Abstract
We introduce a new measure of pseudorandomness, the (periodic) Hamming correlation of order $\ell$ which generalizes the Hamming autocorrelation ($\ell = 2$). We analyze the relation between the Hamming correlation of order $\ell$ and the periodic analog of the correlation measure of order $\ell$ introduced by Mauduit and Sárközy. Roughly speaking, the correlation measure of order $\ell$ is a finer measure than the Hamming correlation of order $\ell$. 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 $\ell$ for some very small $\ell>2$. Thus they have some undesirable linear structures, in particular in view of cryptographic applications such as secure communications.
Highlights
Sequences with ideal pseudorandomness properties have been widely used in wireless communications and cryptography
The Hamming autocorrelation is an important measure for Frequency hopping sequences (FHSs) [20]
The conventional definition of correlation as the sum of products of corresponding sequence components is mostly suitable for phase-modulation techniques
Summary
Sequences with ideal pseudorandomness properties have been widely used in wireless communications and cryptography. For a T -periodic sequence X = (xn) over a given alphabet A of size m, the Hamming autocorrelation function HX (d) of X was proposed by Lempel and Greenberger [20]:. Key words and phrases: Pseudorandomness, Hamming autocorrelation, frequency hopping sequence, correlation measure of order , cryptography. 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.
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