e context of the discussion. Example 5A.16. In GF(16) model of Figure 5A.1, Since a is in GF(4), we may also compute Notice that the variations of trace value with the parameter n are possible. To illustrate variations with the range field of the trace function, consider the element e in GF(16), Tr16(e)=e Pseudonoise Generators We now will prove useful properties of the trace function. Property 1. When a is in GF(q"), Trq(a) has values in GF(q). Proof. (Definition) (Theorem 5A.10) = + a GF(q")=>a = a) = Tra(a). (Definition) Therefore, Trg(a) is a field element whose q-th power equals itself. Only elements of GF(q) have this property. Property 2. Conjugate field elements have the same trace, i.e., GF(q"). (5A.64) The proof of this property follows from the previous property and the fact (Theorem 5A.10) that P(z ) = [P(z)]. Property 3. The trace is linear: for a, b E GF(q) and a, B E GF(q"), (5A.65) Proof. (Definition) (Theorem 5A.10) (a=GF(q)=a=a) = (Definition) Property 4. There are qn-1 elements in GF(q") which have trace value a for each a in GF(q). Appendix 5A. Finite Field Arithmetic Proof. An element a in GF(q") has Trg(a) = a (5A.66) if and only if a is a root of the trace equation +9a  0,aeGF(q) Every element of GF(q") must be the root of exactly one such equation, and each of the q eauations has exactly qn-1 roots. Since all roots of the trace equations are accounted for by elements of GF(q"), there must be exactly qn-1 elements of GF(q") with trace a in GF(q). Property 5. If GF(q)CGF(qk)CGF(qd),then (5A.67) Proof. Consider the nested trace-polynomial expression: (Definitions) k-1 d-1 (Theorem 5A.10) (n = ki + j) = Tra(z) (Definition) Evaluting the above polynomial relation at Z = a, in GF(q), gives the final result. Property 6. Let GF(qd) be the smallest field containing a, and let the min- imum polynomial of ma(z) be denoted by (5A.68) a1 = -Tra(a). (5A.69) Proof. This property follows directly from Theorem 5A.12, whence (5A.70) Pseudonoise Generators APPENDIX 5B. FACTORIZATIONS OF 2N - 1 AND SELECTED P