Computer algebra: symbolic and algebraic computation (2nd ed.)
Computer algebra: symbolic and algebraic computation (2nd ed.)
Analysis of a public key approach based on polynomial substitution
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
Public quadratic polynomial-tuples for efficient signature-verification and message-encryption
Lecture Notes in Computer Science on Advances in Cryptology-EUROCRYPT'88
Efficient computation of zero-dimensional Gro¨bner bases by change of ordering
Journal of Symbolic Computation
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Algebraic aspects of cryptography
Algebraic aspects of cryptography
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
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
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
EUROCRYPT'96 Proceedings of the 15th annual international conference on Theory and application of cryptographic techniques
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
Practical cryptanalysis of SFLASH
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
Tractable rational map signature
PKC'05 Proceedings of the 8th international conference on Theory and Practice in Public Key Cryptography
Cryptanalysis of the tractable rational map cryptosystem
PKC'05 Proceedings of the 8th international conference on Theory and Practice in Public Key Cryptography
Polynomial equivalence problems: algorithmic and theoretical aspects
EUROCRYPT'06 Proceedings of the 24th annual international conference on The Theory and Applications of Cryptographic Techniques
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
Security Analysis of Multivariate Polynomials for Hashing
Information Security and Cryptology
Interactions between computer algebra (Gröbner bases) and cryptology
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
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In this paper, we investigate the security of the Tractable Rationale Maps Signature (TRMS) signature scheme [9] proposed at PKC'05. To do so, we present a hybrid approach for solving the algebraic systems naturally arising when mounting a signature-forgery attack. The basic idea is to compute Gröbner bases of several modified systems rather than a Gröbner basis of the initial system. We have been able to provide a precise bound on the (worst-case) complexity of this approach. For that, we have however assumed a technical condition on the systems arising in our attack; namely the systems are semi-regular [3,5]. This claim is supported by experimental evidences. Finally, it turns out that our approach is efficient. We have obtained a complexity bounded from above by 257 to forge a signature on the parameters proposed by the designers of TRMS [9]. This bound can be improved; assuming an access to 216 processors (which is very reasonable), one can actually forge a signature in approximately 51 hours.