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Page 2225 of Mathematical Reviews Vol. , Issue 91D
[page]

1991
*
Mathematical Reviews
*

Our method

*of**parallelization*is perfect; m*parallel*processors speed the generation*of*pseudorandom bits by a*factor*m, and these*parallel*processors need not communicate. ... The*Boolean*model is a uniform family*of**Boolean*circuits, where “log space” uniformity (uniform family*of*circuits) and “*polynomial*time” uniformity (P-uniform family*of*circuits) are considered. ...##
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Page 480 of Mathematical Reviews Vol. , Issue 94a
[page]

1994
*
Mathematical Reviews
*

Then, assuming GRH, the irreducible

*factors**of*F modulo p can be found in deterministic time*polynomial*in deg F, m, L and log p. ... Assuming the generalized Riemann hypothesis (GRH), determin- istic*polynomial*time methods are known: (i) for*factoring*bino- mials, (ii) for the modulo p reduction*of*an irreducible*polynomial*F € Z[x ...##
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Polynomial division and its computational complexity

1986
*
Journal of Complexity
*

(ii) Then we accelerate

doi:10.1016/0885-064x(86)90001-4
fatcat:ixnugw7k35bbdo4b6rvi7rhhfm
*parallel*division*of*two*polynomials*with integer coefficients*of*degrees at most m by a*factor**of*log m comparing with the*parallel*version*of*the algorithm*of*Sieveking and Kung ... Under the*parallel*model, it attains*Boolean*logarithmic time, which is asymptotically optimum. ... , 656) and gives the improved*parallel*case bounds*of*O(log d log (dm))*parallel**Boolean*steps and O(dm log d log log d) processors. ...##
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Parallelizing Stream with Future
[article]

2013
*
arXiv
*
pre-print

Future is substituted for Lazy in the obtained construct, resulting in possible

arXiv:1305.4367v1
fatcat:yapqbkumfjcxnll4hnyxgwzjjq
*parallelization**of*any algorithm expressible as a Stream computation. ... Stream is re-interpreted in terms*of*a Lazy monad. ... The*polynomial*multiplication example was also run in two versions, stream and stream big, the latter using*polynomials*with bigger coefficients (*of*a*factor*10000000001), in order to increase the footprint ...##
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МАТРИЧНИЙ МЕТОД ПАРАЛЕЛЬНОЇ ДЕКОМПОЗИЦІЇ ДЛЯ МІНІМІЗАЦІЇ СИМЕТРИЧНИХ БУЛЕВИХ ФУНКЦІЙ У ВИГЛЯДІ РОЗШИРЕНОГО ПОЛІНОМА

2018
*
Вісник Черкаського державного технологічного університету. Серія: Технічні науки
*

Вісник Черкаського державного технологічного університету 130 UDC 681.5.004 A matrix method

doi:10.24025/2306-4412.1.2018.162604
fatcat:5byvcislbvcvjbfhzd3mmb6pxy
*of**parallel*decomposition in order to minimize symmetric*Boolean*functions in orthogonal form*of*representation ... Due to this, this method can be used as a powerful component*of*complete matrix method*of**parallel*decomposition for obtaining a complex minimal form*of**Boolean*functions, which has the best indicators ... Construction*of*an expanded*polynomial*based on a*polynomial*(7). ...##
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Page 7337 of Mathematical Reviews Vol. , Issue 2000j
[page]

2000
*
Mathematical Reviews
*

Here, it is shown that the natural extension

*of*their aggressive algorithm to the*parallel*disk case is suboptimal by a*factor*near the number*of*disks in the worst case. ... The kinetic TSP cannot be approximated better than by a*factor**of*2%V") by a*polynomial*time algorithm unless P= NP, even if the maximum velocity is bounded. ...##
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Boolean circuits versus arithmetic circuits

1991
*
Information and Computation
*

We compare the two computational models

doi:10.1016/0890-5401(91)90078-g
fatcat:aydwnfiodfgy3bizzgauso2sei
*of**Boolean*circuits and arithmetic circuits in cases where they both apply, namely the computation*of**polynomials*over the rational numbers or over finite fields ... Over finite lields*of*small characteristic, the two models are equally powerful when size is considered, but*Boolean*circuits are exponentially more powerful than arithmetic circuits with respect to depth ... Section 6 applies this to the problem*of**factoring**polynomials*over large finite fields*of*small characteristic. ...##
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Generation of Efficient Key Bit-Streams Using Sparse Matrix-Vector Multiplication
[article]

2012
*
arXiv
*
pre-print

In this paper we present a matrix based stream cipher, in which a m x n binary matrix single handedly performs the work

arXiv:1207.0806v2
fatcat:znn4nsbnyjeobcpzmkxbfdbk3y
*of*m*parallel*LFSRs. ... Interestingly the output*of*the matrix multiplication can otherwise be used as a*parallel*bit/byte generator, useful for encrypting video streams. ... The*polynomial*collection is denoted by Z 2 [X]. A*polynomial*p(x) ∈ Z 2 [X] is said to be irreducible if it cannot be*factored*into lower degree*polynomials**of*Z 2 [X]. ...##
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Computer algebra on MIMD machine

1989
*
ACM SIGSAM Bulletin
*

Then, different problems are studied, particularly the implementation

doi:10.1145/66062.66065
fatcat:y2pcdfwedrb2hdzjifvf5ncrue
*of*infinite-precision arithmetic, the solution*of*linear systems and*of*Diophantine equations, the*parallelization**of*Buchberger's ... PAC is a computer algebra system, based on MIMD type*parallelism*. It uses*parallelism*as a tool for processing problems wich are too complex for a sequential treatment. ... IV.2 / Implementation on the FPS T20 IV.2.1 / Choice*of*a representation for the*boolean**polynomials*: The choice*of*the structure to represent the*boolean**polynomials*is justified by the fact that we want ...##
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Highly parallel computations modulo a number having only small prime factors

1992
*
Information and Computation
*

Furthermore, the author thanks the unknown reviewer for his careful reading and for many valuable comments on the preparation

doi:10.1016/0890-5401(92)90057-m
fatcat:4o3pczcvufa3fpenzucs2lua54
*of*this paper. ... ACKNOWLEDGMENTS The author is very grateful to Hermann Jung for several helpful discussion and to Frank BauernGppel for simplifying the proof*of*the claim used in proving Theorem 5. ... Moreover, we re-prove the existence*of**polynomial*time uniform*Boolean*circuits taking O(log n) depth and using*polynomially*many processors that solve INV and POW. ...##
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Complexity of parallel matrix computations

1987
*
Theoretical Computer Science
*

We estimate

doi:10.1016/0304-3975(87)90019-3
fatcat:2fcadjkp7fgvbasai26r7jrgdu
*parallel*complexity*of*several matrix computations under both*Boolean*and arithmetic machine models using deterministic and probabilistic approaches. ... In addition, processor &cient algorithms using polylo~arithmi~*parallel*time are devised for some other matrix computations, such as triangufar and QR-*factorizations**of*a matrix and its reduction to Hessenberg ...*polynomial*; (V) its QR-*factorization*; (vi) its LU-*factorization*; (vii) its reduction to the upper I-fessenberg form. ...##
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Page 629 of Mathematical Reviews Vol. , Issue 87b
[page]

1987
*
Mathematical Reviews
*

The author discusses the basic properties

*of*simple circuit*polynomials*and characteristic*polynomials**of*some classes*of*graphs (e.g. wheels), which are similar to those*of*circuit*polynomials*[the author ... The problem*of*finding 1-*factors*in a graph can be reduced to the one*of*find- ing transversals in certain associated set systems. Thus Aharoni characterized the graphs having no 1-*factor*. ...##
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Factorization of polynomials in one variable over the tropical semiring
[article]

2005
*
arXiv
*
pre-print

We show

arXiv:math/0501167v2
fatcat:kj2jhcoigrfi3bzhpeuruyvmcy
*factorization**of**polynomials*in one variable over the tropical semiring is in general NP-complete, either if all coefficients are finite, or if all are either 0 or infinity (*Boolean*case). ... We give algorithms for the*factorization*problem which are not*polynomial*time in the degree, but are*polynomial*time for*polynomials**of*fixed degree. ... There is a*polynomial*time*parallel*algorithm for*factorization*over the tropical semiring, since*parallel**polynomial*time is equivalent to PSPACE which contains NP. ...##
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Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields

1990
*
SIAM journal on computing (Print)
*

an algorithm to interpolate the

doi:10.1137/0219073
fatcat:ms2ohrjgqjddjprlwezcjesqn4
*polynomial*in O(log (nt))*boolean**parallel*time and O(n2t log nt) processors. ... This algorithm yields the first efficient deterministic*polynomial*time algorithm (and moreover*boolean*NC-algorithm) for interpolating t-sparse*polynomials*over finite fields and should be contrasted ... We are grateful to Michael Ben-Or, Johannes Grabmeier, Michael Rabin, Volker Strassen, and Avi Wigderson for a number*of*interesting conversations. ...##
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The synthesis of robust polynomial arithmetic with stochastic logic

2008
*
Proceedings of the 45th annual conference on Design automation - DAC '08
*

As integrated circuit technology plumbs ever greater depths in the scaling

doi:10.1145/1391469.1391636
dblp:conf/dac/QianR08
fatcat:jwvw2zesozai5bmiltca3c2p54
*of*feature sizes, maintaining the paradigm*of*deterministic*Boolean*computation is increasingly challenging. ... The resulting logic processes serial or*parallel*streams that are random at the bit level. ... A*polynomial*g(t) = P n i=0 a n i t i can be*factorized*as g(t) = a n 0 + t(a n 1 + t(a n 2 + · · · + t(a n n−1 + ta n n ))). ...
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