Further investigations on efficient public-key cryptosystems based on discrete logarithm in finite field（exten-sion） were provided

and in case of the degree of field extension being odd

the ordinary results proposed by Wieb Bosma et al were optimized.It was pointed out that even though the degree（k=de） of field extension is odd

the minimal poly nomial overFpd of any element in cyclotomic polynomial subgroup can still be represented with（e?1）/2 elements of Fpd

in the case of e=3

no matter what d is

a cryptosystem with optimization 3 can always be constructed based on the dis-crete logarithm in field extension.Further

it was pointed out that for any e

positive or negative

there exists k=de such that Wieb Bosma’s conjecture is true.