A new discrete Fourier transform randomness test

Abstract

The randomness of random number generators (RNGs) is important for the reliability of cryptographic systems since the outputs of RNGs are usually utilized to construct cryptographic parameters. Statistical tests are employed to evaluate the randomness of the RNG outputs. The discrete Fourier transform (DFT) test is an important test item of the most popular statistical test suite NIST SP800-22. In the standard NIST DFT test and related improved studies, there exist accuracy and efficiency issues. First, the bit sequences generated by known good RNGs have a high probability to be rejected when the sequences are long or the sequence number is large, due to the deviation between the actual distribution of the test statistic values and the assumed normal distribution. Second, the long test time and high memory consumptions of the complex DFT test algorithm also affect its practicability. To solve these problems, we propose a new DFT test method for long sequences ($10^6$ or more bits). Different from the previous DFT test methods focusing on making the distribution of the test statistic values closer to the normal distribution, we reconstruct the statistic to follow the chi-square distribution. Our experiment result shows that our method has higher reliability in the two-level test, and could effectively reduce the test time and the memory consumptions. When applying our method on randomness test, the test efficiency has been increased to about 4 times for $10^6$-bit sequences and 7 times for $10^7$-bit sequences. In conclusion, our method has lower probability of making errors, and is more suitable for practical application scenarios.