Improvements on results of representation of elements in cyclotomic subgroup

Further investigations on efficient public-key cryptosystems based on discrete logarithm in finite field (extension) 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 over F_pd of any element in cyclotomic polynomial subgroup can still be represented with (e-1)/2 elements of F_pd, in the case of e=3, no matter what d is, a cryptosystem with optimization 3 can always be constructed based on the discrete 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.