I don't think your back of the envelope math is any valid. How do you want to find a particular entropy value in a search space of 2
128 in 10-16 years? Let's be generous and say, 100 years. Genuinely curious!
I'm assuming the challenge to find the correct 12 words of a BIP39 wallet, where you know the first derived public address of a standard derivation path. Other addresses of this wallet hold enough funds and you'll loose a lot if you just focus on one public address. So, we really want to get to know the correct sequence of 12 recovery words.
Search space is 2
128. Let's assume you only need to search on average half of it, i.e. search space now approx. 2
127.
BIP39 derivation involves converting a 128bit value to 12 words, because those have to be fed to 2048 rounds of PBKDF2 HMAC-SHA512 hashing, some further derivation to get to first public address of the wallet, which we luckily know.
Let's be generous and we have a computer that can check a billion entropy values per second.
2
127 is 170,141,183,460,469,231,731,687,303,715,884,105,728 possible values to check.
In a year we can check 365*24*3600*1,000,000,000 = 31,536,000,000,000,000 entropy values. But we're fortunate and have a billion of such computers!
How many years we'll have to search?
170,141,183,460,469,231,731,687,303,715,884,105,728 divided by (31,536,000,000,000,000 times 1,000,000,000) equals roughly 5,395,141,535,403 years... Yikes!
I surely made somewhere a mistake. Point me to it, please. Not sure if we have about 380 times the current age of the universe time, totally neglecting that we need also the energy to operate our one billion computers all that time.
