Results 1  10
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25
PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
 SIAM J. on Computing
, 1997
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 1278 (4 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
On The Rapid Computation of Various Polylogarithmic Constants
, 1996
"... We give algorithms for the computation of the dth digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the d ..."
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Cited by 119 (31 self)
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We give algorithms for the computation of the dth digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log (2) or on a modest work station in a few hours run time. We demonstrate this technique by computing the ten billionth hexadecimal digit of, the billionth hexadecimal digits of 2 2 log(2) and log (2), and the ten billionth decimal digit of log(9=10). These calculations rest on the observation that very special types of identities exist for certain numbers like, 2,log(2) and log 2 (2). These are essentially polylogarithmic ladders in an integer base. A number of these identities that we derive in this work appear to be new, for example the critical identity for:
Faster Integer Multiplication
 STOC'07
, 2007
"... For more than 35 years, the fastest known method for integer multiplication has been the SchönhageStrassen algorithm running in time O(n log n log log n). Under certain restrictive conditions there is a corresponding Ω(n log n) lower bound. The prevailing conjecture has always been that the complex ..."
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Cited by 89 (0 self)
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For more than 35 years, the fastest known method for integer multiplication has been the SchönhageStrassen algorithm running in time O(n log n log log n). Under certain restrictive conditions there is a corresponding Ω(n log n) lower bound. The prevailing conjecture has always been that the complexity of an optimal algorithm is Θ(n log n). We present a major step towards closing the gap from above by presenting an algorithm running in time n log n 2 O(log ∗ n). The main result is for boolean circuits as well as for multitape Turing machines, but it has consequences to other models of computation as well.
Univariate polynomials: nearly optimal algorithms for factorization and rootfinding
 In Proceedings of the International Symposium on Symbolic and Algorithmic Computation
, 2001
"... To approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zerofree annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of t ..."
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Cited by 63 (14 self)
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To approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zerofree annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of the nth degree into two factors balanced in the degrees and with the zero sets separated by the basic annulus. Recursive combination of the two algorithms leads to computation of the complete numerical factorization of a polynomial into the product of linear factors and further to the approximation of the roots. The new rootfinder incorporates the earlier techniques of Schönhage, Neff/Reif, and Kirrinnis and our old and new techniques and yields nearly optimal (up to polylogarithmic factors) arithmetic and Boolean cost estimates for the computational complexity of both complete factorization and rootfinding. The improvement over our previous record Boolean complexity estimates is by roughly the factor of n for complete factorization and also for the approximation of wellconditioned (well isolated) roots, whereas the same algorithm is also optimal (under both arithmetic and Boolean models of computing) for the worst case input polynomial, whose roots can be illconditioned, forming
Multidigit Multiplication For Mathematicians
, 2001
"... This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchönhageStr ..."
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Cited by 35 (8 self)
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This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchönhageStrassen trick, Schönhage's trick, Nussbaumer's trick, the cyclic SchönhageStrassen trick, and the CantorKaltofen theorem. It emphasizes the underlying ring homomorphisms.
Nearly optimal computations with structured matrices
"... We propose a nearly optimal algorithm that uses 2n 2 random parameters, O(n) memory space and O((nlog2n)loglogn) operations in a fixed arbitrary field in order to compute the rank and a basis for the null space of a structured n x n matrix X represented with O(n) parameters of its short generator, ..."
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Cited by 16 (10 self)
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We propose a nearly optimal algorithm that uses 2n 2 random parameters, O(n) memory space and O((nlog2n)loglogn) operations in a fixed arbitrary field in order to compute the rank and a basis for the null space of a structured n x n matrix X represented with O(n) parameters of its short generator, as well as to solve a linear system Xy = b or to determine its inconsistency. If rank X = n, the algorithm also computes det X and a short generator of XI. The cost bounds cover correctness verification for the output but not the cost of the generation of random parameters. The algorithm gives a unified treatment of various classes of structured matrices including ones of Toeplitz, Hankel, Vandermonde and Cauchy types.
Partial Fraction Decomposition in C(z) and Simultaneous Newton Iteration for Factorization in C[z]
, 1998
"... The subject of this paper is fast numerical algorithms for factoring univariate polynomials with complex coefficients and for computing partial fraction decompositions (PFDs) of rational functions in C(z). Numerically stable and computationally feasible versions of PFD are specified first for the sp ..."
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Cited by 7 (0 self)
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The subject of this paper is fast numerical algorithms for factoring univariate polynomials with complex coefficients and for computing partial fraction decompositions (PFDs) of rational functions in C(z). Numerically stable and computationally feasible versions of PFD are specified first for the special case of rational functions with all singularities in the unit disk (the ``bounded case'') and then for rational functions with arbitrarily distributed singularities. Two major algorithms for computing PFDs are presented: The first one is an extension of the ``splitting circle method' ' by A. Schonhage (``The Fundamental Theorem of Algebra in Terms of Computational Complexity,' ' Technical Report, Univ. Tubingen, 1982) for factoring polynomials in C[z] to an algorithm for PFD. The second algorithm is a Newton iteration for simultaneously improving the accuracy of all factors in an approximate factorization of a polynomial resp. all partial fractions of an approximate PFD of a rational function. Algorithmically useful starting value conditions for the Newton algorithm are provided. Three subalgorithms are of independent interest. They compute the product of a sequence of polynomials, the sum
Faster polynomial multiplication via multipoint kronecker substitution
 J. Symb. Comput
, 2009
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From Approximate Factorization to Root Isolation with Application to Cylindrical Algebraic Decomposition
, 2013
"... We present an algorithm for isolating the roots of an arbitrary complex polynomial p that also works for polynomials with multiple roots provided that the number k of distinct roots is given as part of the input. It outputs k pairwise disjoint disks each containing one of the distinct roots of p, an ..."
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Cited by 6 (2 self)
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We present an algorithm for isolating the roots of an arbitrary complex polynomial p that also works for polynomials with multiple roots provided that the number k of distinct roots is given as part of the input. It outputs k pairwise disjoint disks each containing one of the distinct roots of p, and its multiplicity. The algorithm uses approximate factorization as a subroutine. In addition, we apply the new root isolation algorithm to a recent algorithm for computing the topology of a real planar algebraic curve specified as the zero set of a bivariate integer polynomial and for isolating the real solutions of a bivariate polynomial system. For input polynomials of degree n and bitsize τ, we improve the currently best running time from Õ(n 9 τ + n 8 τ 2) (deterministic) to Õ(n 6 + n 5 τ) (randomized) for topology computation and from Õ(n 8 + n 7 τ) (deterministic) to Õ(n 6 + n 5 τ) (randomized) for solving bivariate systems. 1