What is the 'base point' (G) from the whitepaper and how is it represented as a single hex value?

Question

The white paper introduces G, aka the base point, as G = (x, −4/5).
I have two questions:

1. what is x? Is this the x from the elliptic curve (is it a variable)? If so, how is a private key aG computed?
2. A point has two coordinates (x and y). How is a point represented as a 256-bit value?

Answer 1

G is the base point of the Ed25519 elliptic curve. The x coordinate is not written explicitly because it can be found back using the y coordinate, the sign of x and the equation of the curve (x is positive for G).

Points on the Ed25519 curve are represented by 32 bytes. These 32 bytes are in fact the little-endian representation of a 256-bit number.

• bits 0 to 254 for the y coordinate
• bit 255 indicates the sign of the x coordinate

Therefore the hex string representing G is 0x5866666666666666666666666666666666666666666666666666666666666666.

There is more information about this base point at https://crypto.stackexchange.com/questions/27392/base-point-in-ed25519