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TitleAlgebraic Cryptanalysis of Hidden Field Equation (HFE) Cryptosystems Using Gröbner Bases
Author(s) Jean-Charles Faugère, Antoine Joux
TypeBook, Chapter in Book, Conference Proceeding
AbstractIn this paper, we review and explain the existing algebraic cryptanalysis of multivariate cryptosystems from the hidden field equation (HFE) family. These cryptanalysis break cryptosystems in the HFE family by solving multivariate systems of equations. In this paper we present a new and efficient attack of this cryptosystem based on fast algorithms for computing Gröbner basis. In particular it was was possible to break the first HFE challenge (80 bits) in only two days of CPU time by using the new algorithm F5 implemented in C.
From a theoretical point of view we study the algebraic properties of the equations produced by instance of the HFE cryptosystems and show why they yield systems of equations easier to solve than random systems of quadratic equations of the same sizes. Moreover we are able to bound the maximal degree occuring in the Gröbner basis computation.
As a consequence, we gain a deeper understanding of the algebraic cryptanalysis against these cryptosystems. We use this understanding to devise a specific algorithm based on sparse linear algebra. In general, we conclude that the cryptanalysis of HFE can be performed in polynomial time. We also revisit the security estimates for existing schemes in the HFE family.
URL dx.doi.org/10.1007/b11817
JournalLecture Notes in Computer Science
PublisherSpringer-Verlag GmbH
Translation No
Refereed No
BookAdvances in Cryptology - CRYPTO 2003
ConferencenameCRYPTO 2003