Distribution of elements of cosets of small subgroups and applications
type: Item
instance of: scholarly article
zbMATH Open document ID: 1290.11017
publication date: 2012-06-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1103.0296
Mathematics Subject Classification ID: 11B30
zbMATH DE Number: 6042732
zbMATH Keywords: multiplicative subgroups
zbMATH Keywords: discrete logarithm
zbMATH Keywords: uniform distribution
author: Jean Bourgain
author: Sergei Konyagin
author: Igor E. Shparlinski
MaRDI profile type: Publication
OpenAlex ID: W2964233528
arXiv ID: 1103.0296
DOI: 10.1093/IMRN/RNR097
published in: IMRN. International Mathematics Research Notices
title: Distribution of elements of cosets of small subgroups and applications
review text: Let \(p\) be a prime number, \(\mathbb F_p\) be the field of the prime order, \(G\) be a multiplicative subgroup of \(\mathbb F_p\setminus\{0\}\), and \(a\in\mathbb F_p\) be an arbitrary element. The authors study the numbers NEWLINE\[NEWLINEU(k,G,a)= \{x: x\in aG,\;|x|\leq k\},\quad V(k,G,a)= \{p: x\in aG,\;\| x\|\leq k\},NEWLINE\]NEWLINE where \(|x|\) and \(\| x\|\) are the least integer and rational heights, correspondingly. The results are effective in the case \(\log\,| G|= o(\log p)\), roughly. There are two applications of the bounds found: to the number of fixed points of the discrete logarithm and to the simultaneous distribution of monomials \(x^{k_1}\), \(x^{k_2}\).
reviewed by: Ilya D. Shkredov
description: scientific article; zbMATH DE number 6042732