Relation between the XL Algorithm and Gröbner Basis Algorithms

We clarify a relation between the XL algorithm and known Grobner basis algorithms. The XL algorithm was proposed to be a more efficient algorithm to solve a system of algebraic equations under a special condition, without calculating a whole Grobner basis. But in our result, it is shown that to solve a system of algebraic equations with a special condition under which the XL algorithm works is equivalent to calculate the reduced Grobner basis of the ideal associated with the system. Moreover we show that the XL algorithm is a Grobner basis algorithm which can be represented as a redundant variant of a known Grobner basis algorithm F4.

[1]  Nicolas Courtois,et al.  The Security of Hidden Field Equations (HFE) , 2001, CT-RSA.

[2]  J. Faugère.  A new efficient algorithm for computing Gröbner bases (F4) , 1999 .

[3]  Heinz Kredel,et al.  Gröbner Bases: A Computational Approach to Commutative Algebra , 1993 .

[4]  Adi Shamir,et al.  Efficient Algorithms for Solving Overdefined Systems of Multivariate Polynomial Equations , 2000, EUROCRYPT.

[5]  Nicolas Courtois,et al.  Higher Order Correlation Attacks, XL Algorithm and Cryptanalysis of Toyocrypt , 2002, ICISC.

[6]  Antoine Joux,et al.  Algebraic Cryptanalysis of Hidden Field Equation (HFE) Cryptosystems Using Gröbner Bases , 2003, CRYPTO.

[7]  Willi Meier,et al.  Fast Algebraic Attacks on Stream Ciphers with Linear Feedback , 2003, CRYPTO.

[8]  Bruno Buchberger,et al.  A criterion for detecting unnecessary reductions in the construction of Groebner bases , 1979, EUROSAM.

[9]  Jacques Patarin,et al.  Cryptanalysis of the Matsumoto and Imai Public Key Scheme of Eurocrypt'88 , 1995, CRYPTO.

[10]  Hideki Imai,et al.  Relation between XL algorithm and Gröbner Bases Algorithms , 2004, IACR Cryptol. ePrint Arch..

[11]  A. Shamir,et al.  Cryptanalysis of the HFE Public Key Cryptosystem , 1999 .

[12]  Jean-Charles Faugère,et al.  Comparison of XL and Gröbner basis algorithms over Finite Fields , 2004 .

[13]  Nicolas Courtois.  Fast Algebraic Attacks on Stream Ciphers with Linear Feedback , 2003, CRYPTO.

[14]  Daniel Lazard,et al.  Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations , 1983, EUROCAL.

[15]  Josef Pieprzyk,et al.  Cryptanalysis of Block Ciphers with Overdefined Systems of Equations , 2002, ASIACRYPT.

[16]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[17]  Frederik Armknecht,et al.  Algebraic Attacks on Combiners with Memory , 2003, CRYPTO.

[18]  Jean Charles Faugère,et al.  A new efficient algorithm for computing Gröbner bases without reduction to zero (F5) , 2002, ISSAC '02.

[19]  Louis Goubin,et al.  Unbalanced Oil and Vinegar Signature Schemes , 1999, EUROCRYPT.