AAECC (2005) 16: 219–228
DOI 10.1007/s00200-005-0181-0
Alev Topuzo˘ glu · Arne Winterhof
On the linear complexity profile of nonlinear
congruential pseudorandom number
generators of higher orders
Received: 7 April 2004 / Published online: 1 July 2005
© Springer-Verlag 2005
Abstract Nonlinear congruential methods are attractive alternatives to the clas-
sical linear congruential method for pseudorandom number generation. Genera-
tors of higher orders are of interest since they admit longer periods. We obtain
lower bounds on the linear complexity profile of nonlinear pseudorandom number
generators of higher orders. The results have applications in cryptography and in
quasi-Monte Carlo methods.
Keywords Linear complexity profile · Nonlinear pseudorandom number gener-
ators · Inversive generators · Sequences over finite fields · Recurrences of higher
order
1 Introduction
The linear complexity profile of a sequence (s
n
) = s
0
,s
1
,... over the field F is
the function L((s
n
),N) defined for every positive integer N , as the least order L
of a linear recurrence relation over F
s
n+L
= c
L-1
s
n+L-1
+ ... + c
0
s
n
,
0 ≤ n ≤ N -L-1, which (s
n
) satisfies. We use the convention that L((s
n
),N) = 0
if the first N elements of (s
n
) are all zero and L((s
n
),N) = N if the first N - 1
elements of (s
n
) are zero and s
N -1
= 0.
A. Topuzo˘ glu
Sabanci University, Orhanli, 34956 Tuzla, Istanbul, Turkey
E-mail: alev@sabanciuniv.edu
A. Winterhof (B )
Johann Radon Institute for Computational and Applied Mathematics,
Austrian Academy of Sciences, Altenbergerstr. 69, 4040 Linz, Austria
E-mail: arne.winterhof@oeaw.ac.at