Fast Spectral Tests for Measuring Nonrandomness and the DES

• Authors: Frank A. Feldman
• Venue: CRYPTO '87 A Conference on the Theory and Applications of Cryptographic Techniques on Advances in Cryptology
• Year: 1987

Two spectral tests for detecting nonrandomness were proposed in 1977. One test, developed by J. Gait [1], considered properties of power spectra obtained from the discrete Fourier transform of finite binary strings. Gait tested the DES [10,11] in output-feedback mode, as a pseudorandom generator. Unfortunately, Gait's test was not properly developed [3,4], nor was his design for testing the DES adequate.Another test, developed by C. Yuen [2], considered analogous properties for the Walsh transform. In estimating the variance of spectral bands, Yuen assumed the spectral components to be independent. Except for the special case of Gaussian random numbers, this assumption introduces a significant error into his estimate.We recently [3,4] constructed a new test for detecting nonrandomness in finite binary strings, which extends and quantifies Gait's test. Our test is based on an evaluation of a statistic, which is a function of Fourier periodograms [5]. Binary strings produced using short-round versions of the DES in output-feedback mode were tested. By varying the number of DES rounds from 1 to 16, it was thought possible to gradually vary the degree of randomness of the resulting strings. However, we found that each of the short-round versions, consisting of 1, 2, 3, 5 and 7 rounds, generated ensembles for which at least 10% of the test strings were rejected as random, at a confidence level approaching certainty.A new test, based on an evaluation of the Walsh spectrum, is presented here. This test extends the earlier test of C. Yuen. Testing of the DES, including short-round versions, has produced results consistent with those previously obtained in [3].We prove that our measure of the Walsh spectrum is equivalent to a measure of the skirts of the logical autocorrelation function. It is clear that an analogous relationship exists between Fourier periodograms and the circular autocorrelation function.