Maximum-order Complexity and Correlation Measures

We estimate the maximum-order complexity of a binary sequence in terms of its correlation measures. Roughly speaking, we show that any sequence with small correlation measure up to a sufficiently large order $k$ cannot have very small maximum-order complexity.


Introduction
the predictability of a sequence and thus its unsuitability in cryptography. For surveys on linear complexity and related measures of pseudorandomness see [6,13,14,17,20,21].
Let k be a positive integer. Mauduit and Sárközy introduced the (Nth) correlation measure of order k of a binary sequence S = (s i ) ∞ i=0 in [10] as where the maximum is taken over all D = (d 1 , d 2 , ..., d k ) with non-negative integers 0 ≤ d 1 < d 2 < ... < d k and U such that U + d k ≤ N. (Actually, [10] deals with finite sequences ((−1) s i ) N −1 i=0 of length N over {−1, +1}.) Brandstätter and the second author [2] 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 L = (ℓ i ) ∞ i=0 defined by where p > 2 is a prime, satisfies and thus (1) implies see [10] and [19, see [8,9,15]. 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 M(S, N) and the correlation measures C k (S, N) of order k. Our main result is the following theorem: Again, any nontrivial bound on C k (S, N) for all k up to a sufficiently large order provides a nontrivial bound on M(S, N). For example, for the Legendre sequence we get immediately see also [19,Theorem 9.3]. (f (N) = O(g (N)) is equivalent to f (N) ≪ g(N).) We prove Theorem 1 in the next section. The expected value of the Nth maximum-order complexity is of order of magnitude log N, see [8] as well as [15,Remark 4] and references therein. Moreover, by [1] for a 'random' sequence of length N the correlation measure C k (S, N) is of order of magnitude √ kN log N and thus by Theorem 1 M(S, N) ≥ 1 2 log N + O(log log N) which is in good correspondence to the result of [8].
In Section 3 we mention some straightforward extensions. If m is a prime, then x → hx is a permutation of Z m for any h ≡ 0 mod m and the sums in (4) can be estimated by the correlation measure C k (S, N) of order k for m-ary sequences as it is defined in [11] and we get 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 analog with bounded lags, see [7] for the analog of (1). For example, the two-prime generator T = (t i ) ∞ i=0 , see [3], of length pq with two odd primes p < q satisfies if gcd(i, pq) = 1 and its correlation measure of order 4 is obviously close to pq, see [16]. However, if we bound the lags d 1 < . . . < d k < p one can derive a nontrivial upper bound on the correlation measure of order k with bounded lags including k = 4 as well as lower bounds on the maximum-order complexity using the analog of Theorem 1 with bounded lags.
Finally, we mention that the lower bound (3) for the Legendre sequence can be extended to Legendre sequences with polynomials using the results of [5] as well as to their generalization using squares in arbitrary finite fields (of odd characteristic) using the results of [12,18]. For sequences defined with a character of order m see [11].

Acknowledgement
The authors are supported by the Austrian Science Fund FWF Projects F5504 and F5511-N26, respectively, which are part of the Special Research Program "Quasi-Monte Carlo Methods: Theory and Applications". L.I. would like to express her sincere thanks for the hospitality during her visit to RICAM.