Bitcoin signatures rely on elliptic curve cryptography, a system in which a private key generates a public key through mathematics that only works one way. Anyone can verify the public key, but no one can go back to the private key. However, Shor’s algorithm, a quantum method published in 1994 for solving problems that ordinary computers cannot solve, can be fed a public key and return the private key that generated it.
Hashing is a different type of problem. A hash scrambles an entry into a fixed-length fingerprint and cannot be run backwards, and the best quantum attack on it, called Grover’s algorithm, only halves the exponent rather than reducing it, taking a 256-bit hash from 2^256 guesses to 2^128.
That’s even more guesses than a machine producing a billion per second could make in the lifetime of the universe.
Modern wallets rely on hashing. A wallet generates addresses in a tree, deriving each key from its parent, and a “hardened” derivation step feeds the parent’s private key through HMAC-SHA512 to produce the child key.
This is a one-way function. An attacker who hijacks an address after Q-day ends up holding exactly the key held and cannot climb the tree to the key it came from.
