IDA: Interactive Document on Algebra

Let s R[X] be a shift polynomial of degree n-1. We present a formula for the output g_k of the shift register at the (k+1)-th shift.

At the heart of the result below, lies the observation that g_k is the coefficient of X^k in the product of s_n-1X + s_n-2X² + ··· + s₀Xⁿ and (g_k-1X^n-1 + g_k-2X^n-2 + ··· + g_k-n)X^k-n.

We need the following two polynomials.

f =

_j=0^n-1 (g_j -

_i=1^j s_n-i · g_j-i)X^j

and

t = s_n-1X + s_n-2X² + ··· + s₀Xⁿ.

Theorem

In a shift register with shift polynomial s and register contents g₀, ..., g_n-1 R, the (k + 1)-st output g_k is the coefficient of X^k in

(1 + t + t² + ··· + t^k) f.