Perhaps it would be wise to already plan for a very long passphrase for important wallets in anticipation of quantum computers, no? Perhaps the vulnerability of current wallets will be earlier than expected?
Not really.
A wallet is a set of key-pairs, they can be either deterministically or non-deterministingally produced.
Let's focus on a single key-pair.
You have the following flow: private key -> public key -> address. It goes only right and can't go backwards.
The private key in Bitcoin is 256 bits long.
Hacking method 1: Brute-forceBrute-force means that you search blindly. Since the key is 256 bits long,
on average, it requires half the total complexity to find a match. This leaves you with a number of 2^255 operations.
Hacking method 2: ECDLP solvingFun part:
Have you ever heard of the birthday paradox? If you have a room with 23 people, there is a probability higher than 50% that you will find 2 of them that have the same birthday. But, the days of the year are 365, so it looks strange, but it's mathematically explained. The rule of the paradox is that for
n objects you need
√n time complexity to have a solid probability of finding a collision.
Now let's go back to bitcoin:
It's more practical to use an algorithm that tries to solve the ECDLP (for example check
this thread).
What these algorithms do, is that they start from a known public key and they try to derive the private key.
In this way, they know which address they try to hack.
You are essentially trying to find a match for
Q = kG, where Q = public key, k = private key and G = a point on the secp256k1 (which is bitcoin's elliptic curve).
So, based on the initial rule, the complexity of this algorithm is √2^256 = 2^128.
A complexity of 2^128 bits is much less than 2^256. It's not the half of it, like many people think, but much less than that!
Final VerdictIs 2^128 still difficult to find? It's almost infeasible, just like 2^256, but between the 2 methods, I think quantum computer users will choose the latters since the search space is vastly smaller.
I highly doubt that quantum computers will even mess with BIP39, passphrases etc. If they have the power they are supposed to have, they will most likely try to search the 2^128 space, without even getting out of their houses.