Thursday 5 November 2015
If is prime and is an irreducible polynomial over , then is a field.
Exercise 2.1 How many elements does have?
What counting principles do you use?
Confusion in AES works on a single byte, i.e. as a function . This is usually implement as a look-up table called an S-box, but it also has an algebraic definition.
AES considers a byte as an element
Let’s have a look at multiplication and addition in this field.
The element is a primitive element in
If you consider the sequence , how many distinct elements do you get?
Problem 2.1 Consider two polynomials over :
Find and to satisfy the division theorem, with .
Exercise 2.2 Consider two polynomials over :
Find and to satisfy the division theorem, with .
Exercise 2.3 Consider two polynomials over :
Find and to satisfy the division theorem, with .
Exercise 2.4 Consider the polynomial
over and the finite field .
- Find a primitive element .
- Tabulate all the powers of with corresponding polynomials.
- Use the table to calculate the following:
Exercise 2.5 Using as defined in the AES standard, i.e. modulo , calculate the following products