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We present an extensive study of symmetric Boolean functions, especially of their cryptographic properties. Our main result establishes the link between the periodicity of the simplified value vector of a symmetric Boolean function and its degree. Besides the reduction of the amount of memory required for representing a symmetric function, this property has some consequences from a cryptographic point of view. For instance, it leads to a new general bound on the order of resiliency of symmetric functions, which improves Siegenthaler's bound. The propagation characteristics of these functions are also addressed and the algebraic normal forms of all their derivatives are given. We finally detail the characteristics of the symmetric functions of degree at most 7, for any number of variables. Most notably, we determine all balanced symmetric functions of degree less than or equal to 7. Index Terms--Boolean functions, correlation immunity, degree, derivation, propagation criterion, resiliency, symmetric functions.

Frequency hopping (FH) sequences have found wide applications in various modern FH spread-spectrum communications and radar systems. In FH spread-spectrum communications, the interference occurs when two distinct transmitters use the same frequency simultaneously. In order to evaluate the goodness of FH sequence design, the Hamming correlation function is used as an important measure. In this correspondence, by considering separately the maximum Hamming autocorrelation sidelobe [H.sub.a], and the maximum Hamming cross correlation [H.sub.c], several new lower bounds on the size p of the frequency slot set F, the sequence length L, the family size M, and correlation properties are established. The new periodic bounds include the known Lempel-Greenberger bounds as special case when M = 2, and are tighter than the Seay bounds under certain conditions when M > 2. Furthermore, the new bounds disclose more information on the relationship between the maximum autocorrelation sidelobe and the maximum cross correlation compared with the Lempel-Greenberger bounds and Seay bounds. Besides, the aperiodic FH bounds which have not yet been previously reported are also presented and discussed in this correspondence. Index Terms--Autocorrelation functions, cross-correlation functions, frequency hopping (FH) sequences, Hamming correlation, lower bounds.

In this paper, we consider the problem of space-time coding over Rician channels in the presence of spatio-temporal correlation. We derive an upper bound on the pairwise word error probability (PWEP) of space-time codes and use it as a basis for a unified approach to analysis and design of space-time codes over any flat, Rician or Rayleigh, block-fading channel. Based on the statistical properties of a Rician channel, the high-signal-to-noise ratio (SNR) behavior of the bound is either exponential or rational. In the former case, design criteria for space-time codes are based on a Euclidean-distance-like measure. However, in the latter case, design criteria are based on a rank criterion together with a coding gain criterion. The analysis of the bound shows that the performance of rank-deficient codes can be highly degraded in the presence of spatio-temporal correlation, but the performance degradation is not severe for full-rank codes. These codes exhibit robustness against the channel correlation profile. Specifically, for channels with a nonsingular covariance matrix, the asymptotic performance of a full-rank code is proportional to its performance in the independent and identically distributed (i.i.d.) case. In the case that the only constraint on the codewords is a maximum energy constraint, we show that the set of rank-deficient codes forms a proper algebraic subset of measure zero in the affine space of space-time codes. Thus, any randomly selected code is full rank with probability one. Hence, in this case, the main challenge in space-time coding is not maximizing diversity but optimizing coding gain. However, optimizing the coding gain is computationally hard and there exists no satisfactory algorithm to solve this problem in general. On the other hand, when we constrain codewords to be from a finite alphabet or to satisfy particular algebraic structures, e.g., group codes, lattice codes, or coset codes, the problem might be tractable, but rank-deficient codes might be highly probable at high spectral efficiency. Index Terms--Correlation, design criteria, fading channels, multiple antennas, performance criterion, space-time codes.