Problem 58: Signatures for Set Equality

Suggested by	Rasmus Pagh
Source	Dortmund 2012
Short link	https://sublinear.info/58

Given $S\subseteq\{1,..n\}$, we would like to construct a fingerprint so that later, given fingerprints of two sets, we can check the equality of the two sets. There are (at least) two possible solutions to the problem:

1. $h(S)=\left(\sum_{i\in S} x^i\right)\bmod p$ for random $x\in \Z_p$. Update time would be roughly $\log p=\Omega(\log n)$. One would like to obtain a better update time.
2. $h(S)=\left(\prod_{i\in S} (x-i)\right) \bmod p$, and random $x$. Insertion can be done in constant time. But the fingerprint is not linear.

Question: Can we construct a fingerprint that achieves constant update time and is linear, while using $O(\log n)$ random bits? Ideally updates would include insertions and deletions.