Say I have created a paper wallet with one bitcoin address in a vault. What are chances that someone else will find the private key for this wallet?
There are around 1,461,501,637,330,902,918,203,684,832,716,283,019,655,932,542,976 possible bitcoin addresses according to BitcoinTalk.org. This makes it a very small possibility of finding another address that is being used. So the chance is around %0.
Essentially zero. They would have to find a private key whose public key hash matched yours. The public key hash is 160 bits long. If they had a billion computers, each of which could try a billion keys a second, and they tried for a billion years, they'd have much, much less than a one in a billion chance of getting it.
There is a guy who has been testing this. He has ran a computer which generates addresses and checks the balances. Out of several years, processing millions of addresses per day, to my knowledge, he has found 3 with small balances. So, you can talk theoretical, where the chances are practically zero, but when there is someone who has attained at least 1, then perhaps it's not as unlikely as the math suggests.
It depends on the sort of computer you use. Current silicon based digital computers are exactly what the elliptic curve signature scheme is designed to beat, so it is highly resistant against an attack with such a computer.
Quantum computers are predicted to be able to implement Shor's algorithm efficiently. If they can be implemented with 4,000 effective Qbits they would be able to find the private key of a wallet with an exposed public key in seconds. In 2017 Aggarwal et al. predicted that it could be completely broken by a quantum computer as early as 2027, by the most optimistic estimates.
Since then research has come on in leaps and bounds, in February-March 2022 81 percent of senior UK executives expect quantum computing to have a significant impact in their industry within seven and a half years, with almost half (48 percent) believing that quantum technology will begin to transform industries as soon as 2025. Andersen Cheng, CEO of London-based cryptography company Post-Quantum says there's been some evidence that it could well be around two years away.
The only true way to defend against this is to avoid large balances with each private key. But if people do that then it will be much easier to find something if adoption happens and active addresses continued to double every 5 years.
Assuming btc is eventually $ 2 million per bitcoin and average person saves say 200k in bitcoin. They would need 2000 keys with 0.00005 to avoid losing more than $100 though a random collision attack.
I guess there should be some kind of insurance against this happening but it goes against this idea consolidating uxtos to save fees is a good idea. It’s another reason why low fees are required for btc to work.