对椭圆曲线来说最流行的有限域是以素数为模的整数域（参见 模运算），或是特征为2的伽罗瓦域GF(2m)。后者在专门的硬件实现上计算更为有效，而前者通常在通用处理器上更为有效。专利的问题也是相关的。一些其他素数的伽罗瓦域的大小和能力也已经提出了，但被密码专家认为有一点问题。



给定一条椭圆曲线E以及一个域，我们考虑具有形式有理数点的阿贝尔群，其中x和y都在中并且定义在这条曲线上的群运算"+"（运算"+"在文章椭圆曲线中描述）。我们然后定义第二个运算"*" | Z×：如果P是上的某个点，那么我们定义等等。注意给定整数 j和k，。椭圆曲线离散对数问题（ECDLP）就是给定点P和Q，确定整数k使。 -- 一般认为在一个有限域乘法群上的离散对数问题（DLP）和椭圆曲线上的离散对数问题（ECDLP）并不等价；ECDLP比DLP要困难的多。



在密码的使用上，曲线和其中一个特定的基点G一起被选择和公布。一个私钥k被作为随机整数来选择；值被作为公钥来公布（注意假设的ECDLP困难性意味着k很难从P中确定）。-Finite field is the most popular for Elliptic Curve primes modulo integer domain (see modular arithmetic), or the characteristics of the Galois field GF (2m). Which is calculated on specialized hardware implementations more effective, while the former often are more effective in a general-purpose processor. The patent issue is also relevant. Some other primes Galois field size and capacity have been proposed, but the password experts think there is a little problem.



Given an elliptic curve E, ​ ​ and a domain, we consider having a form rationals point Abelian group, where x and y are in and define the group operation on this curve "+" (operation "+" in the article elliptic curve described). We then define a second operator "*" | Z ×: if P is a point on, then we define. Note that given the integers j and k. Elliptic curve discrete logarithm problem (ECDLP) is a given point P and Q, determine the integer k so.- Is generally believed that the discrete logarithm problem (DLP)