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Arbiter Physically Unclonable Functions (PUFs) have been proposed as efficient hardware security primitives for generating device-unique authentication responses and cryptographic keys. However, the assumed possibility of modeling their underlying challenge-response behavior causes uncertainty about their actual applicability. In this work, we apply well-known machine learning techniques on challenge-response pairs (CRPs) from 64-stage Arbiter PUFs realized in 65nm CMOS, in order to evaluate the effectiveness of such modeling attacks on a modern silicon implementation. We show that a 90%-accurate model can be built from a training set of merely 500 CRPs, and that 5000 CRPs are sufficient to perfectly model the PUFs. To study the implications of these attacks, there is need for a new methodology to assess the security of PUFs suffering from modeling. We propose such a methodology and apply it to our machine learning results, yielding strict bounds on the usability of Arbiter PUFs. We conclude that plain 64-stage Arbiter PUFs are not secure for challenge-response authentication, and the number of extractable secret key bits is limited to at most 600.