I've been reading about sketches for processing streaming data (the CountMin sketch, the Count sketch, the tug-of-war sketch, FM sketches, etc.). They use hash functions that are required to be 2-independent or 4-independent or $k$-independent for some small $k$.
However from cryptography we have fast hash functions that are very good candidates to essentially perfect: $k$-independent for every value of $k$ you could ever wish for, and so on. We can essentially treat them as uniformly chosen from the set of all functions: as far as we know, no polynomial-time computation can tell that this is not the case. I suspect many practitioners might be happy to use these cryptographic hash functions, if it led to improved performance or better parameters for the sketches.
Does it change anything about the design and analysis of sketches, if we assume we have perfect hash functions like this? Can we get asymptotically better parameters if we assumed our hash function was perfect? Or, can we get simpler sketch constructions?
(For instance, I've noticed that the analysis of existing sketches is often somewhat conservative: e.g., it uses the Markov bound or Chebyshev bound. If we had a perfect hash function we could use a Chernoff-style bound. Are there any sketches where this makes a significant difference, or is it just a small constant factor in the bound?)