scholarly journals On Lower Bounds for Incomplete Character Sums over Finite Fields

1996 ◽  
Vol 2 (2) ◽  
pp. 173-191 ◽  
Author(s):  
Ferruh Özbudak
Keyword(s):  
Finite Fields ◽  
Lower Bounds ◽  
Character Sums ◽  
Incomplete Character Sums

scholarly journals On character sums in finite fields

1939 ◽  
Vol 71 (0) ◽  
pp. 99-121 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Finite Fields ◽  
Character Sums

Weighted Measures of Pseudorandom Binary Lattices

2021 ◽  
Vol 58 (3) ◽  
pp. 319-334
Author(s):  
Huaning Liu ◽  
Yinyin Yang
Keyword(s):  
Finite Fields ◽  
Lower Bounds ◽  
Binary Sequences ◽  
Quadratic Character ◽  
Pseudorandom Sequences ◽  
Even Order

In cryptography one needs pseudorandom sequences whose short subsequences are also pseudorandom. To handle this problem, Dartyge, Gyarmati and Sárközy introduced weighted measures of pseudorandomness of binary sequences. In this paper we continue the research in this direction. We introduce weighted pseudorandom measure for multidimensional binary lattices and estimate weighted pseudorandom measure for truly random binary lattices. We also give lower bounds for weighted measures of even order and present an example by using the quadratic character of finite fields.

scholarly journals A note on character sums in finite fields

2017 ◽  
Vol 46 ◽  
pp. 247-254 ◽  
Author(s):  
Abhishek Bhowmick ◽  
Thái Hoàng Lê ◽  
Yu-Ru Liu
Keyword(s):  
Finite Fields ◽  
Character Sums

scholarly journals Characterization and Enumeration of Good Punctured Polynomials over Finite Fields

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Somphong Jitman ◽  
Aunyarut Bunyawat ◽  
Supanut Meesawat ◽  
Arithat Thanakulitthirat ◽  
Napat Thumwanit
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Complete Characterization ◽  
Binary Field ◽  
Polynomials Over Finite Fields

A family of good punctured polynomials is introduced. The complete characterization and enumeration of such polynomials are given over the binary fieldF2. Over a nonbinary finite fieldFq, the set of good punctured polynomials of degree less than or equal to2are completely determined. Forn≥3, constructive lower bounds of the number of good punctured polynomials of degreenoverFqare given.

The Distribution of Zeros of Quadratic Forms over Finite Fields

2015 ◽  
Vol 7 (2) ◽  
pp. 18
Author(s):  
Ali H. Hakami
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Quadratic Forms ◽  
Distribution Of Zeros ◽  
Linearly Independent ◽  
Set Of Points

Let $m$ be a positive integer with $m < p/2$ and $p$ is a prime. Let $\mathbb{F}_q$ be the finite field in $q = p^f$ elements, $Q({\mathbf{x}})$ be a nonsinqular quadratic form over $\mathbb{F}_q$ with $q$ odd, $V$ be the set of points in $\mathbb{F}_q^n$ satisfying the equation $Q({\mathbf{x}}) = 0$ in which the variables are restricted to a box of points of the type\[\mathcal{B}(m) = \left\{ {{\mathbf{x}} \in \mathbb{F}_q^n \left| {x_i  = \sum\limits_{j = 1}^f {x_{ij} \xi _j } ,\;\left| {x_{ij} } \right| < m,\;1 \leqslant i \leqslant n,\;1 \leqslant j \leqslant f} \right.} \right\},\]where $\xi _1 , \ldots ,\xi _f$ is a basis for $\mathbb{F}_q$ over $\mathbb{F}_p$ and $n > 2$ even. Set $\Delta  = \det Q$ such that $\chi \left( {( - 1)^{n/2} \Delta } \right) = 1.$ We shall motivate work of (Cochrane, 1986) to obtain lower bounds on $m,$ size of the box $\mathcal{B},$ so that $\mathcal{B} \cap V$ is nonempty. For this we show that the box $\mathcal{B}(m)$ contains a zero of $Q({\mathbf{x}})$ provided that $m \geqslant p^{1/2}.$ We also show that the box $\mathcal{B}(m)$ contains $n$ linearly independent zeros of $Q({\mathbf{x}})$ provided that $m \geqslant 2^{n/2} p^{1/2} .$

scholarly journals Solutions of equations over finite fields: Enumeration via bijections

2016 ◽  
Vol 15 (07) ◽  
pp. 1650136 ◽  
Author(s):  
Ioulia N. Baoulina
Keyword(s):  
Recent Result ◽  
Finite Fields ◽  
Simple Proof ◽  
Character Sums ◽  
Alternative Proof ◽  
Number Of Solutions ◽  
Combinatorial Argument ◽  
Solutions Of Equations

We present a simple proof of the well-known fact concerning the number of solutions of diagonal equations over finite fields. In a similar manner, we give an alternative proof of the recent result on generalizations of Carlitz equations. In both cases, the use of character sums is avoided by using an elementary combinatorial argument.

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