How can I evaluate f(x) based on Theorem 3?

[jmomo]

Homework Statement

In F₁₇, 2 is a primitive 8th root of unity. Evaluate f(x) = 7x³+8x²+3x+5 at the eight powers of 2 in F₁₇. Verify that the method requires at most 16 multiplications in F₁₇.

Homework Equations

You can can more clearly see the theorem on page 376-378 and the problem is on page 382 #6:
http://igortitara.files.wordpress.com/2010/04/a-concrete-introduction-to-higher-algebra1.pdf

The Attempt at a Solution

I was able to find that the d=3, but am unclear on how I evaluate f(x) based of Theorem 3.

 

[STEMucator]

Homework Statement

In F₁₇, 2 is a primitive 8th root of unity. Evaluate f(x) = 7x³+8x²+3x+5 at the eight powers of 2 in F₁₇. Verify that the method requires at most 16 multiplications in F₁₇.

Homework Equations

You can can more clearly see the theorem on page 376-378 and the problem is on page 382 #6:
http://igortitara.files.wordpress.com/2010/04/a-concrete-introduction-to-higher-algebra1.pdf

The Attempt at a Solution

I was able to find that the d=3, but am unclear on how I evaluate f(x) based of Theorem 3.

The question states that $2$ is a primitive $2^3$'th root of unity, that is $2^8 = e = 1$.

You need to evaluate $f(2), f(2^2), f(2^3), ... , f(2^8)$. This requires at most $2^r(r-1)$ multiplications, which works out to:

$2^r(r-1) = 2^3(3-1) = 8(2) = 16$