ON THE MINIMAL NUMBER OF SMALL ELEMENTS GENERATING FINITE PRIME FIELDS

2017 ◽  
Vol 96 (2) ◽  
pp. 177-184
Author(s):  
MARC MUNSCH
Keyword(s):  
Finite Field ◽  
General Result ◽  
Upper Bound ◽  
Prime Fields ◽  
Almost All ◽  
Small Elements

In this note, we give an upper bound for the number of elements from the interval $[1,p^{1/4\sqrt{e}+\unicode[STIX]{x1D716}}]$ necessary to generate the finite field $\mathbb{F}_{p}^{\ast }$ with $p$ an odd prime. The general result depends on the distribution of the divisors of $p-1$ and can be used to deduce results which hold for almost all primes.

scholarly journals Roots of sparse polynomials over a finite field

2016 ◽  
Vol 19 (A) ◽  
pp. 196-204 ◽  
Author(s):  
Zander Kelley
Keyword(s):  
Finite Field ◽  
Upper Bound ◽  
Log P ◽  
Diffie Hellman ◽  
Sparse Polynomials ◽  
Prime Fields

For a $t$-nomial $f(x)=\sum _{i=1}^{t}c_{i}x^{a_{i}}\in \mathbb{F}_{q}[x]$, we show that the number of distinct, nonzero roots of $f$ is bounded above by $2(q-1)^{1-\unicode[STIX]{x1D700}}C^{\unicode[STIX]{x1D700}}$, where $\unicode[STIX]{x1D700}=1/(t-1)$ and $C$ is the size of the largest coset in $\mathbb{F}_{q}^{\ast }$ on which $f$ vanishes completely. Additionally, we describe a number-theoretic parameter depending only on $q$ and the exponents $a_{i}$ which provides a general and easily computable upper bound for $C$. We thus obtain a strict improvement over an earlier bound of Canetti et al. which is related to the uniformity of the Diffie–Hellman distribution. Finally, we conjecture that $t$-nomials over prime fields have only $O(t\log p)$ roots in $\mathbb{F}_{p}^{\ast }$ when $C=1$.

scholarly journals Number of Directions Determined by a Set in $$\mathbb {F}_{q}^{2}$$ and Growth in $$\mathrm {Aff}(\mathbb {F}_{q})$$

2021 ◽  
Author(s):  
Daniele Dona
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Upper Bound ◽  
Finite Plane ◽  
Structural Theorem ◽  
Prime Fields ◽  
The One

AbstractWe prove that a set A of at most q non-collinear points in the finite plane $$\mathbb {F}_{q}^{2}$$ F q 2 spans more than $${|A|}/\!{\sqrt{q}}$$ | A | / q directions: this is based on a lower bound by Fancsali et al. which we prove again together with a different upper bound than the one given therein. Then, following the procedure used by Rudnev and Shkredov, we prove a new structural theorem about slowly growing sets in $$\mathrm {Aff}(\mathbb {F}_{q})$$ Aff ( F q ) for any finite field $$\mathbb {F}_{q}$$ F q , generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields.

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

scholarly journals On algebraic closure in pseudofinite fields

2012 ◽  
Vol 77 (4) ◽  
pp. 1057-1066 ◽  
Author(s):  
Özlem Beyarslan ◽  
Ehud Hrushovski
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Algebraic Closure ◽  
Roots Of Unity ◽  
Prime Field ◽  
Almost All ◽  
Pseudofinite Fields

AbstractWe study the automorphism group of the algebraic closure of a substructureAof a pseudo-finite fieldF. We show that the behavior of this group, even whenAis large, depends essentially on the roots of unity inF. For almost all completions of the theory of pseudofinite fields, we show that overA, algebraic closure agrees with definable closure, as soon asAcontains the relative algebraic closure of the prime field.

Introduction

2012 ◽  
Author(s):  
Nicholas M. Katz
Keyword(s):  
Finite Field ◽  
Open Questions ◽  
Prime Fields

This introductory chapter sets out the book's focus, namely equidistribution results over larger and larger finite extensions of a given finite field. Emanuel Kowalski drew attention to the interest of having equidistribution results over, for example, prime fields 𝔽p, that become better and better as p grows. This question is addressed in Chapter 28, where the problem is to make effective the estimates, already given in the equicharacteristic setting of larger and larger extensions of a given finite field. Chapter 29 points out some open questions about “the situation over ℤ” and gives some illustrative examples. The chapter concludes by pointing out two potential ambiguities of notation.

Torsion elements of order 𝑝2 in the Nottingham group

2020 ◽  
Vol 23 (3) ◽  
pp. 489-502
Author(s):  
Chun Yin Hui ◽  
Krishna Kishore
Keyword(s):  
Finite Field ◽  
Conjugacy Class ◽  
Upper Bound ◽  
Conjugacy Classes ◽  
Torsion Element ◽  
Exact Number

AbstractLet κ be a characteristic p finite field of q elements and {\mathfrak{N}_{\kappa}} the Nottingham group over κ. Lubin associated to every conjugacy class of torsion element of {\mathfrak{N}_{\kappa}} a type. We establish an upper bound {B(q;l,m)} on the number of conjugacy classes of order {p^{2}} torsion elements u of {\mathfrak{N}_{\kappa}} of type {\langle l,m\rangle}. In the case where {l<p}, the bound {B(q;l,m)} is the exact number of conjugacy classes. Moreover, we give a criterion on when u and {u^{n}} are conjugate.

BALANCED SETS AND THE VECTOR GAME

2008 ◽  
Vol 04 (03) ◽  
pp. 339-347 ◽  
Author(s):  
ZHIVKO NEDEV ◽  
ANTHONY QUAS
Keyword(s):  
Field Theory ◽  
Lower Bound ◽  
Finite Field ◽  
Upper Bound ◽  
Log P ◽  
Prime Factorization ◽  
Minimum Cardinality ◽  
Player Game

We consider the notion of a balanced set modulo N. A nonempty set S of residues modulo N is balanced if for each x ∈ S, there is a d with 0 < d ≤ N/2 such that x ± d mod N both lie in S. We define α(N) to be the minimum cardinality of a balanced set modulo N. This notion arises in the context of a two-player game that we introduce and has interesting connections to the prime factorization of N. We demonstrate that for p prime, α(p) = Θ( log p), giving an explicit algorithmic upper bound and a lower bound using finite field theory and show that for N composite, α(N) = min p|Nα(p).

scholarly journals On the density and the structure of the Peirce-like formulae

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Antoine Genitrini ◽  
Jakub Kozik ◽  
Grzegorz Matecki
Keyword(s):  
Finite Number ◽  
Upper Bound ◽  
Large Family ◽  
Partial Answer ◽  
International Audience ◽  
Almost All

International audience Within the language of propositional formulae built on implication and a finite number of variables $k$, we analyze the set of formulae which are classical tautologies but not intuitionistic (we call such formulae - Peirce's formulae). We construct the large family of so called simple Peirce's formulae, whose sequence of densities for different $k$ is asymptotically equivalent to the sequence $\frac{1}{ 2 k^2}$. We prove that the densities of the sets of remaining Peirce's formulae are asymptotically bounded from above by $\frac{c}{ k^3}$ for some constant $c \in \mathbb{R}$. The result justifies the statement that in the considered language almost all Peirce's formulae are simple. The result gives a partial answer to the question stated in the recent paper by H. Fournier, D. Gardy, A. Genitrini and M. Zaionc - although we have not proved the existence of the densities for Peirce's formulae, our result gives lower and upper bound for it (if it exists) and both bounds are asymptotically equivalent to $\frac{1}{ 2 k^2}$.

scholarly journals Zero-Knowledge Proofs for Finite Field Arithmetic or: Can Zero-Knowledge be for Free?

1997 ◽  
Vol 4 (27) ◽  
Author(s):  
Ronald Cramer ◽  
Ivan B. Damgård
Keyword(s):  
Finite Field ◽  
Prime Order ◽  
Error Probability ◽  
Communication Complexity ◽  
Boolean Circuit ◽  
Interactive Proof ◽  
Zero Knowledge ◽  
Order Group ◽  
Prime Fields ◽  
Interactive Proof System

We present zero-knowledge proofs and arguments for arithmetic circuits over finite prime fields, namely given a circuit, show in zero-knowledge that inputs can be selected leading to a given output. For a field GF(q), where q is an n-bit prime, a<br />circuit of size O(n), and error probability 2^−n, our protocols require communication of O(n^2) bits. This is the same worst-cast complexity as the trivial (non zero-knowledge)<br />interactive proof where the prover just reveals the input values. If the circuit involves n multiplications, the best previously known methods would in general require communication<br />of  Omega(n^3 log n) bits.<br />Variations of the technique behind these protocols lead to other interesting applications.<br />We first look at the Boolean Circuit Satisfiability problem and give zero-knowledge proofs and arguments for a circuit of size n and error probability 2^−n in which there is an interactive preprocessing phase requiring communication of O(n^2)<br />bits. In this phase, the statement to be proved later need not be known. Later the prover can non-interactively prove any circuit he wants, i.e. by sending only one message, of size O(n) bits.<br />As a second application, we show that Shamirs (Shens) interactive proof system for the (IP-complete) QBF problem can be transformed to a zero-knowledge proof<br />system with the same asymptotic communication complexity and number of rounds. The security of our protocols can be based on any one-way group homomorphism with a particular set of properties. We give examples of special assumptions sufficient for this, including: the RSA assumption, hardness of discrete log in a prime order group, and polynomial security of Die-Hellman encryption. We note that the constants involved in our asymptotic complexities are small enough for our protocols to be practical with realistic choices of parameters.

scholarly journals Double Character Sums over Subgroups and Intervals

2014 ◽  
Vol 90 (3) ◽  
pp. 376-390 ◽  
Author(s):  
MEI-CHU CHANG ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Field ◽  
Upper Bound ◽  
Multiplicative Group ◽  
Character Sums ◽  
Multiplicative Character ◽  
Nontrivial Estimate ◽  
Image Position ◽  
Primitive Roots ◽  
Double Character

AbstractWe estimate double sums $$\begin{eqnarray}S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})=\mathop{\sum }\limits_{x\in {\mathcal{I}}}\mathop{\sum }\limits_{{\it\lambda}\in {\mathcal{G}}}{\it\chi}(x+a{\it\lambda}),\quad 1\leq a<p-1,\end{eqnarray}$$ with a multiplicative character ${\it\chi}$ modulo $p$ where ${\mathcal{I}}=\{1,\dots ,H\}$ and ${\mathcal{G}}$ is a subgroup of order $T$ of the multiplicative group of the finite field of $p$ elements. A nontrivial upper bound on $S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})$ can be derived from the Burgess bound if $H\geq p^{1/4+{\it\varepsilon}}$ and from some standard elementary arguments if $T\geq p^{1/2+{\it\varepsilon}}$, where ${\it\varepsilon}>0$ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters $H$ and $T$. We also estimate double sums $$\begin{eqnarray}T_{{\it\chi}}(a,{\mathcal{G}})=\mathop{\sum }\limits_{{\it\lambda},{\it\mu}\in {\mathcal{G}}}{\it\chi}(a+{\it\lambda}+{\it\mu}),\quad 1\leq a<p-1,\end{eqnarray}$$ and give an application to primitive roots modulo $p$ with three nonzero binary digits.

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