So, yes the fielsfields are are all the same. But (as an engineer would say) practice and theory are different. And itsit is that difference that is important.
Update (edited in from a post below) So I am still looking for insights or suggestions. Thanks.
I should explain what I want to do in more detail andPoncho - you are the reasons. I knowonly one to answer so far that all finite fields ofappears to have a given size are isomorphic. That means that there is effectively only "one" fieldclue about this (I am new to crypto. However, they are NOTstackexchange and don't yet have the same from a different standpoint. There has been a lot of workpoints to find fields that have minimal implementation complexityeither comment or chat). Mainly because ofI am unfortunately (for the obsession with implementing them entirelytime) stuck using a library computer in very small hardwaretown every couple of days and my reference library is not available (minimum gatesand it has been 40 years since abstract algebra in college which didn't even cover this area).
However, from an encryption standpoint I have picked up and read several books in the area (abstract algebra and finite fields), a very high levelbut none of complexity is desired. You want x * y to be highly non-affinethem address this topic or even something close enough. Minimum complexityThey just address the standard theory, which except for information on bases is not optimum herevery useful. So the simple irreducible polynomials that encryption products are using areI have not suitable. They are explictly choosenbeen able to have minimum complexityfind any online references on this direct topic. ItDo you have any suggestions?
Why the 8 4 3 2 0 polynomial? Or is difficult to determinethat just the complexity offirst one you picked? As far as generating a random field from the polynomial. But, by generating random fields the odds of getting a highly complex field are very goodrandom bais for g01, g02, . And it is a good way to make the encryption hard to break becuase the properties of a specific field cannot be directly used in cyptoanalysis.
So., yes the fiels are are all the sameg80 works fine. ButThen I only need to generate another random basis for g02, g03, (as an engineer would say) practice and theory are different. And its that difference that is important.
My algorithm above is not quite correct. It turns outg81. There are some constraints there that I don't currently have enough information to use to directly constrain the sub-algorithm for buildingrandom generator, but brute force would work. I actually think that the main basis products is not rightcompletely constrains the g02, g03, . I have solved some.. basis (the first row / column in the basis multiplication table - that sufficies to ensure distribution and all of it,the other properties) but there are dependencies that I haven't identified yet. I don't know ifdetermined an efficient way to do that approach will work or not.
Possibly a genetic algorithm. However, other approaches mayI would like to be just as good. Since finite fields ofmore intelligent about the same sizeprocess. If you are isomorphiccorrect, an alternative is to buildI could create the basis, determine the affine mapping from the field generated by the 8 4 3 1 0 polynomial and then create the g02, g03, ... basis by applying the affine mapping. If that does, in fact, work then simply generating a random isomorphismaffine mapping should also work. That appears to haveis trivial - just create a lot ofbasis to ensure that the same problemsmatrix is invertable and add a random vector.
So I am still looking for insights or suggestionsdon't know if it will give a better non-affine result, but I suspect that it would (this is research!). ThanksOtherwise there would be no point in using different polynomials to obtain "simple" implementations. I simply want to do the opposite of "simple"!