scholarly journals Character sums and products of factorials modulo p

2005 ◽  
Vol 17 (1) ◽  
pp. 151-160
Author(s):  
Moubariz Z. Garaev ◽  
Florian Luca
Keyword(s):  
Character Sums ◽  
Modulo P

scholarly journals The hybrid power mean of some special character sums of polynomials and two-term exponential sums modulo $ p $

2021 ◽  
Vol 6 (10) ◽  
pp. 10989-11004
Author(s):  
Wenpeng Zhang ◽  
◽  
Jiafan Zhang ◽  
Keyword(s):  
Exponential Sums ◽  
Character Sums ◽  
Gauss Sums ◽  
Asymptotic Formulas ◽  
Analytic Method ◽  
Power Mean ◽  
Hybrid Power ◽  
Special Character ◽  
Modulo P ◽  
Hybrid Power Mean

<abstract><p>We consider the calculation problem of one kind hybrid power mean involving the character sums of polynomials and two-term exponential sums modulo $ p $, an odd prime, and use the analytic method and the properties of classical Gauss sums to give some identities and asymptotic formulas for them.</p></abstract>

scholarly journals Pythagorean triples and quadratic residues modulo an odd prime

2021 ◽  
Vol 7 (1) ◽  
pp. 957-966
Author(s):  
Jiayuan Hu ◽  
◽  
Yu Zhan ◽  
Keyword(s):  
Character Sums ◽  
Pythagorean Triples ◽  
Quadratic Residues ◽  
Elementary Methods ◽  
Modulo P

<abstract><p>In this article, we use the elementary methods and the estimate for character sums to study a problem related to quadratic residues and the Pythagorean triples, and prove the following result. Let $ p $ be an odd prime large enough. Then for any positive number $ 0 &lt; \epsilon &lt; 1 $, there must exist three quadratic residues $ x, \ y $ and $ z $ modulo $ p $ with $ 1\leq x, \ y, \ z\leq p^{1+\epsilon} $ such that the equation $ x^2+y^2 = z^2 $.</p></abstract>

scholarly journals On the Quartic Residues and Their New Distribution Properties

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1337
Author(s):  
Juanli Su ◽  
Jiafan Zhang
Keyword(s):  
Asymptotic Formula ◽  
Fourth Order ◽  
Character Sums ◽  
Exact Calculation ◽  
Calculation Formula ◽  
Analytic Methods ◽  
Counting Functions ◽  
Modulo P ◽  
New Distribution

In this paper, we use the analytic methods, the properties of the fourth-order characters, and the estimate for character sums to study the computational problems of one kind of special quartic residues modulo p, and give an exact calculation formula and asymptotic formula for their counting functions.

scholarly journals The Legendre’s Symbol Modulo p and Its Some Elementary Results

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Jianghua Li ◽  
Yuan Zhang
Keyword(s):  
Character Sums ◽  
Asymptotic Formulas ◽  
Elementary Methods ◽  
Modulo P ◽  
Congruence Equation

The main purpose of this article is using the elementary methods and the properties of the character sums to study the calculating problem of the number of the solutions for one kind congruence equation modulo p (an odd prime) and give some interesting identities and asymptotic formulas for it.

Large families of pseudorandom binary lattices by using the multiplicative inverse modulo P

2019 ◽  
Vol 15 (03) ◽  
pp. 527-546
Author(s):  
Huaning Liu
Keyword(s):  
Exponential Sums ◽  
Character Sums ◽  
Two Dimensional ◽  
Multiplicative Inverse ◽  
Avalanche Effect ◽  
One Dimensional ◽  
Modulo P ◽  
Large Families ◽  
Pseudorandom Measures

Hubert, Mauduit and Sárközy introduced pseudorandom measures for finite pseudorandom binary lattices. Gyarmati, Mauduit, Sárközy and Stewart presented some natural and flexible constructions, which are the two-dimensional extensions and modifications of a few one-dimensional constructions. The upper estimates for the pseudorandom measures of their binary lattices are based on the principle that character sums or exponential sums in two variables can be estimated by fixing one of the variables. In this paper, we constructed two large families of [Formula: see text] dimensional pseudorandom binary lattices by using the multiplicative inverse modulo [Formula: see text], and study the properties: pseudorandom measure, collision and avalanche effect.

scholarly journals The Mean Values of Character Sums and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 318
Author(s):  
Jiafan Zhang ◽  
Yuanyuan Meng
Keyword(s):  
Character Sums ◽  
Gauss Sums ◽  
Mean Values ◽  
Special Polynomials ◽  
Elementary Methods ◽  
The Mean

In this paper, we use the elementary methods and properties of classical Gauss sums to study the calculation problems of some mean values of character sums of special polynomials, and obtained several interesting calculation formulae for them. As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.

scholarly journals Equidistribution Among Cosets of Elliptic Curve Points in Intervals

2020 ◽  
Vol 14 (1) ◽  
pp. 339-345
Author(s):  
Taechan Kim ◽  
Mehdi Tibouchi
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Character Sums ◽  
Fault Analysis ◽  
Signature Schemes ◽  
Large Subgroup

AbstractIn a recent paper devoted to fault analysis of elliptic curve-based signature schemes, Takahashi et al. (TCHES 2018) described several attacks, one of which assumed an equidistribution property that can be informally stated as follows: given an elliptic curve E over 𝔽q in Weierstrass form and a large subgroup H ⊂ E(𝔽q) generated by G(xG, yG), the points in E(𝔽q) whose x-coordinates are obtained from xG by randomly flipping a fixed, sufficiently long substring of bits (and rejecting cases when the resulting value does not correspond to a point in E(𝔽q)) are close to uniformly distributed among the cosets modulo H. The goal of this note is to formally state, prove and quantify (a variant of) that property, and in particular establish sufficient bounds on the size of the subgroup and on the length of the substring of bits for it to hold. The proof relies on bounds for character sums on elliptic curves established by Kohel and Shparlinski (ANTS–IV).

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

scholarly journals On the Mean Values of Certain Character Sums

2013 ◽  
Vol 2013 ◽  
pp. 1-19
Author(s):  
Zhefeng Xu ◽  
Huaning Liu
Keyword(s):  
Character Sums ◽  
Fourth Power ◽  
Power Mean ◽  
Mean Values ◽  
Asymptotic Formulae ◽  
Fourth Power Mean ◽  
The Mean

Letq≥5be an odd number. In this paper, we study the fourth power mean of certain character sums∑χmodq,χ-1=-1*∑1≤a≤q/4aχa4and∑χmodq,χ-1=1*∑1≤a≤q/4aχa4, where∑‍*denotes the summation over primitive characters moduloq, and give some asymptotic formulae.

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