What does it mean to tweak a public key?

Question

Public key tweaks are often talked about in the context of taproot but I cannot find a definition of what precisely "tweaking" means.

Answer 1

The Bitcoin Optech Taproot workshop defines it as:

Tweaking a public key means to alter it with a value (the tweak) such that it remains spendable with knowledge of the original private key and tweak.

A tweak is positive scalar value t where 0 < t < SECP256K1_ORDER. There is an associated tweak point T such that T = t*G. The private key is tweaked by the tweak scalar: x' = x + t and the public key is tweaked by the tweak point: P' = P + T.

The tweaked private key x' can be used to produce a valid signature for the tweaked pubkey P'.