Abstract: We study the problem of differentially private (DP) mechanisms for representingsets of size $k$ from a large universe.Our first construction creates$(\epsilon,\delta)$-DP representations with error probability of $1/(e^\epsilon + 1)$ using space at most $1.05 k \epsilon$ bits wherethe time to construct a representation is $O(k \log(1/\delta))$ while decoding time is $O(\log(1/\delta))$.We also present a second algorithm for pure $\epsilon$-DP representations with the same error using space at most $k \epsilon$ bits, but requiring large decoding times.Our algorithms match the lower bounds on privacy-utility trade-offs (including constants but ignoring $\delta$ factors) and we also present a new space lower boundmatching our constructions up to small constant factors.To obtain our results, we design a new approach embedding sets into random linear systemsdeviating from most prior approaches that inject noise into non-private solutions.