—We discuss the first power and timing side channels on Strong Physical Unclonable Functions (Strong PUFs) in the literature, and describe their efficient exploitation via adapted machine learning (ML) techniques. Our method is illustrated by the example of the two currently most secure (CCS 2010, IEEE T-IFS 2013) electrical Strong PUFs, so-called XOR Arbiter PUFs and Lightweight PUFs. It allows us for the first time to tackle these two architectures with a polynomial attack complexity. In greater detail, our power and timing side channels provide information on the single outputs of the many parallel Arbiter PUFs inside an XOR Arbiter PUF or Lightweight PUF. They indicate how many of these single outputs (in sum) were equal to one (and how many were equal to zero) before the outputs entered the final XOR gate. Taken for itself, this side channel information is of little value, since it does not tell which of the single outputs were zero or one, respectively. But we show that if combined with suitably adapted machine learning techniques, it allows very efficient attacks on the two above PUFs, i.e., attacks that merely use linearly many challenge-response pairs and lowdegree polynomial computation times. Without countermeasures, the two PUFs can hence no longer be called secure, regardless of their sizes. For comparison, the best-performing pure modeling attacks on the above two PUFs are known to have an exponential complexity (CCS 2010, IEEE T-IFS 2013). The practical viability of new our attacks is firstly demonstrated by ML experiments on numerically simulated CRPs. We thereby confirm attacks on the two above PUFs for up to 16 XORs and challenge bitlengths of up to 512. Secondly, we execute a full experimental proof-of-concept for our timing side channel, successfully attacking FPGA-implementations of the two above PUF types for 8, 12, and 16 XORs, and bitlengths 64, 128, 256 and 512. In earlier works (CCS 2010, IEEE T-IFS 2013), 8 XOR architectures with bitlength 512 had been explicitly suggested as secure and beyond the reach of foreseeable attacks. Besides the abovementioned new power and timing side channels, two other central innovations of our paper are our tailormade, polynomial ML-algorithm that integrates the side channel information, and the implementation of Arbiter PUF variants with up to 16 XORs and bitlength 512 in silicon. To our knowledge, such sizes have never been implemented before in the literature. Finally, we discuss efficient countermeasures against our power and timing side channels. They could and should be used to secure future Arbiter PUF generations against the latter.