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It has been shown that adversaries can craft example inputs to neuralnetworks which are similar to legitimate inputs but have been created topurposely cause the neural network to misclassify the input. These adversarialexamples are crafted, for example, by calculating gradients of a carefullydefined loss function with respect to the input. As a countermeasure, someresearchers have tried to design robust models by blocking or obfuscatinggradients, even in white-box settings. Another line of research proposesintroducing a separate detector to attempt to detect adversarial examples. Thisapproach also makes use of gradient obfuscation techniques, for example, toprevent the adversary from trying to fool the detector. In this paper, weintroduce stochastic substitute training, a gray-box approach that can craftadversarial examples for defenses which obfuscate gradients. For those defensesthat have tried to make models more robust, with our technique, an adversarycan craft adversarial examples with no knowledge of the defense. For defensesthat attempt to detect the adversarial examples, with our technique, anadversary only needs very limited information about the defense to craftadversarial examples. We demonstrate our technique by applying it against twodefenses which make models more robust and two defenses which detectadversarial examples.

Stochastic Substitute Training: A Gray-box Approach to Craft Adversarial Examples Against Gradient Obfuscation Defenses