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.
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.
I’d say the roles are reversed. Entropy has to do with uncertainty or surprisal, it is hard to guess a private key than a public one, so I’d argue the private key has more entropy.
If private keys were low entropy then you must be able to guess them randomly, but you can not.
Also, because they’re mathematically linked and many algorithms has a 1:1 relationship between a given private key and public key (especially ECC) the entropy must necessarily be identical. You have a lot more to learn in information theory. The real trick is called trapdoor functions.
You’re wrong about the entropy relationship because you don’t understand the role of the description of the state space and the probability of a given configuration in that space.
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.
Actually, it kinda is the same. The math between information entropy and physics entropy are identical and align when you describe physics in particular ways
But entropy in information theory does not behave the same way after a sequence of operations as entropy in physics, and that’s because operations in information theory does not behave like physical interactions.