Computer algebra: symbolic and algebraic computation (2nd ed.)
Computer algebra: symbolic and algebraic computation (2nd ed.)
Efficient computation of zero-dimensional Gro¨bner bases by change of ordering
Journal of Symbolic Computation
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization
CRYPTO '99 Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology
Cryptanalysis of the TTM Cryptosystem
ASIACRYPT '00 Proceedings of the 6th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Efficient Zero-Knowledge Authentication Based on a Linear Algebra Problem MinRank
ASIACRYPT '01 Proceedings of the 7th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Key recovery on hidden monomial multivariate schemes
EUROCRYPT'08 Proceedings of the theory and applications of cryptographic techniques 27th annual international conference on Advances in cryptology
Computing minimal multi-homogeneous bézout numbers is hard
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
Polynomial equivalence problems: algorithmic and theoretical aspects
EUROCRYPT'06 Proceedings of the 24th annual international conference on The Theory and Applications of Cryptographic Techniques
Solving Structured Polynomial Systems and Applications to Cryptology
CASC '09 Proceedings of the 11th International Workshop on Computer Algebra in Scientific Computing
Geometric Ideas for Cryptographic Equation Solving in Even Characteristic
Cryptography and Coding '09 Proceedings of the 12th IMA International Conference on Cryptography and Coding
FGb: a library for computing Gröbner bases
ICMS'10 Proceedings of the Third international congress conference on Mathematical software
Journal of Symbolic Computation
Algebraic cryptanalysis of curry and flurry using correlated messages
Inscrypt'09 Proceedings of the 5th international conference on Information security and cryptology
CT-RSA'11 Proceedings of the 11th international conference on Topics in cryptology: CT-RSA 2011
Cryptanalysis of multivariate and odd-characteristic HFE variants
PKC'11 Proceedings of the 14th international conference on Practice and theory in public key cryptography conference on Public key cryptography
Algebraic precomputations in differential and integral cryptanalysis
Inscrypt'10 Proceedings of the 6th international conference on Information security and cryptology
Algebraic cryptanalysis of mceliece variants with compact keys
EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques
General fault attacks on multivariate public key cryptosystems
PQCrypto'11 Proceedings of the 4th international conference on Post-Quantum Cryptography
Roots of square: cryptanalysis of double-layer square and square+
PQCrypto'11 Proceedings of the 4th international conference on Post-Quantum Cryptography
On the relation between the MXL family of algorithms and Gröbner basis algorithms
Journal of Symbolic Computation
Improving the complexity of index calculus algorithms in elliptic curves over binary fields
EUROCRYPT'12 Proceedings of the 31st Annual international conference on Theory and Applications of Cryptographic Techniques
Feasibility of position-based multivariate cryptosystems for WSN
International Journal of Internet Technology and Secured Transactions
Cryptanalysis of enhanced TTS, STS and all its variants, or: why cross-terms are important
AFRICACRYPT'12 Proceedings of the 5th international conference on Cryptology in Africa
Quo vadis quaternion? cryptanalysis of rainbow over non-commutative rings
SCN'12 Proceedings of the 8th international conference on Security and Cryptography for Networks
Solving polynomial systems over finite fields: improved analysis of the hybrid approach
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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In this paper, we investigate the difficulty of one of the most relevant problems in multivariate cryptography --- namely MinRank --- about which no real progress has been reported since [9, 19]. Our starting point is the Kipnis-Shamir attack [19]. We first show new properties of the ideal generated by Kipnis-Shamir's equations. We then propose a new modeling of the problem. Concerning the practical resolution, we adopt a Gröbner basis approach that permitted us to actually solve challenges A and B proposed by Courtois in [8]. Using the multi-homogeneous structure of the algebraic system, we have been able to provide a theoretical complexity bound reflecting the practical behavior of our approach. Namely, when r茂戮驴the dimension of the matrices minus the rank of the target matrix in the MinRank problem is constant, then we have a polynomial time attack $\mathcal{O}\left( \ln\left( q\right) \,n^{3\,r^{\prime2}}\right) $. For the challenge C [8], we obtain a theoretical bound of 266.3operations.