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Adversarial attacks have exposed serious vulnerabilities in Deep Neural Networks (DNNs) through their ability to force misclassifications through human-imperceptible perturbations to DNN inputs. We explore a new direction in the field of adversarial attacks by suggesting attacks that aim to degrade the computational efficiency of DNNs rather than their classification accuracy. Specifically, we propose and demonstrate sparsity attacks, which adversarial modify a DNN's inputs so as to reduce sparsity (or the presence of zero values) in its internal activation values. In resource-constrained systems, a wide range of hardware and software techniques have been proposed that exploit sparsity to improve DNN efficiency. The proposed attack increases the execution time and energy consumption of sparsity-optimized DNN implementations, raising concern over their deployment in latency and energy-critical applications. We propose a systematic methodology to generate adversarial inputs for sparsity attacks by formulating an objective function that quantifies the network's activation sparsity, and minimizing this function using iterative gradient-descent techniques. We launch both white-box and black-box versions of adversarial sparsity attacks on image recognition DNNs and demonstrate that they decrease activation sparsity by up to 1.82x. We also evaluate the impact of the attack on a sparsity-optimized DNN accelerator and demonstrate degradations up to 1.59x in latency, and also study the performance of the attack on a sparsity-optimized general-purpose processor. Finally, we evaluate defense techniques such as activation thresholding and input quantization and demonstrate that the proposed attack is able to withstand them, highlighting the need for further efforts in this new direction within the field of adversarial machine learning.