We provide a new proof of Maurer, Renard, and Pietzak's result that the sum of the nCPA advantages of random permutations $P$ and $Q$ bound the CCA advantage of $P^{-1} \circ Q$. Our proof uses probability directly, as opposed to information theory, and has the advantage of providing an alternate sufficient condition of low CCA advantage. Namely, the CCA advantage of a random permutation can be bounded by its separation distance from the uniform distribution. We use this alternate condition to tighten the best known bound on the security of the swap-or-not shuffle in the special case of having fewer queries than the square root of the number of cards.

翻译：我们给出了Maurer、Renard和Pietzak结论的一个新证明，即随机排列$P$和$Q$的nCPA优势之和限制了$P^{-1} \circ Q$的CCA优势。我们的证明直接使用概率方法而非信息论，其优势在于提供了CCA优势低的另一种充分条件：随机排列的CCA优势可以通过其与均匀分布的分离距离来界定。我们利用这一替代条件，在查询次数少于卡片数平方根的特殊情况下，收紧了对swap-or-not洗牌安全性的最佳已知界。