We study streaming algorithms in the white-box adversarial stream model, where the internal state of the streaming algorithm is revealed to an adversary who adaptively generates the stream updates, but the algorithm obtains fresh randomness unknown to the adversary at each time step. We incorporate cryptographic assumptions to construct robust algorithms against such adversaries. We propose efficient algorithms for sparse recovery of vectors, low rank recovery of matrices and tensors, as well as low rank plus sparse recovery of matrices, i.e., robust PCA. Unlike deterministic algorithms, our algorithms can report when the input is not sparse or low rank even in the presence of such an adversary. We use these recovery algorithms to improve upon and solve new problems in numerical linear algebra and combinatorial optimization on white-box adversarial streams. For example, we give the first efficient algorithm for outputting a matching in a graph with insertions and deletions to its edges provided the matching size is small, and otherwise we declare the matching size is large. We also improve the approximation versus memory tradeoff of previous work for estimating the number of non-zero elements in a vector and computing the matrix rank.

翻译：我们研究白盒对抗流模型下的流式计算算法，其中流式算法的内部状态会暴露给自适应生成流更新的对手，但算法在每个时间步均可获取对手未知的新随机性。我们引入密码学假设来构建对抗此类对手的鲁棒算法。针对向量稀疏恢复、矩阵与张量低秩恢复，以及矩阵的低秩加稀疏恢复（即鲁棒主成分分析），我们提出了高效算法。不同于确定性算法，即使在存在此类对抗的情况下，我们的算法也能在输入不具备稀疏性或低秩性时发出报告。我们利用这些恢复算法改进并解决了数值线性代数和组合优化领域中关于白盒对抗流的新问题。例如，我们首次提出高效算法，可在边插入与删除的图中输出匹配（当匹配规模较小时），否则声明匹配规模较大。我们还改进了先前工作关于向量非零元素数量估计与矩阵秩计算的近似精度与存储开销权衡。