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A probabilistic approach to value sets of polynomials over finite fields
[article]

2014
*
arXiv
*
pre-print

In this paper we study the distribution

arXiv:1407.5884v1
fatcat:mc34txft2rgf7afqg7qkoioi2m
*of*the size*of*the*value**set*for*a*random*polynomial*with degree at most q-1*over**a**finite**field*F_q. ... We obtain the exact probability distribution and show that the number*of*missing*values*tends*to**a*normal distribution as q goes*to*infinity. ... Thus this enables us*to*derive the probability distribution*of*the size*of**value**set**of**a*random*polynomial**of*degree d ≤ q − 1*over**a**finite**field*. ...##
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Representation of Markov Functions by Minimal Polynomials over a Finite Field

2018
*
Journal of Physics, Conference Series
*

The method

doi:10.1088/1742-6596/1096/1/012200
fatcat:y7h66dzrt5c7nnwob4elhnjoai
*of*representing Markov functions with minimal characteristic*polynomials**over**a**finite**field*is proposed. These*polynomials*are defined on the basis*of*integrated stochastic matrices. ... The algorithmic implementation*of*the method is shown*to*build*a*sequence*of*the Markov functions class considered, with*a*given linear complexity. ... Acknowledgments This work was supported by RFBR Grant 18-01-00120а "Specialized devices for generating and processing data*sets*in the architecture*of*programmable logic devices class FPGA". ...##
###
On Finite Alphabet Compressive Sensing
[article]

2013
*
arXiv
*
pre-print

This paper considers the problem

arXiv:1303.3943v1
fatcat:obmxdfmzkbaspozmzrukq7oyje
*of*compressive sensing*over**a**finite*alphabet, where the*finite*alphabet may be inherent*to*the nature*of*the data or*a*result*of*quantization. ... designs*of*sensing matrices based on coding-theoretic techniques; (c) enable one*to*solve the exact ℓ_0-minimization problem in*polynomial*time rather than*a**approach**of*convex relaxation followed by ... Given*a**field*F,*a*primitive*polynomial**over*F (and in F[x]) is*a*prime*polynomial**over*F having*a*primitive element*of*an extension*field*, say K, as one*of*its roots, as*a**polynomial*in K[x]. ...##
###
Bearing capacity of spatially random rock masses obeying Hoek–Brown failure criterion

2016
*
Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards
*

The uncertainty propagation methodology employed in the analysis makes use

doi:10.1080/17499518.2016.1232831
fatcat:665vzgwdy5gdhouwjfqzpw62bq
*of**a*non-intrusive*approach**to*build up*a*sparse*polynomial*chaos expansion for the system response. ... In this paper,*a**probabilistic*analysis is presented*to*compute the ultimate bearing capacity*of**a*strip footing resting on*a*spatially varying rock mass. ...*a*given simulation)*to**a**finite*zone which leads*to*several zones with different*values**of*ı c*over*the entire rock mass. ...##
###
Probabilistic Verification over GF(2m) Using Mod2-OBDDs

2010
*
Intelligent Information Management
*

In this paper, we have used

doi:10.4236/iim.2010.22012
fatcat:vg7hdoifuvdedkj76lxwju2ioa
*probabilistic*verification with Galois (or*finite*)*field*GF(2 m ) modifying the CUDD package for the computation*of*signatures in classical OBDDs, and for the construction*of*... Mod2-OBDDs have been constructed with*a*two-level layer*of*-nodes using*a*positive Davio expansion (pDE) for*a*given variable. ... This replacement is determined by an algebraic transformation*of*the Boolean function in terms*of**polynomials**over*the*finite**field*. ...##
###
Coin Flipping in Dynamic Programming Is Almost Useless

2020
*
ACM Transactions on Computation Theory
*

We consider

doi:10.1145/3397476
fatcat:wxtxlgv2qrfmhe6yxgw44ekhtu
*probabilistic*circuits working*over*the real numbers, and using arbitrary semialgebraic functions*of*bounded description complexity as gates. ... We show that*probabilistic*circuits using any*of*these operations as gates can be simulated by deterministic circuits with only about*a*quadratical blowup in size. ... I am thankful*to*Joshua Grochow, Pascal Koiran, Igor Sergeev and Hans Ulrich Simon for inspiring discussions at the initial stages*of*this investigation. ...##
###
Stochastic Finite-Element Analysis of Seismic Soil–Structure Interaction

2002
*
Journal of engineering mechanics
*

by their coordinates with respect

doi:10.1061/(asce)0733-9399(2002)128:1(66)
fatcat:eqsfqgzsnjaoddnbx27w6ohyxq
*to*the*polynomial*chaos basis. ...*A*procedure is presented for the*probabilistic*analysis*of*the seismic soil-structure interaction problem. ... Acknowledgments The financial support*of*the National Science Foundation through the SBIR and Geomechanics programs under Grant Nos. ...##
###
Counting value sets: algorithm and complexity

2013
*
The Open Book Series
*

Given

doi:10.2140/obs.2013.1.235
fatcat:ahvksnn6bvhqxbtlorlwcybyc4
*a**polynomial*in F p m [x]*of*degree d*over*the*finite**field*F p m , one can view it as*a*map from F p m*to*F p m , and examine the image*of*this map, also known as the*value**set*. ... Additionally, we prove that it is NP-hard*to*decide whether*a**polynomial*represented by*a*straight-line program has*a*root in*a*prime-order*finite**field*, thus resolving an open problem proposed by Kaltofen ... Theorem 3 . 3 The problem*of*counting the*value**set**of**a*sparse*polynomial**over**a**finite**field**of*characteristic p = 2 is #P-hard.Corollary 2.The*set**of*sparse permutation*polynomials**over**finite**fields*...##
###
Counting Value Sets: Algorithm and Complexity
[article]

2011
*
arXiv
*
pre-print

Let p be

arXiv:1111.1224v1
fatcat:h52n24p6vfb6lbv6qiwemoucvm
*a*prime. Given*a**polynomial*in _p^m[x]*of*degree d*over*the*finite**field*_p^m, one can view it as*a*map from _p^m*to*_p^m, and examine the image*of*this map, also known as the*value**set*. ... Additionally, we prove that it is NP-hard*to*decide whether*a**polynomial*represented by*a*straight-line program has*a*root in*a*prime-order*finite**field*, thus resolving an open problem proposed by Kaltofen ... Tsuyoshi Ito for pointing out the reference [3]*to*us. ...##
###
On computation of the greatest common divisor of several polynomials over a finite field

2003
*
Finite Fields and Their Applications
*

We propose

doi:10.1016/s1071-5797(03)00022-4
fatcat:uxvfiowt4zcibcaza75snw4yzq
*a**probabilistic*algorithm*to*reduce computing the greatest common divisor*of*m*polynomials**over**a**finite**field*(which requires computing m*À*1 pairwise greatest common divisors)*to*computing ... the greatest common divisor*of*two*polynomials**over*the same*field*. r ... Warm thanks are also due*to*the anonymous referee for his/her careful reading*of*the manuscript, pointing out Refs. [5, 6] , and his/her valuable suggestions. ...##
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Page 834 of Mathematical Reviews Vol. , Issue 97B
[page]

1997
*
Mathematical Reviews
*

In this paper the author analyzes the complexity

*of*constructing*a**finite*extension*of*F*of*composite degree n by implementing*a*tower*of*extension*fields*, using the*probabilistic**approach*. ... Consider the following problem: given*a*positive integer n and*a**finite**field*F*of*g elements, find an irreducible*polynomial**of*degree n*over*F. M. O. Rabin [SIAM J. ...##
###
A Study of Probabilistic FEMs for a Slope Reliability Analysis Using the Stress Fields

2015
*
Open Civil Engineering Journal
*

The results were checked by the Monte Carlo simulation and

doi:10.2174/1874149501509010196
fatcat:5umqhg7lofcb3ilu2kyrklxjda
*a*direct coupling*approach*combining the deterministic*finite*elements code and First Order Reliability Method (FORM) algorithm. ... In this paper, the applicability and the effectiveness*of*the*probabilistic**finite*element methods (FEMs) such as the perturbation method, and the Spectral Stochastic*Finite*Element Method (SSFEM) applied ... Additionally, decomposition*of*the nodal displacement vector solution*of*the problem*over**polynomial*chaos basis is used. ...##
###
Factoring multivariate polynomials over large finite fields

1990
*
Mathematics of Computation
*

*A*simple

*probabilistic*algorithm is presented

*to*find the irreducible factors

*of*

*a*bivariate

*polynomial*

*over*

*a*large

*finite*

*field*. ... One is concerned with exponential sums; the other is related

*to*permutational

*polynomials*

*over*

*finite*

*fields*(

*a*conjecture

*of*Chowla and Zassenhaus). ... Acknowledgment I would like

*to*thank Professors Neal Koblitz, Andrew Odlyzko, and Susan Landau for their comments on the original manuscript, especially Neal Koblitz for his encouragement and many valuable ...

##
###
Factoring Multivariate Polynomials over Large Finite Fields

1990
*
Mathematics of Computation
*

*A*simple

*probabilistic*algorithm is presented

*to*find the irreducible factors

*of*

*a*bivariate

*polynomial*

*over*

*a*large

*finite*

*field*. ... One is concerned with exponential sums; the other is related

*to*permutational

*polynomials*

*over*

*finite*

*fields*(

*a*conjecture

*of*Chowla and Zassenhaus). ... Acknowledgment I would like

*to*thank Professors Neal Koblitz, Andrew Odlyzko, and Susan Landau for their comments on the original manuscript, especially Neal Koblitz for his encouragement and many valuable ...

##
###
Irreducibility of multivariate polynomials

1985
*
Journal of computer and system sciences (Print)
*

*Over*algebraic number

*fields*and

*over*

*finite*

*fields*, we obtain

*polynomial*-time

*probabilistic*algorithms. They are based on an effective version

*of*Hilbert's irreducibility theorem. T ... This includes the important question

*of*testing for irreducibility.

*A*

*probabilistic*reduction from multivariate

*to*bivariate

*polynomials*is given,

*over*an arbitrary (effectively computable)

*field*. ... ACKNOWLEDGMENTS It is

*a*pleasure

*to*acknowledge the many conversations I had with Erich Kaltofen about the factoring problem, and the numerous improvements and corrections he contributed. ...

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