scholarly journals A CLASS OF INCOMPLETE CHARACTER SUMS

2014 ◽  
Vol 65 (4) ◽  
pp. 1195-1211 ◽  
Author(s):  
L. Fu ◽  
D. Wan
Keyword(s):  
Character Sums ◽  
Incomplete Character Sums

scholarly journals Incomplete character sums and a special class of permutations

2001 ◽  
Vol 13 (1) ◽  
pp. 53-63 ◽  
Author(s):  
S. D. Cohen ◽  
H. Niederreiter ◽  
I. E. Shparlinski ◽  
M. Zieve
Keyword(s):  
Special Class ◽  
Character Sums ◽  
Incomplete Character Sums

scholarly journals The Burgess inequality and the least kth power non-residue

2015 ◽  
Vol 11 (05) ◽  
pp. 1653-1678 ◽  
Author(s):  
Enrique Treviño
Keyword(s):  
Upper Bound ◽  
Class Number ◽  
Character Sums ◽  
Explicit Estimate ◽  
Dirichlet Characters ◽  
Cubic Fields ◽  
Incomplete Character Sums

The Burgess inequality is the best upper bound we have for incomplete character sums of Dirichlet characters. In 2006, Booker gave an explicit estimate for quadratic Dirichlet characters which he used to calculate the class number of a 32-digit discriminant. McGown used an explicit estimate to show that there are no norm-Euclidean Galois cubic fields with discriminant greater than 10140. Both of their explicit estimates are on restricted ranges. In this paper, we prove an explicit estimate that works for any range. We also improve McGown's estimates in a slightly narrower range, getting explicit estimates for characters of any order. We apply the estimates to the question of how large must a prime p be to ensure that there is a kth power non-residue less than p1/6.

On Incomplete Character Sums to a Prime-Power Modulus

1987 ◽  
Vol 30 (3) ◽  
pp. 257-266 ◽  
Author(s):  
J. H. H. Chalk
Keyword(s):  
Prime Power ◽  
Short Proof ◽  
Character Sums ◽  
Solution Set ◽  
Character Sum ◽  
Primitive Character ◽  
Image Position ◽  
Incomplete Character Sums ◽  
Main Inequality ◽  
Inequality Theorem

AbstractLet x denote a primitive character to a prime-power modulus k = pα. The expected estimatefor the incomplete character sum has been established for r = 1 and 2 by D. A. Burgess and recently, he settled the case r = 3 for all primes p < 3, (cf. [2] for the proof and for references). Here, a short proof of the main inequality (Theorem 2) which leads to this result is presented; the argument being based upon my characterization in [3] of the solution-set of a related congruence.

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).

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