Correlation sum

Abstract: We propose a new measure to characterize the dimension of complex networks based on the ergodic theory of dynamical systems. This measure is derived from the correlation sum of a trajectory generated by a random walker navigating the network, and extends the classical Grassberger-Procaccia algorithm to the context of complex networks. The method is validated with reliable results for both synthetic networks and real-world networks such as the world air-transportation network or urban networks, and provides a computationally fast way for estimating the dimensionality of networks which only relies on the local information provided by the walkers.

Abstract: The total squared correlation (TSC), maximum squared correlation (MSC), sum capacity (C/sub sum/), and total asymptotic efficiency (TAE) of underloaded signature sets, as well as the TSC and C/sub sum/ of overloaded signature sets are metrics that are optimized simultaneously over the real/complex field. In this present work, closed-form expressions are derived for the MSC, C/sub sum/, and TAE of minimum-TSC binary signature sets. The expressions disprove the general equivalence of these performance metrics over the binary field and establish conditions on the number of signatures and signature length under which simultaneous optimization can or cannot be possible. The sum-capacity loss of the recently designed minimum-TSC binary sets is found to be rather negligible in comparison with minimum-TSC real/complex-valued (Welch-bound-equality) sets.

Abstract: Correlation properties of a general binary combiner with an arbitrary number M of memory bits are derived and novel design criteria proposed. For any positive integer m, the sum of the squares of the correlation coefficients between all nonzero linear functions of m successive output bits and all linear functions of the corresponding m successive inputs is shown to be dependent upon a particular combiner, unlike the memoryless combiners. The minimum and maximum values of the correlation sum as well as the necessary and sufficient conditions for them to be achieved are determined. It turns out that the security of combiners with memory can be considerably improved if M is not small. An efficient linear sequential circuit approximation (LSCA) method is developed for obtaining output and input linear functions with comparatively large correlation coefficients which is feasible for large M and works for any practical scheme. The method consists in deriving and solving a linear sequential circuit with additional nonbalanced inputs that is based on linear approximations of the output and the component next-state functions. The corresponding correlation attack on combiners with linear feedback shift registers is analyzed and it is shown that every such combiner with or without memory is essentially zero-order correlation immune.

Abstract: It is shown that an arbitrary binary keystream generator with M bits of memory can be linearly modeled as a non-autonomous linear feedback shift register of length at most M with an additive input sequence of nonbalanced identically distributed binary random variables. The sum of the squares of input correlation coefficients over all the linear models of any given length proves to be dependent on a keystream generator. The minimum and maximum values of the correlation sum along with the necessary and sufficient conditions for them to be achieved are established. An effective method for the linear model determination based on the linear sequential circuit approximation of autonomous finite-state machines is developed. Linear models for clock controlled shift registers and arbitrary shift register based keystream generators are derived. Several examples including the basic summation generator, the clock-controlled cascade, and the shrinking generator are presented. Linear models are the basis for a general structure-dependent and initial-state-independent statistical test. They may also be used for divide and conquer correlation attacks on the initial state. Security against the corresponding statistical attack appears hard to control in practice and generally hard to achieve with simple keystream generator schemes.