On the Mathematical Aspects of Cryptographic Randomness Tests Using Discrete Fourier Transform

The Discrete Fourier Transform (DFT) test is one of the effective statistical tests used in evaluating randomness. In 2017, Umeno et al. proposed a new standard using the DFT. However, the mathematical analysis still lacks some detailed proofs, and the dependency among the new test and NIST's tests was presented by the authors as an open issue. Within this research, we have thoroughly and comprehensively reviewed the randomness statistical tests based on the DFT. We then provide complete mathematical proofs for the theoretical basis of Umeno et al.'s new DFT test. Furthermore, we conducted several experiments and compared the results obtained for different versions of the DFT tests.