A Worst-Case Entropy Estimation of Oscillator-Based Entropy Sources: When the Adversaries Have Access to the History Outputs

Entropy sources are designed to provide unpredictable random numbers for cryptographic systems. As an assessment of the sources, Shannon entropy is usually adopted to quantitatively measure the unpredictability of the outputs. In several related works about the entropy evaluation of ring oscillator-based (RO-based) entropy sources, authors evaluated the unpredictability with the average conditional Shannon entropy (ACE) of the source, moreover provided a lower bound of the ACE (LBoACE). However, in this paper, we have demonstrated that when the adversaries have access to the history outputs of the entropy source, for example, by some intrusive attacks, the LBoACE may overestimate the actual unpredictability of the next output for the adversaries. In this situation, we suggest to adopt the specific conditional Shannon entropy (SCE) which exactly measures the unpredictability of the future output with the knowledge of previous output sequences and so is more consistent with the reality than the ACE. In particular, to be conservative, we propose to take the lower bound of the SCE (LBoSCE) as an estimation of the worst-case entropy of the sources. We put forward a detailed method to estimate this worst-case entropy of RO-based entropy sources, which we have also verified by experiment on an FPGA device. We recommend to adopt this method to provide a conservative assessment of the unpredictability when the entropy source works in a vulnerable environment and the adversaries might obtain the previous outputs.