We study the space complexity of the two related fields of differential privacy and adaptive data analysis. Specifically, (1) Under standard cryptographic assumptions, we show that there exists a problem P that requires exponentially more space to be solved efficiently with differential privacy, compared to the space needed without privacy. To the best of our knowledge, this is the first separation between the space complexity of private and non-private algorithms. (2) The line of work on adaptive data analysis focuses on understanding the number of samples needed for answering a sequence of adaptive queries. We revisit previous lower bounds at a foundational level, and show that they are a consequence of a space bottleneck rather than a sampling bottleneck. To obtain our results, we define and construct an encryption scheme with multiple keys that is built to withstand a limited amount of key leakage in a very particular way.

翻译：我们研究了差分隐私与自适应数据分析这两个相关领域的空间复杂度。具体而言：（1）在标准密码学假设下，我们证明存在一个问题P，在需满足差分隐私的条件下高效求解所需的空间，比无隐私约束时呈指数级增长。据我们所知，这是首次区分了隐私算法与非隐私算法在空间复杂度上的差异。（2）自适应数据分析领域的研究工作主要聚焦于理解回答自适应查询序列所需的样本数量。我们在基础层面对既有下界结果进行重新审视，发现其根源在于空间瓶颈而非采样瓶颈。为获得上述结论，我们定义并构造了一种支持多密钥的加密方案，该方案能以特定方式抵御有限量密钥泄露。