# Improved algebraic cryptanalysis of QUAD, Bivium and Trivium via graph partitioning on equation systems

Wong, Kenneth Koon-Ho & Bard, Gregory V.
(2010)
Improved algebraic cryptanalysis of QUAD, Bivium and Trivium via graph partitioning on equation systems. In
*Information Security and Privacy : Proceedings of the 15th Australasian Conference, ACISP 2010*, Springer, Macquarie Graduate School of Management, Sydney, pp. 19-36.

Conference Paper
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## Abstract

We present a novel approach for preprocessing systems of polynomial equations via graph partitioning. The variable-sharing graph of a system of polynomial equations is defined. If such graph is disconnected, then the corresponding system of equations can be split into smaller ones that can be solved individually. This can provide a tremendous speed-up in computing the solution to the system, but is unlikely to occur either randomly or in applications. However, by deleting certain vertices on the graph, the variable-sharing graph could be disconnected in a balanced fashion, and in turn the system of polynomial equations would be separated into smaller systems of near-equal sizes. In graph theory terms, this process is equivalent to finding balanced vertex partitions with minimum-weight vertex separators. The techniques of finding these vertex partitions are discussed, and experiments are performed to evaluate its practicality for general graphs and systems of polynomial equations. Applications of this approach in algebraic cryptanalysis on symmetric ciphers are presented: For the QUAD family of stream ciphers, we show how a malicious party can manufacture conforming systems that can be easily broken. For the stream ciphers Bivium and Trivium, we nachieve significant speedups in algebraic attacks against them, mainly in a partial key guess scenario. In each of these cases, the systems of polynomial equations involved are well-suited to our graph partitioning method. These results may open a new avenue for evaluating the security of symmetric ciphers against algebraic attacks.

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ID Code: | 34332 |
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Item Type: | Conference Paper |

Refereed: | Yes |

Additional Information: | Springer Series: Lecture Notes in Computer Science |

Additional URLs: | |

Keywords: | algebraic attacks, stream ciphers, Trivium, QUAD, graph partitioning, vertex separators, polynomial equations |

ISBN: | 9783642140808 |

Subjects: | Australian and New Zealand Standard Research Classification > INFORMATION AND COMPUTING SCIENCES (080000) > COMPUTATION THEORY AND MATHEMATICS (080200) > Applied Discrete Mathematics (080202) Australian and New Zealand Standard Research Classification > INFORMATION AND COMPUTING SCIENCES (080000) > COMPUTATION THEORY AND MATHEMATICS (080200) > Numerical Computation (080205) Australian and New Zealand Standard Research Classification > INFORMATION AND COMPUTING SCIENCES (080000) > DATA FORMAT (080400) > Data Encryption (080402) |

Divisions: | Past > Institutes > Information Security Institute |

Copyright Owner: | Copyright 2010 Springer |

Copyright Statement: | This is the author-version of the work. Conference proceedings published, by Springer Verlag, will be available via Lecture Notes in Computer Science http://www.springer.de/comp/lncs/ |

Deposited On: | 06 Sep 2010 01:31 |

Last Modified: | 18 Jul 2014 05:18 |

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