The whole promise is that reversible computation is theoretically free (in quantum information theory) , quantum computer can exploit it by using quantum superposition to brute force 256bit key (and larger) in a single computation cycle.
There’s a massive fundamental flaw in the reasoning behind this.
It’s the second law of thermodynamics - entropy always increases. While computation is free, reversal of entropy isn’t. What we’re being promised is a 21st century version of perpetual motion machine.
Additional explanation:
Think about private key as low entropy, and public key as high entropy. Owner of the private key has cheat code in the form of missing information which allows reversing the entropy of public key.
However in order to break the cryptography you have to reverse public key back into private key without having access to it. The regualr way is to search for flaws in the cryptography scheme that allow reasoning about missing information and consecutively - private key recovery.
But if you do a brute force attack you are 100% forced to pay full thermodynamical price of reversing the entropy. And 2^256 is way, way more than you could ever afford. Even Planck scale values multiplied by that number grow into universe size and beyond.
In summary, quantum computing can skip time component of brute forcing a key, but it can’t skip the energy costs. So it will never happen.
Why is a private key low entropy and a public one high entropy? Are you talking about Shanon entropy? If so, over what distribution?
Also, as far as I know from the time I was interested in QC, we already can break RSA, you can actually run it on actual quantum computers, although with fewer qubits. That’s why we have quantum safe cryptography.
I really don’t see how any of this violates the second law.
Public key must be high entropy as its sole purpose is to scramble known information, from perspective of a person who doesn’t have a key, scrambled information is a pure random noise.
Private key has to be low entropy, since it has information necessary to revert scrambling. High entropy + information how to undo it isn’t really high anymore.
I’m not deep enough into physics to know what Shanon entropy is. I use definition of entropy as amount of hidden/unreadable information.
Shannon entropy is a concept from information theory. It is mathematically analogous to thermodynamic entropy, but not the same thing. So you’ve identified the Shannon entropy of public and private keys, but I’m also wondering how that relates to their thermodynamic entropy in order for the 2nd law to apply.