Authors:
Hiroki Okada
and
Kazuhide Fukushima
Affiliation:
KDDI Research, Inc., Fujimino-shi, 356-8502 Japan
Keyword(s):
Randomness, RNG, NIST SP 800-22, Discrete Fourier Transformation.
Abstract:
The National Institute of Standards and Technology (NIST) released SP 800-22, which is a test suite for
evaluating pseudorandom number generators for cryptographic applications. The discrete Fourier transform
(DFT) test, which is one of the tests in NIST SP 800-22, was constructed to detect some periodic features of
input sequences. There was a crucial problem in the construction of the DFT test: its reference distribution
of the test statistic was not derived mathematically; instead, it was numerically estimated. Thus, the DFT
test was constructed under the assumption that the pseudorandom number generator (PRNG) used for the
estimation generated “truly” random numbers, which is a circular reasoning. Recently, Iwasaki (Iwasaki,
2020) performed a novel analysis to theoretically derive the correct reference distribution (without numerical
estimation). However, Iwasaki’s analysis relied on some heuristic assumptions.
In this paper, we present theoretical evidence for one of th
e assumptions. Let x0,··· , xn−1 be an n-bit input
sequence. Its Fourier coefficients are defined as F0,...,Fn−1. Iwasaki assumed that Σn2 −1j=0|Fj|2 = n2/2. We
use a quantitative analysis to show that this holds when n is sufficiently large. We also verify that our analysis
is sufficiently accurate with numerical experiments.
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