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Every possible X coordinate on secp256k1 has two corresponding Y coordinates, one of which is the negation of the other. I've also been told that every possible Y coordinate has three corresponding X coordinates (due to the fact that the field size p = 1 mod 3). Is there also a nice relationship between them? Given a point on the curve, how can I obtain the other two points with the same Y coordinate?

1 Answer 1

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Yes, there is!

For a given Y coordinate y, if on the curve, the three corresponding X coordinates are the cube roots of y2 - 7. If you have any such cube root y0, then multiplying it with a cube root of 1 gives you another cube root - for the simple reason that (y0 ∛1)3 = y03(∛1)3 = y031 = y03.

Modulo p = 2256-232-977 (the secp256k1 field size), the three cube roots of 1 are:

  • 1 (obviously)
  • 60197513588986302554485582024885075108884032450952339817679072026166228089408
  • 55594575648329892869085402983802832744385952214688224221778511981742606582254

This is exactly analogous to what's going on for the corresponding Y coordinates given an X coordinate; only there the different Y coordinates are related by multiplication with square roots of 1, namely 1 and -1 = p-1.

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