Linear Complexity

Related Concepts

Berlekamp–Massey Algorithm; Combination Generator; Filter Generator; Linear Feedback Shift Register; Minimal Polynomial; Stream Cipher

Definition

The linear complexity of a semi-infinite sequence $\mathbf{s} = {({s}_{t})}_{t\geq 0}$ of elements of ${\mathbf{F}}_{q}$, $\Lambda (\mathbf{s})$, is the smallest integer $\Lambda$ such that $\mathbf{s}$ can be generated by a linear feedback shift register (LFSR) of length $\Lambda$ over ${\mathbf{F}}_{q}$, and is $\infty$ if no such LFSR exists. By way of convention, the linear complexity of the all-zero sequence is equal to $0$. The linear complexity of a linear recurring sequence corresponds to the degree of its minimal polynomial.

The linear complexity $\Lambda (\mathbf{{s}^{n}})$ of a finite sequence $\mathbf{{s}^{n}} = {s}_{0}{s}_{1}\ldots {s}_{n-1}$ of $n$ elements of ${\mathbf{F}}_{q}$ is the length of the shortest LFSR which produces $\mathbf{{s}^{n}}$ as its first $n$output terms...