中文摘要：

对基于有限（扩）域中离散对数的高效公钥密码体制问题做进一步研究，改进了Wieb Bosma等在扩张次数为奇数时的结果，指出即使在扩张次数为奇数（k=de）的情况下，仍然可以用（e-1）／2个Fp^d。上的元素表示分圆子群中元素在Fp^d上的最小多项式；当取e=3时，无论d为何值，均可以构造基于扩域中的离散对数问题、优化指数为3的密码体制。进一步，对Wieb Bosma等的猜想做了细化分析，指出无论e为奇数或偶数，都存在k=de，使得wieb Bosma等的猜想正确。

英文摘要：

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, of any element in cyclotomic polynomial subgroup can still be represented with （e-1）/2 elements of Fp^d 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.