COMP 165 - Fall 2016 - Homework 2

Due Wednesday, 28 September, 2016 in class

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  1. Write out a list of all fourth-degree polynomials with binary coefficients. For each polynomial, show how it factors into lower-degree irreducible polynomials or write "irreducible".
  2. Sorting the polynomials in the standard order (x4, x4+1, x4+x, ...) call the first irreducible polynomial m(x). Use this to define the multiplication of a finite field of size 16 by calculating a "log table" corresponding to Table 5.5 on page 148.
  3. Using this table, calculate (x2+x) * (x3+x+1), x10 * x11, and the inverse of x3+x mod m(x).
  4. Note that the powers of x may not generate all the elements of the finite field. What happens if you use the last irreducible polynomial in your list as m(x)?