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.

  • $\begingroup$ For the second question: If you say $g:= \beta + 1$, and want to express them as powers of $g$, then this is just $g^1,g^2,\dots,g^{15}=g^0$ $\endgroup$ – tylo Apr 11 '14 at 12:23
  • $\begingroup$ @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. $\endgroup$ – kingmakerking Apr 11 '14 at 15:09
  • $\begingroup$ 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. $\endgroup$ – tylo Apr 11 '14 at 15:23

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.