Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
Solving systems of nonlinear polynomial equations faster
ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Efficient computation of zero-dimensional Gro¨bner bases by change of ordering
Journal of Symbolic Computation
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Certifying inconsistency of sparse linear systems
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
On Wiedemann's Method of Solving Sparse Linear Systems
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization
CRYPTO '99 Proceedings of the 19th 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
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
On solving sparse algebraic equations over finite fields
Designs, Codes and Cryptography
QUAD: A multivariate stream cipher with provable security
Journal of Symbolic Computation
Sparse Algebraic Equations over Finite Fields
SIAM Journal on Computing
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
Phase transition of multivariate polynomial systems
TAMC'07 Proceedings of the 4th international conference on Theory and applications of models of computation
Fast exhaustive search for polynomial systems in F2
CHES'10 Proceedings of the 12th international conference on Cryptographic hardware and embedded systems
Algebraic cryptanalysis of mceliece variants with compact keys
EUROCRYPT'10 Proceedings of the 29th Annual international conference on 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
ASIACRYPT'11 Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security
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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A fundamental problem in computer science is that of finding all the common zeros of m quadratic polynomials in n unknowns over F"2. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity bound was reached by an exhaustive search in 4log"2n2^n operations. We give an algorithm that reduces the problem to a combination of exhaustive search and sparse linear algebra. This algorithm has several variants depending on the method used for the linear algebra step. We show that, under precise algebraic assumptions on the input system, the deterministic variant of our algorithm has complexity bounded by O(2^0^.^8^4^1^n) when m=n, while a probabilistic variant of the Las Vegas type has expected complexity O(2^0^.^7^9^2^n). Experiments on random systems show that the algebraic assumptions are satisfied with probability very close to 1. We also give a rough estimate for the actual threshold between our method and exhaustive search, which is as low as 200, and thus very relevant for cryptographic applications.