Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Fast construction of irreducible polynomials over finite fields
Journal of Symbolic Computation
Fast polynomial factorization over high algebraic extensions of finite fields
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Composing power series over a finite ring in essentially linear time
Journal of Symbolic Computation
Subquadratic-time factoring of polynomials over finite fields
Mathematics of Computation
Fast rectangular matrix multiplication and applications
Journal of Complexity
Modern computer algebra
Efficient computation of minimal polynomials in algebraic extensions of finite fields
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Fast Algorithms for Manipulating Formal Power Series
Journal of the ACM (JACM)
Factoring polynomials over finite fields: a survey
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the second Magma conference
Polynomial factorization: a success story
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Correcting Errors Beyond the Guruswami-Sudan Radius in Polynomial Time
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Explicit capacity-achieving list-decodable codes
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Who was who in polynomial factorization: 1
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Fast computation of special resultants
Journal of Symbolic Computation
Algebraic algorithms and coding theory
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Fast Polynomial Factorization and Modular Composition
SIAM Journal on Computing
Constant-Round multi-party private set union using reversed laurent series
PKC'12 Proceedings of the 15th international conference on Practice and Theory in Public Key Cryptography
Sub-linear root detection, and new hardness results, for sparse polynomials over finite fields
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Modular Composition Modulo Triangular Sets and Applications
Computational Complexity
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We obtain randomized algorithms for factoring degree n univariate polynomials over F_q that use O(n1.5 + o(1) + n1 + o(1)log q) field operations, when the characteristic is at most no(1). When log q The improvements come from a new algorithm for modular composition of degree n univariate polynomials, which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields. The best previous algorithms for modular composition use O(n(omega + 1)/2) field operations, where omega is the exponent of matrix multiplication (Brent & Kung (1978)), with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication (Huang & Pan (1997)). We show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent in the sense that an algorithm for one achieving exponent α implies an algorithm for the other with exponent α + o(1), and vice versa. We then give a new algorithm that requires O(n1 + o(1)) field operations when the characteristic is at most no(1), which is optimal up to lower order terms. Our algorithms do not rely on fast matrix multiplication, in contrast to all previous subquadratic algorithms for these problems. The main operations are fast univariate polynomial arithmetic, multipoint evaluation, and interpolation, and consequently the algorithms could be feasible in practice.