Boolean functions are the main building blocks of most cipher systems. Various aspects of their cryptological characteristics are examined and investigated by many researchers from different fields. This thesis has no claim to obtain original results but consists in an attempt at giving a unified survey of the main results of the subject. In this thesis, the theory of boolean functions is presented in details, emphasizing some important cryptological properties such as balance, nonlinearity, strict avalanche criterion and propagation criterion. After presenting many results about these criteria with detailed proofs, two upper bounds and two lower bounds on the nonlinearity of a boolean function due to Zhang and Zheng are proved. Because of their importance in the theory of boolean functions, construction of Sylvester-Hadamard matrices are shown and most of their properties used in cryptography are proved. The Walsh transform is investigated in detail by proving many properties. By using a property of Sylvester-Hadamard matrices, the fast Walsh transform is presented and its application in finding the nonlinearity of a boolean function is demonstrated. One of the most important classes of boolean functions, so called bent functions, are presented with many properties and by giving several examples, from the paper of Rothaus. By using bent functions, relations between balance, nonlinearity and propagation criterion are presented and it is shown that not all these criteria can be simultaneously satisfied completely. For this reason, several constructions of functions optimizing these criteria which are due to Seberry, Zhang and Zheng are presented.

The operational life of wireless sensor network based distributed sensing systems is limited by the energy provided by a portable battery pack. Owing to the inherently resource constrained nature of wireless sensor networks and nodes, a major research thrust in this field is the search for energy-aware methods of operation. Communication is among the most energy-intensive operations on a wireless device. It is therefore, the focus of our efforts to develop an energy-aware method of communication and to introduce a degree of reconfigurability to ensure autonomous operation of such devices. Given this background, three research tasks have been identified and investigated during the course of this research. 1) Devising an energy-efficient method of communication in a framework of reconfigurable operation: The dependence of the energy consumed during communication on the number of bits transmitted (and received) was identified from prior research work. A novel method of data compression was designed to exploit this dependence. This method uses the time-limited, orthonormal Walsh functions as basis functions for representing signals. The L2 norm of this representation is utilized to further compress the signals. From Parseval’s relation, the square of the L2 norm represents the energy content of a signal. The application of this theorem to our research makes it possible to use the L2 norm as a control knob. The operation of this control knob makes it possible to optimize the number of terms required to represent signals. The time-limited nature of the Walsh functions was leveraged to inject dynamic behaviour into our coding method. This time-limited nature allows decomposition of finite time-segments, without attendant limitations like loss of resolution that are inherent to derived, discrete transforms like the discrete Fourier transform or the discrete time Fourier transform. This decomposition over successive, finite time-segments, coupled with innovative operation of the previously mentioned control knob on every segment, gives us a dynamic scaling technique. The amount of data to be transmitted is in turn based on the magnitude of the coefficients of decomposition of each time-segment, leading to the realization of a variable word length coding method. This dynamic coding method can identify evolving changes or events in the quantity being sensed. The coefficients of decomposition represent features present in successive time-segments of signals and therefore enable identification of evolving events. The ability to identify events as they occur enables the algorithm to react to events as they evolve in the system. In other words the data transmission and the associated energy consumption are imparted a reconfigurable, event-driven nature by implementation of the coding algorithm. Performance evaluation of this method via simulations on machine generated (bearing vibration) and biometric (electro-cardio gram) signals shows it be a viable method for energy-aware communication. 2) Developing a framework for reconfigurable triggering: A framework for completely autonomous triggering of the coding method has been developed. This is achieved by estimating correlations of the signal with the representative Walsh functions. The correlation coefficient of a signal segment with a Walsh function gives a picture of the amount of energy localized by the function. This information is used to autonomously tune the abovementioned control knob or, in more proper terms, the degree of thresholding used in compression. Evaluation of this framework on bearing vibration and electro-cardio gram signals has shown results consistent with those of previous simulations. 3) Devising a computationally compact method of feature classification: A method of investigating time series measurements of dynamic systems in order to classify features buried in the signal measurements was investigated. The approach involves discretizing time-series measurements into strings of pre-defined symbols. These strings are transforms of the original time-series measurements and are a representation of the system dynamics. A method of statistically analyzing the symbol strings is presented and its efficacy is studied through representative simulations and experimental investigation of vibration signals recorded from a rolling bearing element. The method is computationally compact because it obviates the need for local signal processing tasks like denoising, detrending and amplification. Results indicate that the method can effectively classify deteriorating machine health, changing operating conditions and evolving defects. In addition to these major foci, another research task was the design and implementation of a wireless network testbed. This testbed consists of a network of netbooks, connected together wirelessly and was utilized for experimental verification of the variable word length coding method.

<p> In this thesis, a high-speed inverse Walsh transform apparatus was designed and built which sums over the sixteen most dominant coefficients in the time base period. The transform includes a maximum of 64 terms. The Walsh function generator used works with a clock rate up to 10 MHz to produce 64 different sequency terms with accurate timing and hazard free operation. A synchronizing pulse is produced by the circuit to determine the beginning of the Walsh transform period. The final adder stage limits the speed of the apparatus to a 1 MHz square wave. An application of the instrument was made to reconstruct one line of an actual video signal.</p> / Thesis / Master of Engineering (MEngr)

When considering Boolean switching functions with n input variables, there are 2^(2^n) possible functions that can be realized by enumerating all possible combinations of input values and arrangements of output values. As is expected with double exponential growth, the number of functions becomes unmanageable very quickly as n increases. This thesis develops a new approach for computing the spectral classes where the spectral operations are performed by manipulating the truth tables rather than first moving to the spectral domain to manipulate the spectral coefficients. Additionally, a generic approach is developed for modeling these spectral operations within the functional domain. The results of this research match previous for n < or = to 4 but differ when n=5 is considered. This research indicates with a high level of confidence that there are in fact 15 previously unidentified classes, for a total of 206 spectral classes needed to represent all 2^(2^n) Boolean functions.