On the Entropy of Oscillator-Based True Random Number Generators

True random number generators (TRNGs) are essential for cryptographic systems, and they are usually evaluated by the concept of entropy. In general, the entropy of a TRNG is estimated from its stochastic model, and reflected in the statistical results of the generated raw bits. Oscillator-based TRNGs are widely used in practical cryptographic systems due to its elegant structure, and its stochastic model has been studied in different aspects. In this paper, we investigate the applicability of the different entropy estimation methods for oscillator-based TRNGs, including the bit-rate entropy, the lower bound and the approximate entropy. Particularly, we firstly analyze the two existing stochastic models (one of which is phase-based and the other is time-based), and deduce consistent bit-rate entropy results from these two models. Then, we design an approximate entropy calculation method on the output raw bits of a simulated oscillator-based TRNG, and this statistical calculation result well matches the bit-rate entropy from stochastic models. In addition, we discuss the extreme case of tiny randomness where some methods are inapplicable, and provide the recommendations for these entropy evaluation methods. Finally, we design a hardware verification method in a real oscillator-based TRNG, and validate these estimation methods in the hardware platform.