As others have explained, whenever you salt a hash function you're creating a new hash function. "Multiple hash functions" doesn't mean completely different def

As others have explained, whenever you salt a hash function you're creating a new hash function. "Multiple hash functions" doesn't mean completely different definitions of the function, but usually it is just a parametric family. But there's more to this. This is a theoretical paper, the algorithm has to work for any N. Even if the N is larger than the number of particles in the known universe. That's how theory works, you need guarantees and theorems. Then it becomes clear that no "practical" hash function can actually work for this, by a simple pigeonhole argument: you have only 64 bits (or 128 bits, ...) of output, so at some point you'll get too many collisions for anything to work. Then you can start wondering: if I use "salt" (that is, a parameter), how many bits should that salt have, depending on N? How do you guarantee that the parametric family you have always guarantee sufficient independence, and you don't start seeing weird correlations when you start having enough hash functions and running out of entropy? In practice, these considerations are not made, because we don't even know why the hash functions we use actually work. There is very little theory behind them: their quality is mostly empirically assessed by enormous batteries of statistical tests.