For a given integer m, there are at most two points in the secp256k1 curve whose x-value is m.

Have we ever actually found a pair of distinct private keys whose corresponding public keys have the same x-value?

1 Answer 1


Yes, trivially. If private key d has public key X coordinate m, then so does private key n-d, where n is the order of the curve (as n-d equals -d modulo n).

This follows from a basic rule in elliptic curve point arithmetic: -(x, y) is defined as (x, -y). From this is follows that if dG = (x, y) then (-d)G = (n-d)G = (x, -y) = (x, p-y).

  • 1
    Right, the sum of two points with the same x-value is the point at infinity (the identity element of the Elliptic curve group) so the two points are actually inverses of each other, so their preimages by the group isomorphism d--->dG are inverses of each other too. Thank Pieter !
    – mathboi
    Jan 28 at 20:39

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