AMC
Hamming correlation of higher order
Ming Su Arne Winterhof
Advances in Mathematics of Communications 2018, 12(3): 505-513 doi: 10.3934/amc.2018029

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.

keywords: Pseudorandomness Hamming autocorrelation frequency hopping sequence correlation measure of order $\ell$ cryptography
AMC
On the arithmetic autocorrelation of the Legendre sequence
Richard Hofer Arne Winterhof
Advances in Mathematics of Communications 2017, 11(1): 237-244 doi: 10.3934/amc.2017015

The Legendre sequence possesses several desirable features of pseudorandomness in view of different applications such as a high linear complexity (profile) for cryptography and a small (aperiodic) autocorrelation for radar, gps, or sonar. Here we prove the first nontrivial bound on its arithmetic autocorrelation, another figure of merit introduced by Mandelbaum for errorcorrecting codes.

keywords: Arithmetic autocorrelation Legendre sequence pseudorandom sequences pattern distribution cryptography
AMC
On the generalized joint linear complexity profile of a class of nonlinear pseudorandom multisequences
Alina Ostafe Igor E. Shparlinski Arne Winterhof
Advances in Mathematics of Communications 2010, 4(3): 369-379 doi: 10.3934/amc.2010.4.369
Recently, multisequences have gained increasing interest for applications in cryptography and quasi-Monte Carlo methods. We study the (generalized) joint linear complexity of a class of nonlinear pseudorandom multisequences introduced by the first two authors as well as the linear complexity of its coordinate sequences. We prove lower bounds which are much stronger than in the case of single sequences since the multidimensional case brings in new and favourable effects.
keywords: nonlinear pseudorandom number generators. Linear complexity

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