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Let's first define a valid Bitcoin private key as a number in the range of [1, 115792089237316195423570985008687907852837564279074904382605163141518161494336], and a valid Bitcoin public key as a public key derived from a valid Bitcoin private key.

Then my question is: Is there a way to determine whether a specific point on the secp256k1 curve is a valid Bitcoin public key?

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The secp256k1 curve has cofactor one, so every point on the curve can be written as a multiple of the generator. As a result, there are exactly as many points on the curve (excluding the point at infinity) as there are valid (nonzero) private keys.

In short: yes, every point on the curve is a valid public key.

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  • The point K = (9166C289B9F905E55F9E3DF9F69D7F356B4A22095F894F4715714AA4B56606AF, F181EB966BE4ACB5CFF9E16B66D809BE94E214F06C93FD091099AF98499255E7) is part of the secp256k1 curve, but it's derived from an invalid private key (FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF), such that according to the definition in my question, the point is not a valid Bitcoin public key. Question that remains: is there a non-brute-force way to determine such validity of a public key?
    – drogos86
    May 10, 2022 at 9:40
  • @drogos86 That point also has corresponding (valid) private key 0x000000000000000000000000000000014551231950b75fc4402da1732fc9bebe. The restriction for private keys to be in range 1 to 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141 is simply to make sure there is exactly one private key for every public key, but beyond that, private keys act "modulo" the size of the curve.
    – Pieter Wuille
    May 10, 2022 at 12:48

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