(a) Using interpolation, give a polynomial f ∈ F₁₁[x] of degree at most 3 satisfying f(0) = 2; f(2) = 3; f(3) = 1; f(7) = 6

(b) What are all the polynomials in F₁₁[x] which satisfy f(0) = 2, f(2) = 3, f(3) = 1, f(7) = 6?

(2) Hand in your completed worksheets from labs \Fast Multiplication" and \Fast Multiplication II". Hand it in to me by saving the worksheet to a le (after making sure all the cells you want are evaluated) and then emailing it to me.

(3) Let F be a eld and a(x) ∈ F[x] be a polynomial of degree n - 1 = 3^k - 1.

(a) Show that a(x) can be decomposed into

a(x) = b(x³ ) + x .
c(x³) + x².
d(x³);

where b(x), c(x) and d(x) are polynomials in F[x] of degree at most n/3 - 1 = 3^{k - 1} - 1.

(b) Show that if ω ∈ F is a primitive nth root of unity, then a(x) can be evaluated at all the powers of ! by recursively evaluating b(x), c(x) and d(x) at the powers of ω³.

(c) Put all of this together into an algorithm similar to FFT for evaluating a(x) at the powers of ω.

(d) What are the number of additions and number of multiplications in F that this algorithm does on input size n?

(e) The set S = {1, ω, ω² ;ω^{n -1}} has some special properties that make this "3-ary" FFT (and the "binary" FFT from class) work. What properties does a set S need to be used in this way (or in the original FFT algorithm)? Can you find any other sets that have these properties?