這一篇論文主要討論在橢圓曲線上解log問題，我把M-O-V, Semave, 和Voloch的論文作重新的整理。
橢圓曲線上解log問題如下：E是一個在有限體F上的橢圓曲線，給曲線上任兩點有理點P, Q. 我們想算出是否有整數m滿足Q=[m]P.
如果P的order是n,且F的特徵值是p,我們分兩個部分討論
(1)n跟p互質
(2)n跟p不互質
並且我們比較出每個方法的優缺點。

In this master thesis, we talk about the discrete logarithm on an elliptic curve, and this is a reorganization of M-O-V's, Semaev's, and Voloch's papers.
Let E be an elliptic curve over a finite field F and char(F)=p,
the discrete logarithm problem on an elliptic curve is to compute an integer m such that Q=[m]P, where P and Q are rational points on E. If P has order n, we consider two cases:
(1)gcd(n,p)=1
(2)gcd(n,p) is not 1
Finally, we can use the Chinese remainder theorem to compute m

[MOV] A.Menezes, T.Okamoto and S. Vanstone, Reducing elliptic curves logarithms to logarithms in a finite field, IEEE Trans. Info. Theory, 39(1993) 1639-1646.
[M] Alfred Menezes, Elliptic curve public key crytosystems, kluwer Academic Puublishers, 1993.
[Se] I.A. Semave, Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p, Math. Comp.,67(1998),353-356.
[V1] J.F. Voloch, The discrete logarithm problem on elliptic curves and descents, to appear. Available at http://www.ma.utexas.edu/users/voloch
[V2] J.F. Voloch, Explicit p-descent for elliptic curves in character p, Composition Math. 74(1990) 247-258
[V3] J.F. Voloch, An analogue of the Weierstrass z-function in characteristic p, Acta Arithmetica. LXXIX(1997)1-6.
[Si] J.H. Silverman, The arithmetic of elliptic curves, Springer, New York,1986.
[J] N. Jacobson, Basic algebra II, W.H Freeman and Company
[Ser] Jean-Pierre Serre, Local fields, GTM67, Springer, New York, 1979
[Sc1] R. Schoof, Elliptic curves over finite fields and computation of square roots mod p, Math. of Comp.,44(1985),483-494.
[Sc2] R. Schoof, Nonsingular plane cubics over finite field, J Combinant Theory, A46(1987)183-211.
[VW] J.F. Voloch and J.L. Waiker, Euclidean weights of codes from elliptic curves over rings, Preprint, 1997
[H] I.N. Herstein, Topics in algebra, Waltham, Mass.: Blaisdell Publishing Company, 1987.