the Goldreich-Goldwasser-Halevi (GGH) design have the unusual property that each signature leaks information on the signer's secret key, but this does not necessarily imply that such schemes are insecure. At Eurocrypt '03, Szydlo proposed a potential attack by showing that the leakage reduces the key-recovery problem to that of distinguishing integral quadratic forms. He proposed a heuristic method to solve the latter problem, but it was unclear whether his method could attack real-life parameters of GGH and NTRUSIGN. Here, we propose an alternative method to attack signature schemes à la GGH, by studying the following learning problem: given many random points uniformly distributed over an unknown n-dimensional parallelepiped, recover the parallelepiped or an approximation thereof. We transform this problem into a multivariate optimization problem that can be solved by a gradient descent. Our approach is very effective in practice: we present the first succesful key-recovery experiments on NTRUSIGN-251 without perturbation, as proposed in half of the parameter choices in NTRU standards under consideration by IEEE P1363.1. Experimentally, 90,000 signatures are sufficient to recover the NTRUSIGN-251 secret key. We are also able to recover the secret key in the signature analogue of all the GGH encryption challenges, using a number of signatures which is roughly quadratic in the lattice dimension.

Original language | English (US) |
---|---|

Title of host publication | Advances in Cryptology - EUROCRYPT 2006 - 24th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings |

Pages | 271-288 |

Number of pages | 18 |

Volume | 4004 LNCS |

State | Published - 2006 |

Event | 24th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2006 - St. Petersburg, Russian Federation Duration: May 28 2006 → Jun 1 2006 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 4004 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 24th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2006 |
---|---|

Country | Russian Federation |

City | St. Petersburg |

Period | 5/28/06 → 6/1/06 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

*Advances in Cryptology - EUROCRYPT 2006 - 24th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings*(Vol. 4004 LNCS, pp. 271-288). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4004 LNCS).

**Learning a parallelepiped : Cryptanalysis of GGH and NTRU signatures.** / Nguyen, Phong Q.; Regev, Oded.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Advances in Cryptology - EUROCRYPT 2006 - 24th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings.*vol. 4004 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4004 LNCS, pp. 271-288, 24th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2006, St. Petersburg, Russian Federation, 5/28/06.

}

TY - GEN

T1 - Learning a parallelepiped

T2 - Cryptanalysis of GGH and NTRU signatures

AU - Nguyen, Phong Q.

AU - Regev, Oded

PY - 2006

Y1 - 2006

N2 - Lattice-based signature schemes following the Goldreich-Goldwasser-Halevi (GGH) design have the unusual property that each signature leaks information on the signer's secret key, but this does not necessarily imply that such schemes are insecure. At Eurocrypt '03, Szydlo proposed a potential attack by showing that the leakage reduces the key-recovery problem to that of distinguishing integral quadratic forms. He proposed a heuristic method to solve the latter problem, but it was unclear whether his method could attack real-life parameters of GGH and NTRUSIGN. Here, we propose an alternative method to attack signature schemes à la GGH, by studying the following learning problem: given many random points uniformly distributed over an unknown n-dimensional parallelepiped, recover the parallelepiped or an approximation thereof. We transform this problem into a multivariate optimization problem that can be solved by a gradient descent. Our approach is very effective in practice: we present the first succesful key-recovery experiments on NTRUSIGN-251 without perturbation, as proposed in half of the parameter choices in NTRU standards under consideration by IEEE P1363.1. Experimentally, 90,000 signatures are sufficient to recover the NTRUSIGN-251 secret key. We are also able to recover the secret key in the signature analogue of all the GGH encryption challenges, using a number of signatures which is roughly quadratic in the lattice dimension.

AB - Lattice-based signature schemes following the Goldreich-Goldwasser-Halevi (GGH) design have the unusual property that each signature leaks information on the signer's secret key, but this does not necessarily imply that such schemes are insecure. At Eurocrypt '03, Szydlo proposed a potential attack by showing that the leakage reduces the key-recovery problem to that of distinguishing integral quadratic forms. He proposed a heuristic method to solve the latter problem, but it was unclear whether his method could attack real-life parameters of GGH and NTRUSIGN. Here, we propose an alternative method to attack signature schemes à la GGH, by studying the following learning problem: given many random points uniformly distributed over an unknown n-dimensional parallelepiped, recover the parallelepiped or an approximation thereof. We transform this problem into a multivariate optimization problem that can be solved by a gradient descent. Our approach is very effective in practice: we present the first succesful key-recovery experiments on NTRUSIGN-251 without perturbation, as proposed in half of the parameter choices in NTRU standards under consideration by IEEE P1363.1. Experimentally, 90,000 signatures are sufficient to recover the NTRUSIGN-251 secret key. We are also able to recover the secret key in the signature analogue of all the GGH encryption challenges, using a number of signatures which is roughly quadratic in the lattice dimension.

UR - http://www.scopus.com/inward/record.url?scp=33746038898&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33746038898&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:33746038898

SN - 3540345469

SN - 9783540345466

VL - 4004 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 271

EP - 288

BT - Advances in Cryptology - EUROCRYPT 2006 - 24th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings

ER -