Abstract: In this Show-and-Tell Demos paper, progresses on mathematical theories for adversarial deep learning are reported.Firstly, achieving robust memorization for certain neural networks is shown to be an NP-hard problem. Furthermore, neural networks with $O(Nn)$ parameters are constructed for optimal robust memorization of any dataset with dimension $n$ and size $N$ in polynomial time. Secondly, adversarial training is formulated as a Stackelberg game and is shown to result in a network with optimal adversarial accuracy when the Carlini-Wagner's margin loss is used. Finally, the bias classifier is introduced and is shown to be information-theoretically secure against the original-model gradient-based attack.