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

Zhicheng Gao, Qiang Wang
2014 arXiv   pre-print
In this paper we study the distribution 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.  ... 
arXiv:1407.5884v1 fatcat:mc34txft2rgf7afqg7qkoioi2m

Representation of Markov Functions by Minimal Polynomials over a Finite Field

V M Zakharov, S V Shalagin, B F Eminov
2018 Journal of Physics, Conference Series  
The method 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".  ... 
doi:10.1088/1742-6596/1096/1/012200 fatcat:y7h66dzrt5c7nnwob4elhnjoai

On Finite Alphabet Compressive Sensing [article]

Abhik Kumar Das, Sriram Vishwanath
2013 arXiv   pre-print
This paper considers the problem 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].  ... 
arXiv:1303.3943v1 fatcat:obmxdfmzkbaspozmzrukq7oyje

Bearing capacity of spatially random rock masses obeying Hoek–Brown failure criterion

Tamara Al-Bittar, Abdul-Hamid Soubra
2016 Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards  
The uncertainty propagation methodology employed in the analysis makes use 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.  ... 
doi:10.1080/17499518.2016.1232831 fatcat:665vzgwdy5gdhouwjfqzpw62bq

Probabilistic Verification over GF(2m) Using Mod2-OBDDs

J.L. Imana
2010 Intelligent Information Management  
In this paper, we have used 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.  ... 
doi:10.4236/iim.2010.22012 fatcat:vg7hdoifuvdedkj76lxwju2ioa

Coin Flipping in Dynamic Programming Is Almost Useless

Stasys Jukna
2020 ACM Transactions on Computation Theory  
We consider 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.  ... 
doi:10.1145/3397476 fatcat:wxtxlgv2qrfmhe6yxgw44ekhtu

Stochastic Finite-Element Analysis of Seismic Soil–Structure Interaction

Dan M. Ghiocel, Roger G. Ghanem
2002 Journal of engineering mechanics  
by their coordinates with respect 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.  ... 
doi:10.1061/(asce)0733-9399(2002)128:1(66) fatcat:eqsfqgzsnjaoddnbx27w6ohyxq

Counting value sets: algorithm and complexity

Qi Cheng, Joshua Hill, Daqing Wan
2013 The Open Book Series  
Given 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  ... 
doi:10.2140/obs.2013.1.235 fatcat:ahvksnn6bvhqxbtlorlwcybyc4

Counting Value Sets: Algorithm and Complexity [article]

Qi Cheng, Joshua E. Hill, Daqing Wan
2011 arXiv   pre-print
Let p be 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.  ... 
arXiv:1111.1224v1 fatcat:h52n24p6vfb6lbv6qiwemoucvm

On computation of the greatest common divisor of several polynomials over a finite field

Alessandro Conflitti
2003 Finite Fields and Their Applications  
We propose 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.  ... 
doi:10.1016/s1071-5797(03)00022-4 fatcat:uxvfiowt4zcibcaza75snw4yzq

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

Khaled Farah, Mounir Ltifi, Hedi Hassis
2015 Open Civil Engineering Journal  
The results were checked by the Monte Carlo simulation and 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.  ... 
doi:10.2174/1874149501509010196 fatcat:5umqhg7lofcb3ilu2kyrklxjda

Factoring multivariate polynomials over large finite fields

Da Qing Wan
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  ... 
doi:10.1090/s0025-5718-1990-1011448-0 fatcat:ugxvlbiayrax3oe3ficz3yjkmy

Factoring Multivariate Polynomials over Large Finite Fields

Daqing Wan
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  ... 
doi:10.2307/2008511 fatcat:u4d3ted5d5foziuxuzikfaassm

Irreducibility of multivariate polynomials

Joachim von zur Gathen
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.  ... 
doi:10.1016/0022-0000(85)90043-1 fatcat:u6ip3d3tjbdb5jemmc4l4ng4lq
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