Minimal Polynomial

Related Concepts

Berlekamp–Massey Algorithm; Linear Complexity; Linear Feedback Shift Register; Stream Cipher

Definition

The minimal polynomial of a linear recurring sequence $\mathbf{ s} = {({s}_{t})}_{t\geq 0}$ of elements of ${\mathbf F}_{q}$ is the polynomial $P$ in ${\mathbf F}_{q}[X]$ of lowest degree such that ${({s}_{t})}_{t\geq 0}$ is generated by the linear feedback shift register (LFSR) with characteristic polynomial $P$. In other terms, $P ={ \sum \nolimits }_{i=0}^{L-1}{p}_{i}{X}^{i} + {X}^{L}$ is the characteristic polynomial of the linear recurrence relation of least degree satisfied by the sequence:

$${s}_{t+L} + \sum \limits_{i=0}^{L-1}{p}_{ i}{s}_{t+i} = 0,\;t \geq 0.$$

The minimal polynomial of a linear recurring sequence s is monic and unique; it divides the characteristic polynomial of any LFSR which generates s. The degree of the minimal polynomial of s is called its linear complexity. The period of the minimal polynomial of sis equal to the least...