1024 bits seems a bit too much for the birthday problem to be a thing.
I looked at [1] to do the calculation but (2^1024)! is a number too large for any of my tools. If someone has a math shortcut to test this idea properly...
This isn't the birthday problem. That would be the chance of two random links overlapping. The birthday problem scales with n^2, while trying to guess links scales with m * n, number of guesses multiplied by number of links.
(Well, before you apply the logistic taper to it. So you wanted an approximation? There you go. Until you get the chance of a hit to be quite high, it's basically equal to guesses * valid links / 2^1024.)
The chance is less than guessing a random 128 bit username and random 128 bit password. And then guessing a completely different username and password on the very next go.
You'd get far more return on investment breaking bitcoin wallets.
2^1024 is 10^308
Lets say there are 12 billion links per person, and 8 billion people. That's 100 billion billion, or 10^20 links.
10^20 / 10^308 is zero.
Lets say you can test 10 trillion links a second, and started when the big bang happened, you'll have tested 10^30 links so far.
The number of links you'll have found so far is zero.
(Well, before you apply the logistic taper to it. So you wanted an approximation? There you go. Until you get the chance of a hit to be quite high, it's basically equal to guesses * valid links / 2^1024.)