Beat frequency detector-based high-speed true random number generators: Statistical modeling and analysis

True random number generators (TRNGs) are crucial components for the security of cryptographic systems. In contrast to pseudo-random number generators (PRNGs), TRNGs provide higher security by extracting randomness from physical phenomena. To evaluate a TRNG, statistical properties of the circuit model and raw bitstream should be studied. In this article, a model for the beat frequency detector-based high-speed TRNG (BFD-TRNG) is proposed. The parameters of the model are extracted from the experimental data of a test chip. A statistical analysis of the proposed model is carried out to derive mean and variance of the counter values of the TRNG. Our statistical analysis results show that mean of the counter values is inversely proportional to the frequency difference of the two ring oscillators (ROSCs), whereas the dynamic range of the counter values increases linearly with standard deviation of environmental noise and decreases with increase of the frequency difference. Without the measurements from the test data, a model cannot be created; similarly, without a model, performance of a TRNG cannot be predicted. The key contribution of the proposed approach lies in fitting the model to measured data and the ability to use the model to predict performance of BFD-TRNGs that have not been fabricated. Several novel alternate BFD-TRNG architectures are also proposed; these include parallel BFD, cascade BFD, and parallel-cascade BFD. These TRNGs are analyzed using the proposed model, and it is shown that the parallel BFD structure requires less area per bit, whereas the cascade BFD structure has a larger dynamic range while maintaining the same mean of the counter values as the original BFD-TRNG. It is shown that 3.25M and 4M random bits can be obtained per counter value from parallel BFD and parallel-cascade BFD, respectively, where M counter values are computed in parallel. Furthermore, the statistical analysis results illustrate that BFD-TRNGs have better randomness and less cost per bit than other existing ROSC-TRNG designs. For example, it is shown that BFD-TRNGs accumulate 150% more jitter than the original two-oscillator TRNG and that parallel BFD-TRNGs require one-third power and one-half area for same number of random bits for a specified period.