HAMMING CORRELATION OF HIGHER ORDER

. We introduce a new measure of pseudorandomness, the (periodic) Hamming correlation of order (cid:96) which generalizes the Hamming autocorrelation ( (cid:96) = 2). We analyze the relation between the Hamming correlation of order (cid:96) and the periodic analog of the correlation measure of order (cid:96) introduced by Mauduit and S´ark¨ozy. Roughly speaking, the correlation measure of order (cid:96) is a ﬁner measure than the Hamming correlation of order (cid:96) . However, the latter can be much faster calculated and still detects some undesirable linear structures. We analyze examples of sequences with optimal Hamming correlation and show that they have large Hamming correlation of order (cid:96) for some very small (cid:96) > 2. Thus they have some undesirable linear structures, in particular in view of cryptographic applications such as secure communications.


Introduction
Sequences with ideal pseudorandomness properties have been widely used in wireless communications and cryptography. Frequency hopping sequences (FHSs) are an essential part of spread spectrum communication systems such as frequency hopping code division multiple access (FH-CDMA) systems and multiuser radar systems, see for example [4,Chapter 15]. The Hamming autocorrelation is an important measure for FHSs [20].
The conventional definition of correlation as the sum of products of corresponding sequence components is mostly suitable for phase-modulation techniques. For other modulation techniques where large sets of mutual orthogonal signals are employed, the appropriate measure for correlation is the Hamming correlation. More precisely, for a T -periodic sequence X = (x n ) over a given alphabet A of size m, the Hamming autocorrelation function H X (d) of X was proposed by Lempel and Greenberger [20]: is the Kronecker delta. The maximum nontrivial Hamming autocorrelation of X is denoted by Lempel and Greenberger proved the first lower bound on H(X) for a given period T and alphabet size m [20]. Later many FHSs meeting the Lempel-Greenberger bound were constructed, see for example [8,9,17]. We call such sequences attaining the Lempel-Greenberger bound optimal. Mauduit and Sárközy [22,23,30] introduced a pseudorandomness measure for finite sequences. We consider its analog for periodic sequences, see [29].
Let E m = {ε 1 , . . . , ε m } be the set of the complex m-th roots of unity and F be the set of the m! bijections between A and E m . For a T -periodic sequence X = (x 0 , x 1 , . . .) over A the periodic correlation measure of order is defined as where the maximum is taken over all Φ = (ϕ 1 , ϕ 2 , . . . , ϕ ) ∈ F and D = ( In particular, the correlation measure of order detects undesirable linear structures. For other figures of merit for finding linear structures and their relations see [3,6,10,11,15,25] as well as the surveys [24,26,31,32]. In this paper we introduce the Hamming correlation of order , which generalizes the Hamming autocorrelation ( = 2) and compare this new measure with the correlation measure of order , see Theorem 1 below. Roughly speaking, the correlation measure of order is a finer measure but the Hamming correlation of order is easier to calculate since it does not depend on the (m!) choices of Φ. Though many frequency hopping sequences with optimal Hamming autocorrelation have been proposed, they may still have some intrinsic linear structures. Such undesirable structures can be detected by studying the Hamming correlation of order > 2. It is possible to determine the subsequent carrier frequencies from the previous ones during frequency hopping data transmission and predict further data from given partial data, resulting in security risks.
The paper is organized as follows. In Section 2, we define the Hamming correlation of order of a periodic sequence and analyze the relationship between this measure and the correlation measure of order . In Sections 3.1-3.3 we show that some sequences with optimal Hamming autocorrelation have very large Hamming correlation of order for some small > 2. In Section 3.4 we also study some character sequences and show that they have small Hamming correlation of order up to a very large .

Hamming correlation measure of order
In this section we generalize the Hamming autocorrelation to the Hamming correlation of order for studying the intrinsic linear structure of sequences. We restrict ourselves to sequences over A = Z m . However, this definition can be easily extended to an arbitrary alphabet A of size m using any fixed bijection between A and Z m .
Let X = {x n } be a T -periodic sequence over Z m and be a positive integer. Now we define using Kronecker's delta where the maximum is taken over all (Note that the sums inside the large brackets in (1) are considered modulo m.) For a = 0, we denote the Hamming correlation of order of X by H (X) = H ,0 (X). Note that the Hamming autocorrelation is H(X) = H 2 (X).
Next we state and prove a relation between H ,a (X) and the correlation measure of order .
By (1) and (2) we obtain Separating the summand for i = 0 we obtain If m is a prime, x → ω ±ix is a bijection between Z m and the set of complex m-th roots of unity E m for any i = 1, . . . , m−1 and the maximum on the right hand side of (3) is bounded by Γ (X). For composite m the result follows from [6, Propositions 1 and 2].
Remark 1. The correlation measure of order of a random sequence is small up to a sufficiently large , see [1,2]. Hence, the Hamming correlation of order of a pseudorandom sequence should be close to T m for all small . We present a simple corollary of the Lempel-Greenberger bound [20,Lemma 4] stated in [14, Corollary 1.2]: The following result reveals that the Hamming correlation of some order attains high peak values if there is some approximate linear structure in a given sequence.  Conversely, if H (X) is large, then there is a linear recurrence of − 1 summands with coefficients ±1 satisfied by many sequence elements.

Examples
In this section, we study H ,a (X) for some sequences with optimal Hamming autocorrelation. We show that their Hamming correlation of order can be large for some small > 2 if the parameters are not carefully chosen.

A construction of Lempel and Greenberger.
We study a sequence construction Y of [20] over Z p k of period p n − 1 with H(Y ) = p n−k − 1 meeting the bound (4) by [20,Theorem 1]. We show that in some cases there is a small > 2 and some a with large H ,a (Y ). It is constructed as follows. Let p be a prime. We start with a maximum length sequence X over Z p of period p n − 1, that is, X = (x j ) satisfies a linear recurrence over Z p of length n, x j+t c t (mod p), j ≥ 0.
For 1 ≤ k < n we derive from each k consecutive elements of X an element of Z p k and derive a sequence Y = (y j ) over Z p k of period p n − 1 which meets the Lempel-Greenberger bound, see also [20,Lemma 3].
For 0 ≤ j < p n − 1 we have x j+t+i c t )p i (mod p k ),  (6) and (5). Note that for k ≥ 2 x j+t c t (mod p 2 ) by (5) again. Hence, by the pigeonhole principle there exists b ∈ Z p k−2 such that size at most (n + 1)(p − 1) + 1 ≤ (n + 1)p and is divisible by p because of (5). Therefore, there exists some a ∈ [bp 2 , (b + 1)p 2 ) (divisible by p) such that for at least p n−k+2 n + 1 different 0 ≤ j < p n − 1. If c t = 1 for at least one 0 ≤ t < n, we have for = n + 1 − |{t : c t = 0}|, which is larger than the desired value close to p n−k , see Theorem 1, if n is small with respect to p. Now we study the concrete example given above [20,Theorem 1] in more detail: We start with a linear recurrence of order 3, and derive via (6) a sequence over Z 9 of period 26 satisfying Note that x j+3 − x j+2 + x j ∈ {0, 3} by (7) and thus y j+3 − y j+2 + y j ≡ 0, 3 (mod 3 2 ). Since the period of X is 26 we have and otherwise x j+3 − x j+2 + x j = 0. The first case occurs 9 times and the latter 17 times, consequently, H 3 (Y ) ≥ 17 which is much larger than the desired value T m = 26 9 . 3.2. FHSs based on discrete logarithms of Z p n . In [21], some classes of FHSs meeting the Lempel-Greenberger bound were constructed based on discrete logarithms in Z p n (or generalized cyclotomy). Let p = ef + 1 be an odd prime such that f is odd, n a positive integer, and g a primitive root modulo p n . For our purposes it is enough to define X = (x j ) only for x j with j ∈ Z * p n by x j = log g (j) (mod ep n−1 ) or equivalently x j = i if and only if j = g i+tep n−1 , 0 ≤ i < ep n−1 .
Hence (8) is satisfied for at least ϕ(p n ) = p n − p n−1 different j ∈ Z * p n and thus H 4 (X) ≥ p n − p n−1 for n ≥ 4 or n = 2, 3 and p > 3.
In [21, Method A] a method is described how to destroy the linear structure of X.

3.3.
Sequences derived from Fermat quotients. Next we give another very interesting sequence of optimal Hamming autocorrelation but with Hamming correlation measure of order 4 equal to its period.
For a prime p and an integer u with gcd(u, p) = 1 the Fermat quotient q p (u) modulo p is defined as and we also define q p (kp) = 0 for k ∈ Z. Some well-known properties of Fermat quotients are discussed for example in [13]. For any integers k and u we have Some pseudorandomness measures of the binary sequences derived from Fermat quotients are discussed in [5,7,12,16,18,27]. A collection of optimal families of perfect polyphase sequences using the Fermat quotient sequences is proposed in [28]. Note that (q p (u)) is a p 2 -periodic sequence over Z p , and we have the following result.
Proof. First we proof H Q (τ ) = p for any 0 < τ < p 2 . The following equations are all over F p . In order to investigate the number of solutions for the equation over F p for some τ , we need to discuss two cases. (i) p |τ From (9) and (10) we derive Therefore, by the definition of Hamming correlation we obtain H Q (τ ) = p.