Basic principles of mechanical theorem proving in elementary geometrics
Journal of Automated Reasoning
Mechanical geometry theorem proving
Mechanical geometry theorem proving
Refutational proofs of geometry theorems via characteristic set computation
ISSAC '90 Proceedings of the international symposium on Symbolic and algebraic computation
A new method for solving algebraic systems of positive dimension
Discrete Applied Mathematics - Special volume on applied algebra, algebraic algorithms, and error-correcting codes
A generalized Euclidean algorithm for computing triangular representations of algebraic varieties
Journal of Symbolic Computation
Some results on theorem proving in geometry over finite fields
ISSAC '93 Proceedings of the 1993 international symposium on Symbolic and algebraic computation
Zero-suppressed BDDs for set manipulation in combinatorial problems
DAC '93 Proceedings of the 30th international Design Automation Conference
An elimination method for polynomial systems
Journal of Symbolic Computation
Representation for the radical of a finitely generated differential ideal
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
On the theories of triangular sets
Journal of Symbolic Computation - Special issue on polynomial elimination—algorithms and applications
Factorization-free decomposition algorithms in differential algebra
Journal of Symbolic Computation - Special issue on symbolic computation in algebra, analysis and geometry
Unmixed-dimensional decomposition of a finitely generated perfect differential ideal
Journal of Symbolic Computation
Handbook of Applied Cryptography
Handbook of Applied Cryptography
Ciphertext Only Reconstruction of Stream Ciphers Based on Combination Generators
FSE '00 Proceedings of the 7th International Workshop on Fast Software Encryption
Ritt-Wu's Decomposition Algorithm and Geometry Theorem Proving
Proceedings of the 10th International Conference on Automated Deduction
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Lifting techniques for triangular decompositions
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
A pommaret division algorithm for computing Grobner bases in boolean rings
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
A characteristic set method for ordinary difference polynomial systems
Journal of Symbolic Computation
An equational approach to theorem proving in first-order predicate calculus
IJCAI'85 Proceedings of the 9th international joint conference on Artificial intelligence - Volume 2
Logical Foundations of Proof Complexity
Logical Foundations of Proof Complexity
EUROCRYPT'96 Proceedings of the 15th annual 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
Decomposing polynomial sets into simple sets over finite fields: The positive-dimensional case
Theoretical Computer Science
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Efficient characteristic set methods for computing zeros of polynomial equation systems in a finite field are proposed. The concept of proper triangular sets is introduced and an explicit formula for the number of zeros of a proper and monic triangular set is given. An improved zero decomposition algorithm is proposed to reduce the zero set of an equation system to the union of zero sets of monic proper triangular sets. The bitsize complexity of this algorithm is shown to be O(l^n) for Boolean polynomials, where n is the number of variables and l=2 is the number of equations. We also give a multiplication free characteristic set method for Boolean polynomials, where the sizes of the polynomials occurred during the computation do not exceed the sizes of the input polynomials and the bitsize complexity of algorithm is O(n^d) for input polynomials with n variables and degree d. The algorithms are implemented in the case of Boolean polynomials and extensive experiments show that they are quite efficient for solving certain classes of Boolean equations raising from stream ciphers.