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 *A polynomial basis is specified by an irreducible polynomial modulo 2, called the field polynomial. The bit string (a$m-1$ … a$2$ a$1$ a$0$) is taken to represent the polynomial $$ over GF(2). The field arithmetic is implemented as polynomial arithmetic modulo p(t), where p(t) is the field polynomial.
 * A normal basis is specified by an element θ of a particular kind. The bit string (a$0$ a$1$ a$2$ … a$m-1$) is taken to represent the element $$ Normal basis field arithmetic is not easy to describe or efficient to implement in general, but is for a special class called Type T low-complexity normal bases. For a given field degree m, the choice of T specifies the basis and the field arithmetic (see Appendix 6.2).

There are many polynomial bases and normal bases from which to choose. The following procedures are commonly used to select a basis representation.


 * Polynomial Basis: If an irreducible trinomial t$m$ + t$k$ + 1 exists over GF(2), then the field polynomial p(t) is chosen to be the irreducible trinomial with the lowest-degree middle term t$k$. If no irreducible trinomial exists, then one selects instead a pentanomial t$m$ + t$a$ + t$b$ + t$c$ + 1. The particular pentanomial chosen has the following properties: the second term t$a$ has the lowest degree m; the third term t$b$ has the lowest degree among all irreducible pentanomials of degree m and second term t$a$; and the fourth term t$c$ has the lowest degree among all irreducible pentanomials of degree m, second term t$a$, and third term t$b$.