Distributed learning is central for large-scale training of deep-learning models. However, it is exposed to a security threat in which Byzantine participants can interrupt or control the learning process. Previous attack models assume that the rogue participants (a) are omniscient (know the data of all other participants), and (b) introduce large changes to the parameters. Accordingly, most defense mechanisms make a similar assumption and attempt to use statistically robust methods to identify and discard values whose reported gradients are far from the population mean. We observe that if the empirical variance between the gradients of workers is high enough, an attacker could take advantage of this and launch a non-omniscient attack that operates within the population variance. We show that the variance is indeed high enough even for simple datasets such as MNIST, allowing an attack that is not only undetected by existing defenses, but also uses their power against them, causing those defense mechanisms to consistently select the byzantine workers while discarding legitimate ones. We demonstrate our attack method works not only for preventing convergence but also for repurposing of the model behavior (``backdooring''). We show that less than 25\% of colluding workers are sufficient to degrade the accuracy of models trained on MNIST, CIFAR10 and CIFAR100 by 50\%, as well as to introduce backdoors without hurting the accuracy for MNIST and CIFAR10 datasets, but with a degradation for CIFAR100.