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Constructing high order elements through subspace polynomials

Authors:

Qi Cheng,

Shuhong Gao,

Daqing WanAuthors Info & Claims

SODA '12: Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete algorithms

Pages 1457 - 1463

Published: 17 January 2012 Publication History

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Abstract

Every finite field has many multiplicative generators. However, finding one in polynomial time is an important open problem. In fact, even finding elements of high order has not been solved satisfactorily. In this paper, we present an algorithm that for any positive integer c and prime power q, finding an element of order exp(Ω(√q^c)) in the finite field [EQUATION] in deterministic time (q^c)^O(1). We also show that there are exp(Ω(√q^c)) many weak keys for the discrete logarithm problems in those fields with respect to certain bases.

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Cited By

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Chen BLiu H(2018)Constructions of cyclic constant dimension codesDesigns, Codes and Cryptography10.1007/s10623-017-0394-986:6(1267-1279)Online publication date: 1-Jun-2018
https://dl.acm.org/doi/10.1007/s10623-017-0394-9
Brochero Martnez FGarefalakis TReis LTzanaki E(2017)On the multiplicative order of the roots of bXqr+1aXqr+dXcFinite Fields and Their Applications10.1016/j.ffa.2017.05.00247:C(33-45)Online publication date: 1-Sep-2017
https://dl.acm.org/doi/10.1016/j.ffa.2017.05.002

Index Terms

Constructing high order elements through subspace polynomials
1. Mathematics of computing
  1. Discrete mathematics
  2. Mathematical analysis
    1. Numerical analysis
      1. Computations on polynomials

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SODA '12: Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete algorithms

January 2012

1764 pages

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SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

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Society for Industrial and Applied Mathematics

United States

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Published: 17 January 2012

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Kyoto University

SODA '12: ACM-SIAM Symposium on Discrete Algorithms

January 17 - 19, 2012

Kyoto, Japan

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Chen BLiu H(2018)Constructions of cyclic constant dimension codesDesigns, Codes and Cryptography10.1007/s10623-017-0394-986:6(1267-1279)Online publication date: 1-Jun-2018
https://dl.acm.org/doi/10.1007/s10623-017-0394-9
Brochero Martnez FGarefalakis TReis LTzanaki E(2017)On the multiplicative order of the roots of bXqr+1aXqr+dXcFinite Fields and Their Applications10.1016/j.ffa.2017.05.00247:C(33-45)Online publication date: 1-Sep-2017
https://dl.acm.org/doi/10.1016/j.ffa.2017.05.002

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A General Asymptotic Expansion Formula for Integrals Involving High-Order Orthogonal Polynomials

Low-Order Preconditioning of High-Order Triangular Finite Elements

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A General Asymptotic Expansion Formula for Integrals Involving High-Order Orthogonal Polynomials

Low-Order Preconditioning of High-Order Triangular Finite Elements

Higher-order matching polynomials and d-orthogonality

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