Computer algebra: symbolic and algebraic computation (2nd ed.)
Computer algebra: symbolic and algebraic computation (2nd ed.)
Public quadratic polynomial-tuples for efficient signature-verification and message-encryption
Lecture Notes in Computer Science on Advances in Cryptology-EUROCRYPT'88
Polynomial decomposition algorithms
Journal of Symbolic Computation
Functional decomposition ofpolynomials: the tame case
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Functional decomposition of polynomials: The wild case
Journal of Symbolic Computation
Efficient computation of zero-dimensional Gro¨bner bases by change of ordering
Journal of Symbolic Computation
Algebraic Methods for Constructing Asymmetric Cryptosystems
AAECC-3 Proceedings of the 3rd International Conference on Algebraic Algorithms and Error-Correcting Codes
Asymmetric cryptography with S-Boxes
ICICS '97 Proceedings of the First International Conference on Information and Communication Security
Trapdoor one-way permutations and multivariate polynominals
ICICS '97 Proceedings of the First International Conference on Information and Communication Security
Cryptanalysis of ``2 R'' Schemes
CRYPTO '99 Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology
Cryptoanalysis of the Matsumoto and Imai Public Key Scheme of Eurocrypt'88
CRYPTO '95 Proceedings of the 15th Annual International Cryptology Conference on 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
Analytic models for token-ring networks
Analytic models for token-ring networks
The functional decomposition of polynomials
The functional decomposition of polynomials
Cryptanalysis of Patarin's 2-round public key system with S boxes (2R)
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
Cryptanalysis of Rational Multivariate Public Key Cryptosystems
PQCrypto '08 Proceedings of the 2nd International Workshop on Post-Quantum Cryptography
Public-Key identification schemes based on multivariate cubic polynomials
PKC'12 Proceedings of the 15th international conference on Practice and Theory in Public Key Cryptography
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In this paper, we study the security of 2R− schemes [17,18], which are the “minus variant” of two-round schemes. This variant consists in removing some of the n polynomials of the public key, and permits to thwart an attack described at Crypto'99 [25] against two-round schemes. Usually, the “minus variant” leads to a real strengthening of the considered schemes. We show here that this is actually not true for 2R− schemes. We indeed propose an efficient algorithm for decomposing 2R− schemes. For instance, we can remove up to $\left \lfloor\frac{n}{2} \right \rfloor$ equations and still be able to recover a decomposition in O(n12). We provide experimental results illustrating the efficiency of our approach. In practice, we have been able to decompose 2R− schemes in less than a handful of hours for most of the challenges proposed by the designers [18]. We believe that this result makes the principle of two-round schemes, including 2R− schemes, useless.