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Feb 6 at 20:09 comment added Mark Schultz-Wu @ArghyaBhattacharjee also I would suggest using $f'_k(x) :=H(k||x)$ instead. Prepending (rather than post-pending) strings is the typical way that domain separation is achieved, which seems to be what you want.
Feb 6 at 19:40 comment added Mark Schultz-Wu @ArghyaBhattacharjee it is implied by the assumption that $H$ is a random oracle, so it is implied by a standard assumption. It's not clear to me how meaningfully better of an assumption it is to assume $H$ is 2-universal versus just a random oracle directly though --- I think that really depends on your tastes/applications.
Feb 6 at 18:15 comment added Arghya Bhattacharjee Thanks for the response. For a good cryptographic hash function $H$, would you say it's fair to assume that $\{f_k(x) := H(x||k) \mid k \in \{0,1\}^n\}$ is a family of 2-way universal hash functions? Then we can directly use the Leftover Hash Lemma.
Feb 5 at 23:14 history answered Mark Schultz-Wu CC BY-SA 4.0