Function: elllog
Section: elliptic_curves
C-Name: elllog
Prototype: GGGDG
Help: elllog(E,P,G,{o}): return the discrete logarithm of the point P of
 the elliptic curve E in base G. If present, o represents the order of G.
 If not present, assume that G generates the curve.
Doc: given two points $P$ and $G$ on the elliptic curve $E/\F_q$, returns the
 discrete logarithm of $P$ in base $G$, i.e. the smallest non-negative
 integer $n$ such that $P = [n]G$.
 See \tet{znlog} for the limitations of the underlying discrete log algorithms.
 If present, $o$ represents the order of $G$, see \secref{se:DLfun};
 the preferred format for this parameter is \kbd{[N, factor(N)]}, where $N$
 is  the order of $G$.

 If no $o$ is given, assume that $G$ generates the curve.
 The function also assumes that $P$ is a multiple of $G$.
 \bprog
 ? a = ffgen(ffinit(2,8),'a);
 ? E = ellinit([a,1,0,0,1]);  \\ over F_{2^8}
 ? x = a^3; y = ellordinate(E,x)[1];
 ? P = [x,y]; G = ellmul(E, P, 113);
 ? ord = [242, factor(242)]; \\ P generates a group of order 242. Initialize.
 ? ellorder(E, G, ord)
 %4 = 242
 ? e = elllog(E, P, G, ord)
 %5 = 15
 ? ellmul(E,G,e) == P
 %6 = 1
 @eprog
