# Open Problems on the Cross-correlation of *m*-Sequences

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## Abstract

Pseudorandom sequences are important for many applications in communication systems, in coding theory, and in the design of stream ciphers. Maximum-length linear sequences (or *m*-sequences) are popular in sequence designs due to their long period and excellent pseudorandom properties. In code-division multiple-access (CDMA) applications, there is a demand for large families of sequences having good correlation properties. The best families of sequences in these applications frequently use *m*-sequences in their constructions. Therefore, the problem of determining the correlation properties of *m*-sequences has received a lot of attention since the 1960s, and many interesting theoretical results of practical interest have been obtained. The cross-correlation of *m*-sequences is also related to other important problems, such as almost perfect nonlinear functions (APN) and almost bent functions (AB), and to the nonlinearity of S-boxes in many block ciphers including AES. This chapter gives an updated survey of the cross-correlation of *m*-sequences and describes some of the most important open problems that still remain in this area.

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