S-box is one of the core components of symmetric cryptographic algorithms, but differential distribution table (DDT) is an important tool to research some properties of S-boxes to resist differential attacks. In this paper, we give a relationship between the sum-of-squares of DDT and the sum-of-squares indicator of (n, m)-functions based on the autocorrelation coefficients. We also get some upper and lower bounds on the sum-of-squares of DDT of balanced (n, m)-functions, and prove that the sum-of-squares of DDT of (n, m)-functions is affine invariant under affine affine equivalent. Furthermore, we obtain a relationship between the sum-of-squares of DDT and the signal-to-noise ratio of (n, m)-functions. In addition, we calculate the distributions of the sum-of-squares of DDT for all 3-bit S-boxes, the 4-bit optimal S-boxes and all 302 balanced S-boxes (up to affine equivalence), data experiments verify our results.

In this work, a statistical analysis is performed for a simple constrained high-order Yule-Walker (YW) tone frequency estimator obtained from the first equation of the constrained high-order YW equations. Explicit expressions for its estimation bias and variance are efficiently derived by virtue of a Taylor series expansion technique. Especially, being explicit in terms of frequency, data length and Signal-to-Noise Ratio (SNR) value, the resulting bias expression can not be obtained by using the asymptotic analyses used for the parameter estimation methods. The obtained expressions are compared with their counterparts of the Pisarenko tone frequency estimator. Simulations are performed to support the theoretical results.

In this paper, the Pisarenko and the Constrained Yule-Walker (CYW) estimators of a tone frequency are first newly derived from the viewpoint of using directly the autocorrelation coefficients. Then, simulation of these two estimators is carried out in some detail. The simulation results show that compared with the Pisarenko estimator the CYW estimator, which has not been adequately studied, works poorly for low and moderate Signal-to-Noise Ratio (SNR) values. However, in case of high SNR value, it yields very small bias and comparable estimation variance, and thus produces more accurate tone frequency estimates.