### Abstract

The “ Gluing Algorithm” of Semaev (Des. Codes Cryptogr. 49:47–60, 2008)-that finds all solutions of a sparse system of linear equations over the Galois field GF(q)-has average running time O(mq^{maxǀUk1Xjǀ-k}) where m is the total number of equations, and U^{k} _{1}X_{j} is the set of all unknowns actively occurring in the first k equations. In order to make the implementation of the algorithm faster, our goal here is to minimize the exponent of q in the case where every equation contains at most three unknowns. The main result states that if the total number ǀU^{m} _{1}X_{j}ǀ of unknowns is equal to m, then the best achievable exponent is between c_{1}m and c_{2}m for some positive constants c_{1} and c_{2}.

Original language | English |
---|---|

Pages (from-to) | 175-185 |

Number of pages | 11 |

Journal | Designs, Codes, and Cryptography |

Volume | 75 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

### Fingerprint

### Keywords

- Hypergraph ordering
- Sparse systems of Boolean equations

### ASJC Scopus subject areas

- Applied Mathematics
- Computer Science Applications

### Cite this

*Designs, Codes, and Cryptography*,

*75*(1), 175-185. https://doi.org/10.1007/s10623-013-9899-z

**Speeding up deciphering by hypergraph ordering.** / Horak, Peter; Tuza, Zsolt.

Research output: Contribution to journal › Article

*Designs, Codes, and Cryptography*, vol. 75, no. 1, pp. 175-185. https://doi.org/10.1007/s10623-013-9899-z

}

TY - JOUR

T1 - Speeding up deciphering by hypergraph ordering

AU - Horak, Peter

AU - Tuza, Zsolt

PY - 2015

Y1 - 2015

N2 - The “ Gluing Algorithm” of Semaev (Des. Codes Cryptogr. 49:47–60, 2008)-that finds all solutions of a sparse system of linear equations over the Galois field GF(q)-has average running time O(mqmaxǀUk1Xjǀ-k) where m is the total number of equations, and Uk 1Xj is the set of all unknowns actively occurring in the first k equations. In order to make the implementation of the algorithm faster, our goal here is to minimize the exponent of q in the case where every equation contains at most three unknowns. The main result states that if the total number ǀUm 1Xjǀ of unknowns is equal to m, then the best achievable exponent is between c1m and c2m for some positive constants c1 and c2.

AB - The “ Gluing Algorithm” of Semaev (Des. Codes Cryptogr. 49:47–60, 2008)-that finds all solutions of a sparse system of linear equations over the Galois field GF(q)-has average running time O(mqmaxǀUk1Xjǀ-k) where m is the total number of equations, and Uk 1Xj is the set of all unknowns actively occurring in the first k equations. In order to make the implementation of the algorithm faster, our goal here is to minimize the exponent of q in the case where every equation contains at most three unknowns. The main result states that if the total number ǀUm 1Xjǀ of unknowns is equal to m, then the best achievable exponent is between c1m and c2m for some positive constants c1 and c2.

KW - Hypergraph ordering

KW - Sparse systems of Boolean equations

UR - http://www.scopus.com/inward/record.url?scp=84925292339&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84925292339&partnerID=8YFLogxK

U2 - 10.1007/s10623-013-9899-z

DO - 10.1007/s10623-013-9899-z

M3 - Article

VL - 75

SP - 175

EP - 185

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

SN - 0925-1022

IS - 1

ER -