# How do I express each element in a field F as a power of a primitive element?

I have a field $\mathbb F_{2^4}$, and it is represented as a residue ring of the
polynomials over $\mathbb F_2$ modulo the polynomial $\beta^4 + \beta^3 + \beta^2 + \beta + 1$.

I want to express each element in this field as a power of a primitive element $β+1$.

My questions are:

1. What are the elements of the fields?
2. How do I express them as powers of the given primitive element?

Any hints will be very helpful.

• For the second question: If you say $g:= \beta + 1$, and want to express them as powers of $g$, then this is just $g^1,g^2,\dots,g^{15}=g^0$ – tylo Apr 11 '14 at 12:23
• @tylo : Hello... Thanks that helps... I figured out that for g^15 = g^8... Is it something weird? Am I doing something wrong or it is possible. I did not try after g15. – kingmakerking Apr 11 '14 at 15:09
• Why would it be $g^8$? If your element is primitive and the polynomial is irreducible (I did not check), the multiplicative group generated by $g$ is cyclic and has exactly 15 distinct elements. – tylo Apr 11 '14 at 15:23

## 1 Answer

Just enumerate all powers of your primitive element. There are only 15 of them. That will give you a lookup table that allows you to translate from field element to its expression as a power of the primitive element.

• tallows->allows; Personally I would write "There are only 15 of them before you reach an expression equal to 1". – Cryptographeur Apr 11 '14 at 15:39