Bitcoin private key, location on ECC curve In the ECDSA algorithm, the Bitcoin private key is supposedly a point on the graph (or is it?).  But the private key is a single integer, and not x,y coordinates.  Is the integer, by itself, the x value or the y value?  If it is x, then what is y? If it is y, then what is x?
The basic elliptic curve operation is addition of points.
The operation of applying this addition repeatedly is called the scalar multiplication of a point by an integer.

The private key is the 'scalar', the point being multiplied is the 'Generator' point, the result is the public key.

Scalar multiplication is basically repeated addition. Multiplying the Generator point by 5 means: calculating G+G+G+G+G.

You calculate this by first calculating  G2= G+G,  then G4=G2+G2, then G5=G4+G.

The curve formula

The formula for the curve used by bitcoin calculations is as follows:

y^2 == x^3 + 7   ( mod p )


where p = 2^256 - 2^32 - 977

Points on the curve

a point (x,y) is on the curve if it matches the above equation

Curve Addition

Curve addition is best visualized geometrically



image from certicom

Elliptic curve cryptography does not use floating point values for it's coordinates, all calculations are done in integers modulo a large prime ( mentioned above, named p ). But the method of calculating the sum of 2 points remains the same.

adding points

Add points P1=(x1,y1) and P2=(x2,y2),  resulting in Psum= (xsum, ysum)

slope = (y1-y2)/(x1-x2)
xsum = slope^2 - (x1+x2)
ysum = slope*(x1-xsum)-y1


point doubling

if P1 and P2 are the same point, the above adding formula would
involve a division by zero, so a different formula is needed to
calculate P+P

slope = 3 * x^2 / (2*y)
xdbl = slope^2 - 2*x
ydbl = slope *(x-xdbl)-y


ecdsa keys

For ECDSA a generator point G was chosen.

The private key is just an integer, lets name it k.
The public key is the generator point added to itself k number of times. In other words, multiplied by k.

If you choose your privatekey unwisely, say 1, your public key would equal the generator point, this address: 1EHNa6Q4Jz2uvNExL497mE43ikXhwF6kZm

As you can see, it was even used recently.

What makes ECDSA a useful crypto system, is that it is easy to calculate a public key from a privatekey, but not the other way around.
Another way of putting this is that multiplication is easy, but there is no (easy) division algorithm on an elliptic curve.

Example code

See this gist for an example in python