Linear Complexity, k-Error Linear Complexity, and the Discrete Fourier Transform

Complexity measures for sequences of elements of a finite field play an important role in cryptology. We focus first on the linear complexity of periodic sequences. By means of the discrete Fourier transform, we determine the number of periodic sequences S with given prime period length N and linear complexity LN, 0(S)=c as well as the expected value of the linear complexity of N-periodic sequences. Cryptographically strong sequences should not only have a large linear complexity, but also the change of a few terms should not cause a significant decrease of the linear complexity. This requirement leads to the concept of the k-error linear complexity LN, k(S) of sequences S with period length N. For some k and c we determine the number of periodic sequences S with given period length N and LN, k(S)=c. For prime N we establish a lower bound on the expected value of the k-error linear complexity.