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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 , .

Fm) and computing the associated variety V ( / ) . The Grobner basis of / provides implicit solutions to the equation system over the algebraic closure of the field F. A particularly useful monomial ordering for finding solutions to this polynomial equation system in F is the lex ordering, which is an example of an elimination ordering. It is worth noting that equation systems arising in cryptography often display many properties. Cryptographic equation systems are often defined over a small finite field GF((7) and the solutions of cryptographic interest lie in this field.

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