In a backdoor attack, an attacker injects corrupted examples into the training set. The goal of the attacker is to cause the final trained model to predict the attacker's desired target label when a predefined trigger is added to test inputs. Central to these attacks is the trade-off between the success rate of the attack and the number of corrupted training examples injected. We pose this attack as a novel bilevel optimization problem: construct strong poison examples that maximize the attack success rate of the trained model. We use neural tangent kernels to approximate the training dynamics of the model being attacked and automatically learn strong poison examples. We experiment on subclasses of CIFAR-10 and ImageNet with WideResNet-34 and ConvNeXt architectures on periodic and patch trigger attacks and show that NTBA-designed poisoned examples achieve, for example, an attack success rate of 90% with ten times smaller number of poison examples injected compared to the baseline. We provided an interpretation of the NTBA-designed attacks using the analysis of kernel linear regression. We further demonstrate a vulnerability in overparametrized deep neural networks, which is revealed by the shape of the neural tangent kernel.