Physical Unclonable Functions (PUFs) have established themselves in the scientific literature, and are also gaining ground in commercial applications. Recently, however, several attacks on PUF core properties have been reported. They concern their physical and digital unclonability, as well as their assumed resilience against invasive or side channel attacks. In this paper, we join some of these techniques in order to further improve their effectiveness. The combination of machine-learning based modeling techniques with side channel information allows us to attack so-called XOR Arbiter PUFs and Lightweight PUFs up to a size and complexity that was previously out of reach. For Lightweight PUFs, for example, we report successful attacks for bitlengths of 64, 128 and 256, and for up to nine single Arbiter PUFs whose output is XORed. Previous work at CCS 2010 and IEEE TIFS 2013, which provides the currently most efficient modeling results, had only been able to attack this structure for up to five XORs and bitlength 64. Our attack employs the first power side channel (PSC) for Strong PUFs in the literature. This PSC tells the attacker the number of single Arbiter PUF within an XOR Arbiter PUF or Lightweight PUF architecture that are zero or one. This PSC is of little value if taken by itself, but strongly improves an attacker’s capacity if suitably combined with modeling techniques. At the end of the paper, we discuss efficient and simple countermeasures against this PSC, which could be used to secure future PUF generations.