<p>Yes, trivially. If private key <em>d</em> has public key X coordinate <em>m</em>, then so does private key <em>n-d</em>, where <em>n</em> is the order of the curve (as <em>n-d</em> equals <em>-d</em> modulo <em>n</em>).</p>
<p>This follows from a basic rule in elliptic curve point arithmetic: <em>-(x, y)</em> is defined as <em>(x, -y)</em>. From this is follows that if <em>dG = (x, y)</em> then <em>(-d)G = (n-d)G = (x, -y) = (x, p-y)</em>.</p>

CC BY-SA 4.0
2024-01-28T20:39:40.283Z
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
CC BY-SA 4.0
2024-01-28T13:42:04.437Z
2024-01-28T13:42:04.437Z
Pieter Wuille
<p>For a given integer m, there are at most two points in the secp256k1 curve whose x-value is m.</p>
<p>Have we ever actually found a pair of distinct private keys whose corresponding public keys have the same x-value?</p>

<private-key><public-key><secp256k1>
Two private keys whose corresponding public keys have the same x-value?
mathboi
# Two private keys whose corresponding public keys have the same x-value?

(Asked by: username_0 on 2024-01-28)

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?

**Tags:** private-key, public-key, secp256k1


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## 1 answers

### Answer by username_1 on 2024-01-28. Score: 4

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)*.

> **Comments:**
> **username_0**: 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 !  

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