  Let $X$ be a smooth projective curve over a finite field $\mathbb{F}$ with
$q$ elements. For $m\geq 1,$ let $X_m$ be the curve $X$ over the finite field
$\mathbb{F}_m$, the $m$-th extension of $\mathbb{F}.$ Let $K_n(m)$ be the
$K$-group $K_n(X_m)$ of the smooth projective curve $X_m.$
  In this paper, we study the structure of the groups $K_n(m).$ If $l$ is a
prime, we establish an analogue of Iwasawa theorem in algebraic number theory
for the orders of the $l$-primary part $K_n(l^m)\{l\}$ of $K_n(l^m)$. In
particular, when $X$ is an elliptic curve $E$ defined over $\mathbb{F},$ our
method determines the structure of $K_n(E).$ Our results can be applied to
construct an efficient {\bf DL} system in elliptic cryptography.
