1. Introduction
For a positive integer
N, the
Nth linear complexity of a binary sequence
is the smallest positive integer
L such that there are constants
with
(We use the convention
if
and
if
.) The
Nth linear complexity is a measure for the predictability of a sequence and thus its unsuitability in cryptography. For surveys on linear complexity and related measures of pseudorandomness see [
1,
2,
3,
4,
5,
6].
Let
k be a positive integer. Mauduit and Sárközy introduced the (
Nth)
correlation measure of order k of a binary sequence
in [
7] as
where the maximum is taken over all
with non-negative integers
and
U such that
. (Actually, [
7] deals with finite sequences
of length
N over
.)
Brandstätter and the second author [
8] proved the following relation between the
Nth linear complexity and the correlation measures of order
k:
Roughly speaking, any sequence with small correlation measure up to a sufficiently large order k must have a high Nth linear complexity as well.
For example, the Legendre sequence
defined by
where
is a prime, satisfies
and thus (
1) implies
Using
for any
we get
see [
7,
9] (Theorem 9.2). (Here
is equivalent to
for some absolute constant
c.)
The
Nth
maximum-order complexity of a binary sequence
is the smallest positive integer
M such that there is a polynomial
with
see [
10,
11,
12]. Obviously we have
and the maximum-order complexity is a finer measure of pseudorandomness than the linear complexity.
In this paper we analyze the relationship between maximum-order complexity and the correlation measures of order k. Our main result is the following theorem:
Theorem 1. For any binary sequence we have Again, any nontrivial bound on
for all
k up to a sufficiently large order provides a nontrivial bound on
. For example, for the Legendre sequence we get immediately from (
2)
Now we have either
and the bound (
4) below is trivial or
which implies
see also [
9] (Theorem 9.3). (Here
is equivalent to
.)
We prove Theorem 1 in the next section.
The expected value of the
Nth maximum-order complexity is of order of magnitude
, see [
10] as well as [
12] (Remark 4) and references therein. Moreover, by [
13] for a sequence of length
N with very high probability the correlation measure
is of order of magnitude
and thus by Theorem 1
which is in good correspondence to the result of [
10].
In
Section 3 we mention some straightforward extensions.
3. Further Remarks
Theorem 1 can be easily extended to
m-ary sequences with
along the lines of [
14]:
Let
be a primitive
mth root of unity. Then we have
As in the proof of Theorem 1 we get
We have one term of absolute value
and
terms of the form
with
,
,
and
.
If
m is a prime, then
is a permutation of
for any
and the sums in (
5) can be estimated by the correlation measure
of order
k for
m-ary sequences as it is defined in [
15] and we get
If
m is composite,
is not a permutation of
if
and we have to substitute the correlation measure of order
k by the power correlation measure of order
k introduced in [
14].
Now we return to the case .
Even if the correlation measure of order
k is large for some small
k, we may be still able to derive a nontrivial lower bound on the maximum-order complexity by substituting the correlation measure of order
k by its analogue with bounded lags, see [
16] for the analogue of (
1). For example, the two-prime generator
, see [
17], of length
with two odd primes
satisfies
if
and its correlation measure of order 4 is obviously close to
, see [
18]. However, if we bound the lags
one can derive a nontrivial upper bound on the correlation measure of order
k with bounded lags including
as well as lower bounds on the maximum-order complexity using the analogue of Theorem 1 with bounded lags.
Finally, we mention that the lower bound (
4) for the Legendre sequence can be extended to Legendre sequences with polynomials using the results of [
19] as well as to their generalization using squares in arbitrary finite fields (of odd characteristic) using the results of [
20,
21]. For sequences defined with a character of order
m see [
15].