## Cryptology ePrint Archive: Report 2005/241

On the binary sequences with high $GF(2)$ linear complexities and low $GF(p)$ linear complexities

Hao Chen and Liqing Xu

Abstract: Klapper [1] showed that there are binary sequences of period $q^n-1$ ($q$ is a prime power $p^m$, $p$ is an odd prime) with the maximal possible linear complexity $q^n-1$ when considered as sequences over $GF(2)$, while the sequences have very low linear complexities when considered as sequences over $GF(p)$. This suggests that the binary sequences with high $GF(2)$ linear complexities and low $GF(p)$ linear complexities are note secure in cryptography. In this note we give some simple constructions of the binary sequences with high $GF(2)$ linear complexities and low $GF(p)$ linear complexities. We also prove some lower bounds on the $GF(p)$ linear complexities of binary sequences and a lower bound on the number of the binary sequences with high $GF(2)$ linear complexities and low $GF(p)$ linear complexities .

Category / Keywords: secret-key cryptography / Cryptography, stream cipher, $GF(2)$ linear complexity, $GF(p)$ linear complexity