Base Point on elliptic curve

Question

How can i define the base point of the elliptic curve group element used as the elliptic curve domain parameter over Fp (prime)

Answer 1

The "base point" is conventional. When using El Gamal or similar algorithms with an elliptic curve, you want to work inside a group with a prime order (either the complete curve has a prime order, or you work in a subgroup of the curve). Within a group of prime order, any element (except the neutral element, which is the "point at infinity" for an elliptic curve) is a generator for the group, so any point will be appropriate. The only requirement here is that all parties agree on the point to use.

It can easily be shown that any base point is as good as any other with regards to resistance to discrete logarithm: if you can solve DL efficiently in base $G$, then you can solve it efficiently in any other base $G'$ by first solving $G'$ with regards to $G$: first, use your efficient DL solving algorithm to compute $k$ such that $G'$ = $kG$; then, given $xG'$, solve it in base $G$ to get $xG' = rG$, then the solution will be $x = r/k \pmod n$ (where $n$ is the group order).

In practice, the "base point" is considered to be part of the curve definition, along with the other curve parameters. For instance, FIPS 186-4 defines the "NIST curves" (in annex D); for each of them, it lists the field modulus ($p$), the curve order ($n$), the curve equation parameters (NIST curves in a prime field use $y^2 = x^3 -3x + b$, so the only equation parameter is $b$), and the two coordinates $G_x$ and $G_y$ of the conventional base point.