We implement and analyze highly stable PUFs using two random gate oxide breakdown mechanisms: plasma induced breakdown and voltage stressed breakdown. These gate oxide breakdown PUFs can be easily implemented in commercial silicon processes, and they are highly stable. We fabricated bit generation units for the stable PUFs on 99 testchips with 65nm CMOS bulk technology. Measurement results show that the plasma induced breakdown can generate complete stable responses. For the voltage stressed breakdown, the responses are with 0.12\% error probability at a worst case corner, which can be effectively accommodated by taking the majority vote from multiple measurements. Both PUFs show significant area reduction compared to SRAM PUF. We compare methods for evaluating the security level of PUFs such as min-entropy, mutual information and guesswork as well as inter- and intra-FHD, and the popular NIST test suite. We show that guesswork can be viewed as a generalization of min-entropy and mutual information. In addition, we analyze our testchip data and show through various statistical distance measures that the bits are independent. Finally, we propose guesswork as a new statistical measure for the level of statistical independence that also has an operational meaning in terms of security.