Reference Work Entry

Encyclopedia of Cryptography and Security

pp 277-277

# Cross-Correlation

• Tor HellesethAffiliated withThe Selmer Center, Department of Informatics, University of Bergen

## Definition

Let {at} and {bt} be two sequences of period n (so at = at + n and bt = bt + n for all values of t) over an alphabet being the integers mod q (Modular Arithmetic). The cross-correlation between the sequences {at} and {bt} at shift τ is defined as
$$C(\tau ) ={ \sum \limits _{t=0}^{n-1}}{\omega }^{{a}_{t+\tau }-{b}_{t} }$$
where $$\omega$$ is a complex q-th root of unity. Note that in the special case of binary sequences, then q = 2 and $$\omega = -1$$.

In the special case when the two sequences are the same, the cross-correlation is the same as the autocorrelation.

## Applications

Many applications in stream ciphers and communication systems require large families of cyclically distinct sequences with a low maximal nontrivial value of the auto- and cross-correlation between any two sequences in the family.

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