Gro¨bner bases: a computational approach to commutative algebra
Gro¨bner bases: a computational approach to commutative algebra
Efficient computation of zero-dimensional Gro¨bner bases by change of ordering
Journal of Symbolic Computation
The Gröbner basis algorithm and subresultant theory
ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations
EUROCAL '83 Proceedings of the European Computer Algebra Conference on Computer Algebra
Cryptanalysis of Block Ciphers with Overdefined Systems of Equations
ASIACRYPT '02 Proceedings of the 8th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
CRYPTO 2008 Proceedings of the 28th Annual conference on Cryptology: Advances in Cryptology
MXL2: Solving Polynomial Equations over GF(2) Using an Improved Mutant Strategy
PQCrypto '08 Proceedings of the 2nd International Workshop on Post-Quantum Cryptography
PolyBoRi: A framework for Gröbner-basis computations with Boolean polynomials
Journal of Symbolic Computation
Algebraic Attack on the MQQ Public Key Cryptosystem
CANS '09 Proceedings of the 8th International Conference on Cryptology and Network Security
EUROCRYPT'96 Proceedings of the 15th annual international conference on Theory and application of cryptographic techniques
Efficient algorithms for solving overdefined systems of multivariate polynomial equations
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
Algebraic attacks on stream ciphers with linear feedback
EUROCRYPT'03 Proceedings of the 22nd international conference on Theory and applications of cryptographic techniques
About the XL algorithm over GF(2)
CT-RSA'03 Proceedings of the 2003 RSA conference on The cryptographers' track
Parallel Gaussian elimination for Gröbner bases computations in finite fields
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation
MXL3: an efficient algorithm for computing gröbner bases of zero-dimensional ideals
ICISC'09 Proceedings of the 12th international conference on Information security and cryptology
Algebraic cryptanalysis of curry and flurry using correlated messages
Inscrypt'09 Proceedings of the 5th international conference on Information security and cryptology
Algebraic precomputations in differential and integral cryptanalysis
Inscrypt'10 Proceedings of the 6th international conference on Information security and cryptology
An analysis of the XSL algorithm
ASIACRYPT'05 Proceedings of the 11th international conference on Theory and Application of Cryptology and Information Security
Flexible partial enlargement to accelerate gröbner basis computation over F2
AFRICACRYPT'10 Proceedings of the Third international conference on Cryptology in Africa
An analysis of XSL Applied to BES
FSE'07 Proceedings of the 14th international conference on Fast Software Encryption
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The computation of Grobner bases remains one of the most powerful methods for tackling the Polynomial System Solving (PoSSo) problem. The most efficient known algorithms reduce the Grobner basis computation to Gaussian eliminations on several matrices. However, several degrees of freedom are available to generate these matrices. It is well known that the particular strategies used can drastically affect the efficiency of the computations. In this work, we investigate a recently-proposed strategy, the so-called ''Mutant strategy'', on which a new family of algorithms is based (MXL, MXL"2 and MXL"3). By studying and describing the algorithms based on Grobner basis concepts, we demonstrate that the Mutant strategy can be understood to be equivalent to the classical Normal Selection Strategy currently used in Grobner basis algorithms. Furthermore, we show that the ''partial enlargement'' technique can be understood as a strategy for restricting the number of S-polynomials considered in an iteration of the F"4 Grobner basis algorithm, while the new termination criterion used in MXL"3 does not lead to termination at a lower degree than the classical Gebauer-Moller installation of Buchberger's criteria. We claim that our results map all novel concepts from the MXL family of algorithms to their well-known Grobner basis equivalents. Using previous results that had shown the relation between the original XL algorithm and F"4, we conclude that the MXL family of algorithms can be fundamentally reduced to redundant variants of F"4.