Physical unclonable functions are not easy to integrate into cryptographic systems because they age, and are sensitive to environmental interferences. Excellent error correcting schemes were developed to handle such drifts, however the computing power needed at the client level can leak information to opponents, and are difficult to deploy to networks of ultra-low power Internet of Things. Response-based cryptography methods, which are server based, use search engines to uncover the erratic keys generated by the physical unclonable functions, minimizing the consumption of electric power at the client level. However, when the defect densities are high, the latencies associated with search engines can be prohibitive. The statistical analysis presented in this paper shows how the fragmentation of the cryptographic keys can significantly reduce the latencies of the search engine, even when error rates are high. The statistical model developed, with Poisson distribution, shows that the level of fragmentation in sub-keys can handle up to 15% error rates. The methodology is generic, and can be applied to any type of physically unclonable functions with defects in the 15% range, or lower.