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 improve 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结果的一个新证明，即随机排列$P$和$Q$的nCPA优势之和界定了$P^{-1} \circ Q$的CCA优势。我们的证明直接使用概率而非信息论方法，其优势在于提供了CCA优势较低的另一个充分条件：随机排列的CCA优势可以通过其与均匀分布间的分离距离来界定。利用这一替代条件，我们改进了在查询次数少于卡片数的平方根这一特殊情形下，"交换或不交换"(Swap or Not)洗牌方案的最佳已知安全性界。