Number of solutions of systems of homogeneous polynomial equations over finite fields

Mrinmoy Datta; Sudhir R. Ghorpade

doi:10.1090/proc/13239

On the Number of Zeros Over a Finite Field of Certain Symmetric Polynomials

Canadian Mathematical Bulletin ◽

10.4153/cmb-1980-046-5 ◽

1980 ◽

Vol 23 (3) ◽

pp. 327-332

Author(s):

P. V. Ceccherini ◽

J. W. P. Hirschfeld

Keyword(s):

Finite Field ◽

Finite Fields ◽

Polynomial Equations ◽

Symmetric Polynomials ◽

Number Of Solutions ◽

Number Of Zeros

A variety of applications depend on the number of solutions of polynomial equations over finite fields. Here the usual situation is reversed and we show how to use geometrical methods to estimate the number of solutions of a non-homogeneous symmetric equation in three variables.

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Number of Solutions of Equations over Finite Fields and a Conjecture of Lang and Weil

Number Theory and Discrete Mathematics ◽

10.1007/978-3-0348-8223-1_27 ◽

2002 ◽

pp. 269-291 ◽

Cited By ~ 4

Author(s):

Sudhir R. Ghorpade ◽

Gilles Lachaud

Keyword(s):

Finite Fields ◽

Number Of Solutions ◽

Solutions Of Equations

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Quantum computing and polynomial equations over Z_2

Quantum Information and Computation ◽

10.26421/qic5.2-2 ◽

2005 ◽

Vol 5 (2) ◽

pp. 102-112

Author(s):

C.M. Dawson ◽

H.L. Haselgrove ◽

A.P. Hines ◽

D. Mortimer ◽

M.A. Nielsen ◽

...

Keyword(s):

Quantum Computing ◽

Finite Field ◽

Quantum Computation ◽

Quantum Computer ◽

Complexity Classes ◽

Polynomial Equations ◽

Computational Power ◽

Number Of Solutions

What is the computational power of a quantum computer? We show that determining the output of a quantum computation is equivalent to counting the number of solutions to an easily computed set of polynomials defined over the finite field Z_2. This connection allows simple proofs to be given for two known relationships between quantum and classical complexity classes, namely BQP/P/\#P and BQP/PP.

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Finding roots in with the successive resultants algorithm

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157014000138 ◽

2014 ◽

Vol 17 (A) ◽

pp. 203-217 ◽

Cited By ~ 2

Author(s):

Christophe Petit

Keyword(s):

Finite Field ◽

Elliptic Curve ◽

Finite Fields ◽

Coding Theory ◽

Discrete Logarithm ◽

Discrete Logarithm Problem ◽

Polynomial Equations ◽

Preliminary Results ◽

Characteristic Case ◽

Practical Security

AbstractThe problem of solving polynomial equations over finite fields has many applications in cryptography and coding theory. In this paper, we consider polynomial equations over a ‘large’ finite field with a ‘small’ characteristic. We introduce a new algorithm for solving this type of equations, called the successive resultants algorithm (SRA). SRA is radically different from previous algorithms for this problem, yet it is conceptually simple. A straightforward implementation using Magma was able to beat the built-in Roots function for some parameters. These preliminary results encourage a more detailed study of SRA and its applications. Moreover, we point out that an extension of SRA to the multivariate case would have an important impact on the practical security of the elliptic curve discrete logarithm problem in the small characteristic case.Supplementary materials are available with this article.

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Solutions of equations over finite fields: Enumeration via bijections

Journal of Algebra and Its Applications ◽

10.1142/s021949881650136x ◽

2016 ◽

Vol 15 (07) ◽

pp. 1650136 ◽

Cited By ~ 1

Author(s):

Ioulia N. Baoulina

Keyword(s):

Finite field and group algorithms for orthogonal sequence search

Information and Control Systems ◽

10.31799/1684-8853-2021-4-2-17 ◽

2021 ◽

pp. 2-17

Author(s):

Nikolay Balonin ◽

Alexander Sergeev ◽

Olga Sinitshina

Keyword(s):

Finite Fields ◽

Practical Importance ◽

Video Data ◽

Polynomial Equations ◽

Ideal Object ◽

Domestic Literature ◽

Finite Dimensional ◽

Sequence Search ◽

Lack Of Information ◽

First Time

Introduction: Hadamard matrices consisting of elements 1 and –1 are an ideal object for a visual application of finite dimensional mathematics operating with a finite number of addresses for –1 elements. The notation systems of abstract algebra methods, in contrast to the conventional matrix algebra, have been changing intensively, without being widely spread, leading to the necessity to revise and systematize the accumulated experience. Purpose: To describe the algorithms of finite fields and groups in a uniform notation in order to facilitate the perception of the extensive knowledge necessary for finding orthogonal and suborthogonal sequences. Results: Formulas have been proposed for calculating relatively unknown algorithms (or their versions) developed by Scarpis, Singer, Szekeres, Goethal — Seidel, and Noboru Ito, as well as polynomial equations used to prove the theorems about the existence of finite-dimensional solutions. This replenished the significant lack of information both in the domestic literature (most of these issues are published here for the first time) and abroad. Practical relevance: Orthogonal sequences and methods for their effective finding via the theory of finite fields and groups are of direct practical importance for noise-immune coding, compression and masking of video data.

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Power trace functions over finite fields

Journal of Algebra and Its Applications ◽

10.1142/s0219498820501960 ◽

2019 ◽

Vol 19 (10) ◽

pp. 2050196

Author(s):

Robert W. Fitzgerald ◽

Yasanthi Kottegoda

Keyword(s):

Finite Fields ◽

Trace Function ◽

Authentication Code ◽

Number Of Solutions ◽

Successful Attack

We count the number of solutions to a power trace function equal to a constant and use this to find the probability of a successful attack on an authentication code proposed by Ding et al. (2005) [C. Ding, A. Salomaa, P. Solé and X. Tian, Three constructions of authentication/secrecy codes, J. Pure Appl. Algebra 196 (2005) 149–168].

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