d under addition and multiplication modulo p where p is the number of distinct integer elements of the field. Example 5A.9. In the addition and multiplication tables of GF(16) (see Figure 5A.1), 1 + 1 = 0 and hence {0, 1} = GF(2). Now consider the case in which a subfield GF(q) (not necessarily an inte- ger subfield) exists within a larger field, e.g., as GF(2) or GF(4) exist within GF(16) in Figure 5A.1. Let x be a field element in the larger field but not in GF(q). Find all elements in the larger field which are equal to linear combinations of powers of X with coefficients in GF(q), i.e., elements of the form (5A.26) where a = is a coefficient vector of elements from GF(q). The right side above will be referred to as a pseudopolynomial in X, since x is not an indeterminate and the sum is simply a field element. For a given value of d, the number of possible coefficient vectors a is qd. If the larger finite field contains n elements, we shall choose d SO that qd > n. This means Va=Vb (5A.27) for some a # b. Hence, (5A.28) (5A.29) and the coefficients (a - bi) cannot all be zero, though the non-zero coef- ficient with largest subscript need not be (ad-1 - bd-1). Replacing X by an indeterminate Z above, it is apparent that the set of polynomials over GF(q) which has x as a root, is not empty. One polynomial totally characterizes the properties of X and the way it interacts with elements of GF(q). DEFINITION 5A.6. The monic polynomial mx(z) over GF(q), of minimum degree dm with respect to all polynomials over GF(q) having X as a root, is called the minimum polynomial mx(z) of x over GF(q). Pseudonoise Generators +01abcdefghjkmnpq 001bcdefghjkmnpq 110 bafjqcpkdhnmg aabo1 idcmjkpfgeqhn bba10 jfndhgcpqekm cfdjlaklmqbegpnh ddjcfalgben1mkhg eqmnkg0hdfpcabj 1 ffcjdbhneaqpgmk g g pkhmednobqacf1 hkpgqnfeb0mjda j jdfcblpaqmnhke k hgpemqalndjbj mneqgkapcjhdlf nmqephbgfdkjl0ca p pghknqjmlaebfc0d qenmhplkjcgfbad0 . 01abcdefghjkmnpq 0000000000000000 101abcdefghjkmnpq a 10ab1ghjkmnpqcdef Obla mnpqcdefghji 0cgmdefahjkbnpq1 ddhnef