5

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?
6

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 5866666666666666666666666666666666666666666666666666666666666666.

There are more info about this base point in https://crypto.stackexchange.com/questions/27392/base-point-in-ed25519

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