Finite fields
Handbook of Applied Cryptography
Handbook of Applied Cryptography
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
QUARTZ, 128-Bit Long Digital Signatures
CT-RSA 2001 Proceedings of the 2001 Conference on Topics in Cryptology: The Cryptographer's Track at RSA
FLASH, a Fast Multivariate Signature Algorithm
CT-RSA 2001 Proceedings of the 2001 Conference on Topics in Cryptology: The Cryptographer's Track at RSA
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 Oil & Vinegar Signature Scheme
CRYPTO '98 Proceedings of the 18th Annual International Cryptology Conference on Advances in Cryptology
Unbalanced oil and vinegar signature schemes
EUROCRYPT'99 Proceedings of the 17th 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
Essential Algebraic Structure within the AES
CRYPTO '02 Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology
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
Generic Attacks and the Security of Quartz
PKC '03 Proceedings of the 6th International Workshop on Theory and Practice in Public Key Cryptography: Public Key Cryptography
Survey and benchmark of block ciphers for wireless sensor networks
ACM Transactions on Sensor Networks (TOSN)
On the security of stepwise triangular systems
Designs, Codes and Cryptography
Multivariates Polynomials for Hashing
Information Security and Cryptology
Learning Fuzzy Rules with Evolutionary Algorithms -- An Analytic Approach
Proceedings of the 10th international conference on Parallel Problem Solving from Nature: PPSN X
QUAD: A multivariate stream cipher with provable security
Journal of Symbolic Computation
l-invertible cycles for multivariate quadratic (MQ) public key cryptography
PKC'07 Proceedings of the 10th international conference on Practice and theory in public-key cryptography
Efficient GF(pm) arithmetic architectures for cryptographic applications
CT-RSA'03 Proceedings of the 2003 RSA conference on The cryptographers' track
On building hash functions from multivariate quadratic equations
ACISP'07 Proceedings of the 12th Australasian conference on Information security and privacy
Analysis of multivariate hash functions
ICISC'07 Proceedings of the 10th international conference on Information security and cryptology
Building secure tame-like multivariate public-key cryptosystems: the new TTS
ACISP'05 Proceedings of the 10th Australasian conference on Information Security and Privacy
Tractable rational map signature
PKC'05 Proceedings of the 8th international conference on Theory and Practice in Public Key Cryptography
Efficient cryptanalysis of RSE(2)PKC and RSSE(2)PKC
SCN'04 Proceedings of the 4th international conference on Security in Communication Networks
A study of the security of unbalanced oil and vinegar signature schemes
CT-RSA'05 Proceedings of the 2005 international conference on Topics in Cryptology
QUAD: a practical stream cipher with provable security
EUROCRYPT'06 Proceedings of the 24th annual international conference on The Theory and Applications of Cryptographic Techniques
Implementing minimized multivariate PKC on low-resource embedded systems
SPC'06 Proceedings of the Third international conference on Security in Pervasive Computing
Solving underdetermined systems of multivariate quadratic equations revisited
PKC'12 Proceedings of the 15th international conference on Practice and Theory in Public Key Cryptography
Hidden bits approach for authentication in RFID systems
RFIDSec'12 Proceedings of the 8th international conference on Radio Frequency Identification: security and privacy issues
Most Tensor Problems Are NP-Hard
Journal of the ACM (JACM)
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The security of several recent digital signature schemes is based on the difficulty of solving large systems of quadratic multivariate polynomial equations over a finite field F. This problem, sometimes called MQ, is known to be NP-hard. When the number m of equations is equal to the number n of variables, and if n n 驴 m, the techniques of relinearization and XL, due to A. Shamir et. al., have shown to be successful for solving MQ. In signature schemes, we usually have n 驴 m. For example signature schemes Flash and Sflash submitted to Nessie call for primitives or the UOV scheme published at Eurocrypt 1999. Little is known about the security of such underdefined systems.In this paper, three new and different methods are presented for solving underdefined multivariate systems of quadratic equations. As already shown at Eurocrypt 1999, the problem MQ becomes polynomial when n 驴 m(m+1) for fields F of characteristic 2. We show that for any field, for about n 驴 2m/7(m + 1), exponential but quite small in practice, the problem becomes polynomial in n.When n 驴 m the complexity of all our 3 algorithms tends to qm. However for practical instances of cryptosystems with n 驴 O(m), we show how to achieve complexities significantly lower than exhaustive search. For example we are able break Unbalanced Oil and Vinegar signature schemes for some "bad" choices of the parameters (but not for the parameters proposed in [4]).