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57
Subquadratictime factoring of polynomials over finite fields
 Math. Comp
, 1998
"... Abstract. New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree n over a finite field of constant cardinality in time O(n 1.815). Previous algorithms required time Θ(n 2+o(1)). The new algorithms rely on fast ..."
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Cited by 75 (11 self)
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Abstract. New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree n over a finite field of constant cardinality in time O(n 1.815). Previous algorithms required time Θ(n 2+o(1)). The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree n over the finite field Fq with q elements, the algorithms use O(n 1.815 log q) arithmetic operations in Fq. The new “baby step/giant step ” techniques used in our algorithms also yield new fast practical algorithms at superquadratic asymptotic running time, and subquadratictime methods for manipulating normal bases of finite fields. 1.
A New Polynomial Factorization Algorithm and its Implementation
 Journal of Symbolic Computation
, 1996
"... We consider the problem of factoring univariate polynomials over a finite field. We demonstrate that the new baby step/giant step factoring method, recently developed by Kaltofen & Shoup, can be made into a very practical algorithm. We describe an implementation of this algorithm, and present th ..."
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Cited by 66 (5 self)
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We consider the problem of factoring univariate polynomials over a finite field. We demonstrate that the new baby step/giant step factoring method, recently developed by Kaltofen & Shoup, can be made into a very practical algorithm. We describe an implementation of this algorithm, and present the results of empirical tests comparing this new algorithm with others. When factoring polynomials modulo large primes, the algorithm allows much larger polynomials to be factored using a reasonable amount of time and space than was previously possible. For example, this new software has been used to factor a "generic" polynomial of degree 2048 modulo a 2048bit prime in under 12 days on a Sun SPARCstation 10, using 68 MB of main memory. 1 Introduction We consider the problem of factoring a univariate polynomial of degree n over the field F p of p elements, where p is prime. This problem has been wellstudied, and many algorithms for its solution have been proposed. In general, the running tim...
Nearly Optimal Algorithms For Canonical Matrix Forms
, 1993
"... A Las Vegas type probabilistic algorithm is presented for finding the Frobenius canonical form of an n x n matrix T over any field K. The algorithm requires O~(MM(n)) = MM(n) (log n) ^ O(1) operations in K, where O(MM(n)) operations in K are sufficient to multiply two n x n matrices over K. This nea ..."
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Cited by 58 (11 self)
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A Las Vegas type probabilistic algorithm is presented for finding the Frobenius canonical form of an n x n matrix T over any field K. The algorithm requires O~(MM(n)) = MM(n) (log n) ^ O(1) operations in K, where O(MM(n)) operations in K are sufficient to multiply two n x n matrices over K. This nearly matches the lower bound of \Omega(MM(n)) operations in K for this problem, and improves on the O(n^4) operations in K required by the previously best known algorithms. We also demonstrate a fast parallel implementation of our algorithm for the Frobenius form, which is processorefficient on a PRAM. As an application we give an algorithm to evaluate a polynomial g(x) in K[x] at T which requires only O~(MM(n)) operations in K when deg g < n^2. Other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix, the rational Jordan form of a matrix, for testing whether two matrices are similar, and for matrix powering, which are substantially faster than those previously known.
Efficient Computation of Minimal Polynomials in Algebraic Extensions of Finite Fields
 In Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation (Vancouver, BC
, 1999
"... New algorithms are presented for computing the minimal polynomial over a finite field K of a given element in an algebraic extension of K of the form K[ff] or K[ff][fi]. The new algorithms are explicit and can be implemented rather easily in terms of polynomial multiplication, and are much more effi ..."
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Cited by 32 (0 self)
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New algorithms are presented for computing the minimal polynomial over a finite field K of a given element in an algebraic extension of K of the form K[ff] or K[ff][fi]. The new algorithms are explicit and can be implemented rather easily in terms of polynomial multiplication, and are much more efficient than other algorithms in the literature. 1 Introduction In this paper, we consider the problem of computing the minimal polynomial over a finite field K of a given element oe in an algebraic extension of K of the form K[ff] or K[ff][fi]. The minimal polynomial of oe is defined to be the unique monic polynomial OE oe=K 2 K[x] of least degree such that OE oe=K (oe) = 0. In the first case, we assume that the ring K[ff] is given as K[x]=(f) where f 2 K[x] is a monic polynomial of degree n, and that elements in K[ff] are represented in the natural way as elements of K[x] !n (the set of polynomials of degree less than n). Similarly, in the second case, we assume that K[ff] is given as a...
Fast Polynomial Factorization Over High Algebraic Extensions of Finite Fields
 In Kuchlin [1997
, 1997
"... New algorithms are presented for factoring polynomials of degree n over the finite field of q elements, where q is a power of a fixed prime number. When log q = n 1+a , where a ? 0 is constant, these algorithms are asymptotically faster than previous known algorithms, the fastest of which require ..."
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Cited by 22 (5 self)
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New algorithms are presented for factoring polynomials of degree n over the finite field of q elements, where q is a power of a fixed prime number. When log q = n 1+a , where a ? 0 is constant, these algorithms are asymptotically faster than previous known algorithms, the fastest of which required time \Omega\Gamma n(log q) 2 ), y or \Omega\Gamma n 3+2a ) in this case, which corresponds to the cost of computing x q modulo an n degree polynomial. The new algorithms factor an arbitrary polynomial in time O(n 3+a+o(1) +n 2:69+1:69a ). All measures are in fixed precision operations, that is in bit complexity. Moreover, in the special case where all the irreducible factors have the same degree, the new algorithms run in time O(n 2:69+1:69a ). In particular, one may test a polynomial for irreducibility in O(n 2:69+1:69a ) bit operations. These results generalize to the case where q = p k , where p is a small prime number relative to q. 1 Introduction The expected run...
Fast modular composition in any characteristic
, 2008
"... We give an algorithm for modular composition of degree n univariate polynomials over a finite field Fq requiring n 1+o(1) log 1+o(1) q bit operations; this had earlier been achieved in characteristic n o(1) by Umans (2008). As an application, we obtain a randomized algorithm for factoring degree n p ..."
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Cited by 18 (1 self)
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We give an algorithm for modular composition of degree n univariate polynomials over a finite field Fq requiring n 1+o(1) log 1+o(1) q bit operations; this had earlier been achieved in characteristic n o(1) by Umans (2008). As an application, we obtain a randomized algorithm for factoring degree n polynomials over Fq requiring (n 1.5+o(1) + n 1+o(1) log q) log 1+o(1) q bit operations, improving upon the methods of von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998). Our results also imply algorithms for irreducibility testing and computing minimal polynomials whose running times are bestpossible, up to lower order terms. As in Umans (2008), we reduce modular composition to certain instances of multipoint evaluation of multivariate polynomials. We then give an algorithm that solves this problem optimally (up to lower order terms), in arbitrary characteristic. The main idea is to lift to characteristic 0, apply a small number of rounds of multimodular reduction, and finish with a small number of multidimensional FFTs. The final evaluations are then reconstructed using the Chinese Remainder Theorem. As a bonus, we obtain a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithm uses techniques which are commonly employed in practice, so it may be competitive for real problem sizes. This contrasts with previous asymptotically fast methods relying on fast matrix multiplication. Supported by NSF DMS0545904 (CAREER) and a Sloan Research Fellowship.
List Decoding of qary ReedMuller Codes
 IEEE Trans. Inform. Theory
, 2004
"... The qary ReedMuller codes RMq(u, m) of length n = qm are a generalization of ReedSolomon codes, which allow polynomials in m variables to encode the message. Using an idea of reducing the multivariate case to univariate case, randomized listdecoding algorithms for ReedMuller codes were given in ..."
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Cited by 18 (1 self)
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The qary ReedMuller codes RMq(u, m) of length n = qm are a generalization of ReedSolomon codes, which allow polynomials in m variables to encode the message. Using an idea of reducing the multivariate case to univariate case, randomized listdecoding algorithms for ReedMuller codes were given in [1] and [27]. The algorithm in [27] is an improvement of the algorithm in [1], it works for up to E < n(1 − √ 2u/q) errors but is applicable only to codes RMq(u, m) with u < q/2. In this paper, we will propose some deterministic listdecoding algorithms for qary ReedMuller codes. Viewing qary ReedMuller codes as codes from order domains, we present a listdecoding algorithm for qary ReedMuller codes, which is a straightforward generalization of the listdecoding algorithm of ReedSolomon codes in [9]. The algorithm works for up to n(1 − m+1 √ u/q) m − 1 errors, and it is applicable to codes RMq(u, m) with u < q. The algorithm can be implemented to run in time polynomial in the length of the codes. Following [12], we show that qary ReedMuller codes are subfield subcodes of ReedSolomon codes. We then present a second listdecoding algorithm for qary ReedMuller codes. This algorithm works for codes with any rates, and achieves an errorcorrection bound n(1 − √ (n − d)/n) − 1. So the second algorithm achieves a better errorcorrection bound than the algorithm in [27], since when u is small, n(1 − √ (n − d)/n) = n(1 − √ u/q). The implementation of the second algorithm requires O(n) field operations in Fq and O(n3) field operations in Fqm under some assumption. Also, we prove that qary ReedMuller codes can be described as onepoint AG codes. And using the algorithm of AG codes in [9], we give a third listdecoding
Fast algorithms for zerodimensional polynomial systems using duality
 APPLICABLE ALGEBRA IN ENGINEERING, COMMUNICATION AND COMPUTING
, 2001
"... Many questions concerning a zerodimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A = k[X1,..., Xn]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the q ..."
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Many questions concerning a zerodimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A = k[X1,..., Xn]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A. We present new formulæ for the rational parametrizations, extending those of Rouillier, and algorithms extending ideas introduced by Shoup in the univariate case. Our approach is based on the Amodule structure of the dual space � A. An important feature of our algorithms is that we do not require � A to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 n D 5/2) and O(n2 n D 5/2) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.