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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
Minwise Independent Permutations
 Journal of Computer and System Sciences
, 1998
"... We define and study the notion of minwise independent families of permutations. We say that F ⊆ Sn is minwise independent if for any set X ⊆ [n] and any x ∈ X, when π is chosen at random in F we have Pr(min{π(X)} = π(x)) = 1 X . In other words we require that all the elements of any fixed set ..."
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Cited by 276 (14 self)
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We define and study the notion of minwise independent families of permutations. We say that F ⊆ Sn is minwise independent if for any set X ⊆ [n] and any x ∈ X, when π is chosen at random in F we have Pr(min{π(X)} = π(x)) = 1 X . In other words we require that all the elements of any fixed set X have an equal chance to become the minimum element of the image of X under π. Our research was motivated by the fact that such a family (under some relaxations) is essential to the algorithm used in practice by the AltaVista web index software to detect and filter nearduplicate documents. However, in the course of our investigation we have discovered interesting and challenging theoretical questions related to this concept – we present the solutions to some of them and we list the rest as open problems.
Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity
, 1988
"... , Introduction and References only) Benny Chor Oded Goldreich MIT \Gamma Laboratory for Computer Science Cambridge, Massachusetts 02139 ABSTRACT \Gamma A new model for weak random physical sources is presented. The new model strictly generalizes previous models (e.g. the Santha and Vazirani model [2 ..."
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Cited by 217 (6 self)
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, Introduction and References only) Benny Chor Oded Goldreich MIT \Gamma Laboratory for Computer Science Cambridge, Massachusetts 02139 ABSTRACT \Gamma A new model for weak random physical sources is presented. The new model strictly generalizes previous models (e.g. the Santha and Vazirani model [24]). The sources considered output strings according to probability distributions in which no single string is too probable. The new model provides a fruitful viewpoint on problems studied previously as: ffl Extracting almost perfect bits from sources of weak randomness: the question of possibility as well as the question of efficiency of such extraction schemes are addressed. ffl Probabilistic Communication Complexity: it is shown that most functions have linear communication complexity in a very strong probabilistic sense. ffl Robustness of BPP with respect to sources of weak randomness (generalizing a result of Vazirani and Vazirani [27]). The paper has appeared in SIAM Journal o...
Bulk SpinResonance Quantum Computation
 Computation,” Science
, 1997
"... This article presents a new approach to quantum computing based on using bulk samples rather than isolated degrees of freedom. The problem, of course, is that such samples microscopically are in a thermal distribution of states, and it is impractical to hope to cool macroscopic materials to their gr ..."
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Cited by 119 (7 self)
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This article presents a new approach to quantum computing based on using bulk samples rather than isolated degrees of freedom. The problem, of course, is that such samples microscopically are in a thermal distribution of states, and it is impractical to hope to cool macroscopic materials to their ground state; furthermore, bulk samples are macroscopic ensembles whose members cannot be addressed individually for readout. We present here solutions to these problems. First, we report on a procedure to take advantage of the structure present in thermal equilibrium to introduce into the system's large density matrix a perturbation that acts exactly like a much smaller dimensional effective pure state. We then show how quantum computation can be performed using this ensemble system in such a way that the result is deterministic and can be read out efficiently. One great advantage of this approach is that, because of the massive redundancy provided by having a large ensemble of identical copies of the system, environmental interactions or intentional measurements only weakly perturb the computer's state. Thus, quantum computation becomes experimentally accessible in many naturally existing materials. From the perspective of the NMR chemist, our scheme is unusual in a fundamental respect. In NMR spectroscopy, the primary purpose is to elucidate molecular structure and chemical dynamics, and great efforts are made to enhance the desired signal and render the detected spectra into a form that reflects properties of the system under study. Our purpose here is very different we view each molecule as a single computer, whose state is determined by the orientations of its spins. Sequences of rf pulses, which manipulate spin orientations and couplings, constitute quantum logic gate...
An Introduction to Quantum Computing for NonPhysicists
 Los Alamos Physics Preprint Archive http://xxx.lanl.gov/abs/quantph/9809016
, 2000
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