Matching a Value To An Elliptic Curve Point

Elliptic Curve Cryptography (ECC) is taking over in the world of cryptography, and you will see it in Tor, Bitcoin, Ethereum, IoT, and in many other places where public key is used. Normally we use ECC to sign for things (with ECDSA) and also to create shared keys (with ECDH). So, if we need to apply a piece of data onto our elliptic curve, how would we match it?

Within ECC, we take a point on the elliptic curve (G) and then can multiply it by a scalar (priv). The public key point is then priv multiplied by G. At the current time, if the scalar (priv) is a large number, it will not be possible to determine the value of priv, even if we know the public key and the G points:

In many applications we will take a value and then hash it. Then we must fit the hashed value onto an elliptic curve point (such as with the Schnorr signature). Obviously we need integer values for the x,y points, so we need to find an x,y point which has integer values for both x and y. So let’s take a common elliptic curve:

y² =x³ +7