Distributional properties of powers of matrices

Fernando Chamizo; Dulcinea Raboso

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 3, page 801-817
  • ISSN: 0011-4642

Abstract

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We apply the larger sieve to bound the number of 2 × 2 matrices not having large order when reduced modulo the primes in an interval. Our motivation is the relation with linear recursive congruential generators. Basically our results establish that the probability of finding a matrix with large order modulo many primes drops drastically when a certain threshold involving the number of primes and the order is exceeded. We also study, for a given prime and a matrix, the existence of nearby non-similar matrices having large order. In this direction we find matrices of large order when the trace is restricted to take values in a short interval.

How to cite

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Chamizo, Fernando, and Raboso, Dulcinea. "Distributional properties of powers of matrices." Czechoslovak Mathematical Journal 64.3 (2014): 801-817. <http://eudml.org/doc/262197>.

@article{Chamizo2014,
abstract = {We apply the larger sieve to bound the number of $2\times 2$ matrices not having large order when reduced modulo the primes in an interval. Our motivation is the relation with linear recursive congruential generators. Basically our results establish that the probability of finding a matrix with large order modulo many primes drops drastically when a certain threshold involving the number of primes and the order is exceeded. We also study, for a given prime and a matrix, the existence of nearby non-similar matrices having large order. In this direction we find matrices of large order when the trace is restricted to take values in a short interval.},
author = {Chamizo, Fernando, Raboso, Dulcinea},
journal = {Czechoslovak Mathematical Journal},
keywords = {larger sieve; pseudorandom number; finite field; special linear group of degree 2; general linear group of degree 2; larger sieve; pseudorandom number; finite field; special linear group of degree 2; general linear group of degree 2},
language = {eng},
number = {3},
pages = {801-817},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Distributional properties of powers of matrices},
url = {http://eudml.org/doc/262197},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Chamizo, Fernando
AU - Raboso, Dulcinea
TI - Distributional properties of powers of matrices
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 3
SP - 801
EP - 817
AB - We apply the larger sieve to bound the number of $2\times 2$ matrices not having large order when reduced modulo the primes in an interval. Our motivation is the relation with linear recursive congruential generators. Basically our results establish that the probability of finding a matrix with large order modulo many primes drops drastically when a certain threshold involving the number of primes and the order is exceeded. We also study, for a given prime and a matrix, the existence of nearby non-similar matrices having large order. In this direction we find matrices of large order when the trace is restricted to take values in a short interval.
LA - eng
KW - larger sieve; pseudorandom number; finite field; special linear group of degree 2; general linear group of degree 2; larger sieve; pseudorandom number; finite field; special linear group of degree 2; general linear group of degree 2
UR - http://eudml.org/doc/262197
ER -

References

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