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Motivated by a fundamental paradigm in cryptography, we consider a recentvariant of the classic problem of bounding the distinguishing advantage betweena random function and a random permutation. Specifically, we consider theproblem of deciding whether a sequence of $q$ values was sampled uniformly withor without replacement from $[N]$, where the decision is made by a streamingalgorithm restricted to using at most $s$ bits of internal memory. In thiswork, the distinguishing advantage of such an algorithm is measured by the KLdivergence between the distributions of its output as induced under the twocases. We show that for any $s=\Omega(\log N)$ the distinguishing advantage isupper bounded by $O(q \cdot s / N)$, and even by $O(q \cdot s / N \log N)$ when$q \leq N^{1 - \epsilon}$ for any constant $\epsilon > 0$ where it is nearlytight with respect to the KL divergence.

An Information-Theoretic Proof of the Streaming Switching Lemma for Symmetric Encryption