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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.  ... 

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  ... 

Polynomial division and its computational complexity

Dario Bini, Victor Pan
1986 Journal of Complexity  
(ii) Then we accelerate 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.  ... 
doi:10.1016/0885-064x(86)90001-4 fatcat:ixnugw7k35bbdo4b6rvi7rhhfm

Parallelizing Stream with Future [article]

Raphaël Jolly
2013 arXiv   pre-print
Future is substituted for Lazy in the obtained construct, resulting in possible 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  ... 
arXiv:1305.4367v1 fatcat:yapqbkumfjcxnll4hnyxgwzjjq

МАТРИЧНИЙ МЕТОД ПАРАЛЕЛЬНОЇ ДЕКОМПОЗИЦІЇ ДЛЯ МІНІМІЗАЦІЇ СИМЕТРИЧНИХ БУЛЕВИХ ФУНКЦІЙ У ВИГЛЯДІ РОЗШИРЕНОГО ПОЛІНОМА

S. V. Burmistrov, O. M. Panasco, N. V. Kovalska
2018 Вісник Черкаського державного технологічного університету. Серія: Технічні науки  
Вісник Черкаського державного технологічного університету 130 UDC 681.5.004 A matrix method 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).  ... 
doi:10.24025/2306-4412.1.2018.162604 fatcat:5byvcislbvcvjbfhzd3mmb6pxy

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.  ... 

Boolean circuits versus arithmetic circuits

Joachim von zur Gathen, Gadiel Seroussi
1991 Information and Computation  
We compare the two computational models 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.  ... 
doi:10.1016/0890-5401(91)90078-g fatcat:aydwnfiodfgy3bizzgauso2sei

Generation of Efficient Key Bit-Streams Using Sparse Matrix-Vector Multiplication [article]

M. Sivasankar, T. R. Padmanabhan
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 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].  ... 
arXiv:1207.0806v2 fatcat:znn4nsbnyjeobcpzmkxbfdbk3y

Computer algebra on MIMD machine

J.-L. Roch, P. Senechaud, F. Sievert-Rich, G. Villard
1989 ACM SIGSAM Bulletin  
Then, different problems are studied, particularly the implementation 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  ... 
doi:10.1145/66062.66065 fatcat:y2pcdfwedrb2hdzjifvf5ncrue

Highly parallel computations modulo a number having only small prime factors

Thomas Zeugmann
1992 Information and Computation  
Furthermore, the author thanks the unknown reviewer for his careful reading and for many valuable comments on the preparation 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.  ... 
doi:10.1016/0890-5401(92)90057-m fatcat:4o3pczcvufa3fpenzucs2lua54

Complexity of parallel matrix computations

Victor Pan
1987 Theoretical Computer Science  
We estimate 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.  ... 
doi:10.1016/0304-3975(87)90019-3 fatcat:2fcadjkp7fgvbasai26r7jrgdu

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.  ... 

Factorization of polynomials in one variable over the tropical semiring [article]

Ki Hang Kim, Fred W. Roush
2005 arXiv   pre-print
We show 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.  ... 
arXiv:math/0501167v2 fatcat:kj2jhcoigrfi3bzhpeuruyvmcy

Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields

Dima Yu. Grigoriev, Marek Karpinski, Michael F. Singer
1990 SIAM journal on computing (Print)  
an algorithm to interpolate the 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.  ... 
doi:10.1137/0219073 fatcat:ms2ohrjgqjddjprlwezcjesqn4

The synthesis of robust polynomial arithmetic with stochastic logic

Weikang Qian, Marc D. Riedel
2008 Proceedings of the 45th annual conference on Design automation - DAC '08  
As integrated circuit technology plumbs ever greater depths in the scaling 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 ))).  ... 
doi:10.1145/1391469.1391636 dblp:conf/dac/QianR08 fatcat:jwvw2zesozai5bmiltca3c2p54
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