fm S lF[a;o,a;i,... ,x„] is a projective variety of P G ( n , F ) , of which the affine portion is W nU = V. Mathematical Background 31 The above definitions of affine and projective varieties arc given in terms of a finite set of polynomials. 76 sliows that varieties are in fact defined by polynomial ideals. 76 Let / be an ideal of F [ a ; i , . . ,a;„]. If V(/) denotes the set { ( a i , .

Is a primitive clement of K and a = /3'' (0 < i < q" — 1). The discrete logarithm is a function a = log fj/3''' = i. We can thus log defined by Z„ represent the nonzero elements a £ K by log^a £ Z^n_i. If we adopt the convention t h a t the discrete logarithm of 0 is denoted by oo, then we can represent an element of K by an element of Zgn_i = Z^n_i U {oo}. T h e Zech or Jacobi logarithm offers another logarithmic method for describing a finite field element. The Zech logarithm is based on the function Z: Z g n _ l Z g i - i given by Z{n) = log^(/3" + 1) so /3^(") = /3" + 1 with the convention that /3°° = 0.

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