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53
Factoring polynomials with rational coefficients
 MATH. ANN
, 1982
"... In this paper we present a polynomialtime algorithm to solve the following problem: given a nonzero polynomial fe Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive polynomia ..."
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Cited by 982 (11 self)
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In this paper we present a polynomialtime algorithm to solve the following problem: given a nonzero polynomial fe Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive polynomials feZ[X] into irreducible factors in Z[X]. Here we call f ~ Z[X] primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. [8]. Its running time, measured in bit operations, is O(nl2+n9(log[fD3). Here f~Tl[X] is the polynomial to be factored, n = deg(f) is the degree of f, and for a polynomial ~ a ~ i with real coefficients a i. i An outline of the algorithm is as follows. First we find, for a suitable small prime number p, a padic irreducible factor h of f, to a certain precision. This is done with Berlekamp's algorithm for factoring polynomials over small finite fields, combined with Hensel's lemma. Next we look for the irreducible factor h o of f in
Solving lowdensity subset sum problems
 in Proceedings of 24rd Annu. Symp. Foundations of comput. Sci
, 1983
"... Abstract. The subset sum problem is to decide whether or not the O1 integer programming problem C aixi = M, Vi,x,=O or 1, il has a solution, where the ai and M are given positive integers. This problem is NPcomplete, and the difficulty of solving it is the basis of publickey cryptosystems of kna ..."
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Cited by 127 (5 self)
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Abstract. The subset sum problem is to decide whether or not the O1 integer programming problem C aixi = M, Vi,x,=O or 1, il has a solution, where the ai and M are given positive integers. This problem is NPcomplete, and the difficulty of solving it is the basis of publickey cryptosystems of knapsack type. An algorithm is proposed that searches for a solution when given an instance of the subset sum problem. This algorithm always halts in polynomial time but does not always find a solution when one exists. It converts the problem to one of finding a particular short vector v in a lattice, and then uses a lattice basis reduction algorithm due to A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovasz to attempt to find v. The performance of the proposed algorithm is analyzed. Let the density d of a subset sum problem be defined by d = n/log2(maxi ai). Then for “almost all ” problems of density d c 0.645, the vector v we searched for is the shortest nonzero vector in the lattice. For “almost all ” problems of density d < l/a it is proved that the lattice basis reduction algorithm locates v. Extensive computational tests of the algorithm suggest that it works for densities d < de(n), where d=(n) is a cutoff value that is substantially larger than I/n. This method gives a polynomial time attack on knapsack publickey cryptosystems that can be expected to break them if they transmit information at rates below d=(n), as n+ 01.
Structure of three interval exchange transformations II: A combinatorial description of . . .
, 2002
"... We describe an algorithm for generating the symbolic sequences which code the orbits of points under an interval exchange transformation on three intervals. The algorithm has two components. The first is an arithmetic division algorithm applied to the lengths of the intervals. This arithmetic const ..."
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Cited by 38 (7 self)
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We describe an algorithm for generating the symbolic sequences which code the orbits of points under an interval exchange transformation on three intervals. The algorithm has two components. The first is an arithmetic division algorithm applied to the lengths of the intervals. This arithmetic construction was originally introduced by the authors in an earlier paper and may be viewed as a twodimensional generalization of the regular continued fraction. The second component is a combinatorial algorithm which generates the bispecial factors of the associated symbolic subshift as a function of the arithmetic expansion. As a consequence we obtain a complete characterization of those sequences of block complexity 2n + 1 which are natural codings of orbits of threeinterval exchange transformations, thereby answering an old question of Rauzy.
Extended gcd and Hermite normal form algorithms via lattice basis reduction
 Experimental Mathematics
, 1998
"... Extended gcd calculation has a long history and plays an important role in computational number theory and linear algebra. Recent results have shown that finding optimal multipliers in extended gcd calculations is difficult. We present an algorithm which uses lattice basis reduction to produce small ..."
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Cited by 34 (6 self)
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Extended gcd calculation has a long history and plays an important role in computational number theory and linear algebra. Recent results have shown that finding optimal multipliers in extended gcd calculations is difficult. We present an algorithm which uses lattice basis reduction to produce small integer multipliers x1,..., xm for the equation d = gcd (d1,..., dm) = x1d1 + · · · + xmdm, where d1,..., dm are given integers. The method generalises to produce small unimodular transformation matrices for computing the Hermite normal form of an integer matrix. 1
Theory and applications of the doublebase number system
 IEEE Transactions on Computers
, 1999
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Geodesic multidimensional continued fractions
 Proc. London Math. Soc
, 1994
"... A multidimensional continued fraction expansion is given which finds provably good Diophantine approximations for all 6 e U'1. For any Q> 1 it finds some approximation (p, q) e Z ' with 1 s £ q = £ Q satisfying \\q % p  = £ V j + 1 Q~xld. This expansion consists of a sequence of redu ..."
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Cited by 25 (1 self)
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A multidimensional continued fraction expansion is given which finds provably good Diophantine approximations for all 6 e U'1. For any Q> 1 it finds some approximation (p, q) e Z ' with 1 s £ q = £ Q satisfying \\q % p  = £ V j + 1 Q~xld. This expansion consists of a sequence of reduced lattice bases for a parametrized series of lattice bases B,(8) (of different lattices) in GL(d + 1,0?), where the positive real parameter / varies. This parametrized family B,(&) forms a geodesic in GL(d + 1,0?), and also projects to a geodesic ge in the Riemannian symmetric space 9>dJrX of all positive definite symmetric matrices. The multidimensional continued fraction expansion is a 'cutting sequence ' expansion for g0 using a Minkowski fundamental domain of GL(d + 1, 2) \ ^ r f + 1. This method generalizes to give continued fraction expansions finding good Diophantine approximations to an arbitrary set of linear forms. 1.
Solving exponential Diophantine equations using lattice basis reduction algorithms
 J. NUMBER THEORY
, 1987
"... Let S be the set of all positive integers with prime divisors from a fixed finite set of primes. Algorithms are given for solving the diophantine inequality 0 < x y < y” in x, y E S for Iixed 6 E (0, 1), and for the diophantine equation x + y = z in x, y, 2 G S. The method is based on multid ..."
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Cited by 20 (2 self)
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Let S be the set of all positive integers with prime divisors from a fixed finite set of primes. Algorithms are given for solving the diophantine inequality 0 < x y < y” in x, y E S for Iixed 6 E (0, 1), and for the diophantine equation x + y = z in x, y, 2 G S. The method is based on multidimensional diophantine approximation, in the real and padic case, respectively. The main computational tool is the L³Basis Reduction Algorithm. Elaborate examples are presented.
Static Tiling for Heterogeneous Computing Platforms
, 1999
"... In the framework of fully permutable loops, tiling has been extensively studied as a sourceto source program transformation. However, little work has been devoted to the mapping and scheduling of the tiles on physical processors. Moreover, targeting heterogeneous computing platforms has, to the bes ..."
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Cited by 17 (7 self)
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In the framework of fully permutable loops, tiling has been extensively studied as a sourceto source program transformation. However, little work has been devoted to the mapping and scheduling of the tiles on physical processors. Moreover, targeting heterogeneous computing platforms has, to the best of our knowledge, never been considered. In this paper we extend static tiling techniques to the context of limited computational resources with dierentspeed processors. In particular, we present eÆcient scheduling and mapping strategies that are asymptotically optimal. The practical usefulness of these strategies is fully demonstrated by MPI experiments on a heterogeneous network of workstations. Key words: tiling, communicationcomputation overlap, mapping, limited resources, dierentspeed processors, heterogeneous networks Corresponding author: Yves Robert LIP, Ecole Normale Superieure de Lyon, 69364 Lyon Cedex 07, France Phone: + 33 4 72 72 80 37, Fax: + 33 4 72 72 80 80 Email: Y...
Cryptanalytic attacks on the multiplicative knapsack cryptosystem and on Shamir’s fast signature scheme
 IEEE Truns. Inforni. Theory
, 1984
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A Simple Proof of the Exponential Convergence of the Modified JacobiPerron Slgorithm
"... It has recently been shown in Ito et al. (1993) that the modified JacobiPerron algorithm is strongly convergent (in the sense of Brentjes 1981) almost everywhere with exponential rate. Their proof relies on very complicated computations. In this paper we will show that the original paper of Podsypa ..."
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Cited by 9 (0 self)
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It has recently been shown in Ito et al. (1993) that the modified JacobiPerron algorithm is strongly convergent (in the sense of Brentjes 1981) almost everywhere with exponential rate. Their proof relies on very complicated computations. In this paper we will show that the original paper of Podsypanin (1977) on the modified JacobiPerron algorithm almost contains a proof of this convergence with exponential rate. The only ingredients missing in that paper are some ergodictheoretical facts about the transformation generating the approximations. This leads to a very simple proof of the beforementioned exponential convergence in the modified JacobiPerron algorithm. Mathematics Subject Classification: 11J70, 11K50, 28D05. 1 Background of the problem The archetypal example of a multidimensional generalisation of the regular continued fraction expansion is the JacobiPerron algorithm (JPA), see Bernstein (1971), Brentjes (1981) and Schweiger (1973). This algorithm produces, for almost ...