Boolean functions play an important role in the design and analysis of symmetric-key cryptosystems, as well as having applications in other fields such as coding theory. Boolean functions acting on large number of inputs introduces the problem of computing the cryptographic properties. Traditional methods of computing these properties involve transformations which require computation and memory resources exponential in the number of input variables. When the number of inputs is large, Boolean functions are usually defined by the algebraic normal form (ANF) representation. In this thesis, methods for computing the weight and nonlinearity of Boolean functions from the ANF representation are investigated. The relation between the ANF coefficients and the weight of a Boolean function was introduced by Carlet and Guillot. This expression allows the weight to be computed in $mathcal{O}(2^p)$ operations for a Boolean function containing $p$ monomials in its ANF. In this work, a more efficient algorithm for computing the weight is proposed, which eliminates the unnecessary calculations in the weight expression. By generalizing the weight expression, a formulation of the distances to the set of linear functions is obtained. Using this formulation, the problem of computing the nonlinearity of a Boolean function from its ANF is reduced to an associated binary integer programming problem. This approach allows the computation of nonlinearity for Boolean functions with high number of input variables and consisting of small number of monomials in a reasonable time.

As Field-Programmable Gate Array (FPGA) capacity can now support several processors on a single device, the scalability of FPGA design tools and methods has emerged as a major obstacle for the wider use of FPGAs. For example, logic synthesis, which has traditionally been the fastest step in the FPGA Computer-Aided Design (CAD) flow, now takes several hours to complete in a typical FPGA compile. In this work, we address this problem by focusing on two areas. First, we revisit FPGA logic synthesis and attempt to improve its scalability. Specifically, we look at a binary decision diagram (BDD) based logic synthesis flow, referred to as FBDD, where we improve its runtime by several fold with a marginal impact to the resulting circuit area. We do so by speeding up the classical cut generation problem by an order-of-magnitude which enables its application directly at the logic synthesis level. Following this, we introduce a guided partitioning technique using a fast global budgeting formulation, which enables us to optimize individual “pockets” within the circuit without degrading the overall circuit performance. By using partitioning we can significantly reduce the solution space of the logic synthesis problem and, furthermore, open up the possibility of parallelizing the logic synthesis step. The second area we look at is the area of Engineering Change Orders (ECOs). ECOs are incremental modifications to a design late in the design flow. This is beneficial since it is minimally disruptive to the existing circuit which preserves much of the engineering effort invested previously in the design. In a design flow where most of the steps are fully automated, ECOs still remain largely a manual process. This can often tie up a designer for weeks leading to missed project deadlines which is very detrimental to products whose life-cycle can span only a few months. As a solution to this, we show how we can leverage existing logic synthesis techniques to automatically modify a circuit in a minimally disruptive manner. This can significantly reduce the turn-around time when applying ECOs.

As Field-Programmable Gate Array (FPGA) capacity can now support several processors on a single device, the scalability of FPGA design tools and methods has emerged as a major obstacle for the wider use of FPGAs. For example, logic synthesis, which has traditionally been the fastest step in the FPGA Computer-Aided Design (CAD) flow, now takes several hours to complete in a typical FPGA compile. In this work, we address this problem by focusing on two areas. First, we revisit FPGA logic synthesis and attempt to improve its scalability. Specifically, we look at a binary decision diagram (BDD) based logic synthesis flow, referred to as FBDD, where we improve its runtime by several fold with a marginal impact to the resulting circuit area. We do so by speeding up the classical cut generation problem by an order-of-magnitude which enables its application directly at the logic synthesis level. Following this, we introduce a guided partitioning technique using a fast global budgeting formulation, which enables us to optimize individual “pockets” within the circuit without degrading the overall circuit performance. By using partitioning we can significantly reduce the solution space of the logic synthesis problem and, furthermore, open up the possibility of parallelizing the logic synthesis step. The second area we look at is the area of Engineering Change Orders (ECOs). ECOs are incremental modifications to a design late in the design flow. This is beneficial since it is minimally disruptive to the existing circuit which preserves much of the engineering effort invested previously in the design. In a design flow where most of the steps are fully automated, ECOs still remain largely a manual process. This can often tie up a designer for weeks leading to missed project deadlines which is very detrimental to products whose life-cycle can span only a few months. As a solution to this, we show how we can leverage existing logic synthesis techniques to automatically modify a circuit in a minimally disruptive manner. This can significantly reduce the turn-around time when applying ECOs.

Block and stream ciphers are widely used to protect the privacy of digital information. A variety of attacks against block and stream ciphers exist; the most recent being the algebraic attacks. These attacks reduce the cipher to a simple algebraic system which can be solved by known algebraic techniques. These attacks have been very successful against a variety of stream ciphers and major efforts (for example eSTREAM project) are underway to design and analyze new stream ciphers. These attacks have also raised some concerns about the security of popular block ciphers. In this thesis, apart from designing new stream ciphers, we focus on analyzing popular nonlinear transformations (Boolean functions and S-boxes) used in block and stream ciphers for various cryptographic properties, in particular their resistance against algebraic attacks. The main contribution of this work is the design of two new stream ciphers and a thorough analysis of the algebraic immunity of Boolean functions and S-boxes based on power mappings. First we present WG, a family of new stream ciphers designed to obtain a keystream with guaranteed randomness properties. We show how to obtain a mathematical description of a WG stream cipher for the desired randomness properties and security level, and then how to translate this description into a practical hardware design. Next we describe the design of a new RC4-like stream cipher suitable for high speed software applications. The design is compared with original RC4 stream cipher for both security and speed. The second part of this thesis closely examines the algebraic immunity of Boolean functions and S-boxes based on power mappings. We derive meaningful upper bounds on the algebraic immunity of cryptographically significant Boolean power functions and show that for large input sizes these functions have very low algebraic immunity. To analyze the algebraic immunity of S-boxes based on power mappings, we focus on calculating the bi-affine and quadratic equations they satisfy. We present two very efficient algorithms for this purpose and give new S-box constructions that guarantee zero bi-affine and quadratic equations. We also examine these S-boxes for their resistance against linear and differential attacks and provide a list of S-boxes based on power mappings that offer high resistance against linear, differential, and algebraic attacks. Finally we investigate the algebraic structure of S-boxes used in AES and DES by deriving their equivalent algebraic descriptions.

In this thesis several random Boolean networks are simulated. Both completely computer generated network and models for biological networks are simulated. Several different tools are used to gain knowledge about the robustness. These tools are Derrida plots, noise analysis and mean probability for canalizing rules. Some simulations on how entropy works as an indicator on if a network is robust are also included. The noise analysis works by measuring the hamming distance between the state of the network when noise is applied and when no noise is applied. For many of the simulated networks two types of rules are applied: nested canalizing and flat distributed rules. The computer generated networks consists of two types of networks: scale-free and ER-networks. One of the conclusions in this report is that nested canalizing rules are often more robust than flat distributed rules. Another conclusion is that the mean probability for canalizing rules has, for flat distributed rules, a very dominating effect on if the network is robust or not. Yet another conclusion is that the probability distribution for indegrees, for flat distributed rules, has a strong effect on if a network is robust due to the connection between the probability distribution for indegrees and the mean probability for canalizing rules.

Block and stream ciphers are widely used to protect the privacy of digital information. A variety of attacks against block and stream ciphers exist; the most recent being the algebraic attacks. These attacks reduce the cipher to a simple algebraic system which can be solved by known algebraic techniques. These attacks have been very successful against a variety of stream ciphers and major efforts (for example eSTREAM project) are underway to design and analyze new stream ciphers. These attacks have also raised some concerns about the security of popular block ciphers. In this thesis, apart from designing new stream ciphers, we focus on analyzing popular nonlinear transformations (Boolean functions and S-boxes) used in block and stream ciphers for various cryptographic properties, in particular their resistance against algebraic attacks. The main contribution of this work is the design of two new stream ciphers and a thorough analysis of the algebraic immunity of Boolean functions and S-boxes based on power mappings. First we present WG, a family of new stream ciphers designed to obtain a keystream with guaranteed randomness properties. We show how to obtain a mathematical description of a WG stream cipher for the desired randomness properties and security level, and then how to translate this description into a practical hardware design. Next we describe the design of a new RC4-like stream cipher suitable for high speed software applications. The design is compared with original RC4 stream cipher for both security and speed. The second part of this thesis closely examines the algebraic immunity of Boolean functions and S-boxes based on power mappings. We derive meaningful upper bounds on the algebraic immunity of cryptographically significant Boolean power functions and show that for large input sizes these functions have very low algebraic immunity. To analyze the algebraic immunity of S-boxes based on power mappings, we focus on calculating the bi-affine and quadratic equations they satisfy. We present two very efficient algorithms for this purpose and give new S-box constructions that guarantee zero bi-affine and quadratic equations. We also examine these S-boxes for their resistance against linear and differential attacks and provide a list of S-boxes based on power mappings that offer high resistance against linear, differential, and algebraic attacks. Finally we investigate the algebraic structure of S-boxes used in AES and DES by deriving their equivalent algebraic descriptions.

A new approach to the process of Directed Evolution is proposed, which utilizes different machine learning algorithms. Directed Evolution is a process of improving a protein for catalytic purposes by introducing random mutations in its sequence to create variants. Through these mutations, Directed Evolution explores the sequence space, which is defined as all the possible sequences for a given number of amino acids. Each variant sequence is divided into one of two classes, positive or negative, according to their activity or stability. By employing machine learning algorithms for feature selection on the sequence of these variants of the protein, attributes or amino acids in its sequence important for the classification into positive or negative, can be identified. Support Vector Machines (SVMs) were utilized to identify the important individual amino acids for any protein, which have to be preserved to maintain its activity. The results for the case of beta-lactamase show that such residues can be identified with high accuracy while using a small number of variant sequences. Another class of machine learning problems, Boolean Learning, was used to extend this approach to identifying interactions between the different amino acids in a proteins sequence using the variant sequences. It was shown through simulations that such interactions can be identified for any protein with a reasonable number of variant sequences. For experimental verification of this approach, two fluorescent proteins, mRFP and DsRed, were used to generate variants, which were screened for fluorescence. Using Boolean Learning, an interacting pair was identified, which was shown to be important for the fluorescence. It was also shown through experiments and simulations that knowing such pairs can increase the fraction active variants in the library. A Boolean Learning algorithm was also developed for this application, which can learn Boolean functions from data in the presence of classification noise.

Understanding the mechanisms of gene regulation in the field of systems biology is a very important issue. With the development of bio-information technology, we can capture large quantities of gene¡¦s expression data from DNA microarray data. In order to discover the relationship of gene regulation, the simulation of gene regulatory networks have been proposed. Among these simulations methods, the S-system model is the most widely used in non-linear differential equations. It can simulate the dynamic behavior of gene regulatory networks and gene expression, but can¡¦t explain the structure and orientation of gene regulatory networks. Therefore, we propose a Boolean knowledge-based approach to assist the S-system modeling of gene regulatory networks. In this study, we derive the positive and negative regulatory relationships between genes from the regulation of S-system parameters, and use the structure of Boolean networks as our knowledge base. According to the results of the experiment, we can verify our assumptions for the regulation of the S-system parameters, and also has a better understanding of the regulatory relationship between genes.

With the advances in very large scale integration (VLSI) technology, hardware is going parallel. Software, which was traditionally designed to execute on single core microprocessors, now faces the tough challenge of taking advantage of this parallelism, made available by the scaling of hardware. The work presented in this dissertation studies the acceleration of electronic design automation (EDA) software on several hardware platforms such as custom integrated circuits (ICs), field programmable gate arrays (FPGAs) and graphics processors. This dissertation concentrates on a subset of EDA algorithms which are heavily used in the VLSI design flow, and also have varying degrees of inherent parallelism in them. In particular, Boolean satisfiability, Monte Carlo based statistical static timing analysis, circuit simulation, fault simulation and fault table generation are explored. The architectural and performance tradeoffs of implementing the above applications on these alternative platforms (in comparison to their implementation on a single core microprocessor) are studied. In addition, this dissertation also presents an automated approach to accelerate uniprocessor code using a graphics processing unit (GPU). The key idea is to partition the software application into kernels in an automated fashion, such that multiple instances of these kernels, when executed in parallel on the GPU, can maximally benefit from the GPU?s hardware resources. The work presented in this dissertation demonstrates that several EDA algorithms can be successfully rearchitected to maximally harness their performance on alternative platforms such as custom designed ICs, FPGAs and graphic processors, and obtain speedups upto 800X. The approaches in this dissertation collectively aim to contribute towards enabling the computer aided design (CAD) community to accelerate EDA algorithms on arbitrary hardware platforms.

In this thesis we obtain several interesting algebraic properties of the operations used in the block cipher IDEA which are important for cryptographic analyzes. We view each of these operations as a function from $mathbb Z_{2}^n times mathbb Z_{2}^n to mathbb Z_{2}^n$. By fixing one of variables $v(z)=mathbf Z$ in $mathbb Z_{2}^n times mathbb Z_{2}^n$, we define functions $mathbf {f}_z$ and $mathbf {g}_z$ from $mathbb Z_{2}^n$ to $mathbb Z_{2}^n$ for the addition $BIGboxplus$ and the multiplication $BIGodot$ operations, respectively. We first show that the nonlinearity of $mathbf {g}_z$ remains the same under some transformations of $z$. We give an upper bound for the nonlinearity of $mathbf {g}_{2^k}$, where $2leq k &lt / n-1$. We list all linear relations which make the nonlinearity of $mathbf {f}_z$ and $mathbf {g}_z$ zero and furthermore, we present all linear relations for $mathbf {g}_z$ having a high probability. We use these linear relations to derive many more linear relations for 1-round IDEA. We also devise also a new algorithm to find a set of new linear relations for 1-round IDEA based on known linear relations. Moreover, we extend the largest known linear class of weak keys with cardinality $2^{23}$ to two classes with cardinality $2^{24}$ and $2^{27}$. Finally, we obtain several interesting properties of the set $ { ({mathbf X},{mathbf X} BIGoplus {mathbf A}) in mathbb Z_2^n times mathbb Z_2^n ,|, (mathbf {X}BJoin {mathbf Z})BIGoplus( ({mathbf X} BIGoplus {mathbf A} ) BJoin mathbf {Z} ) = {mathbf B} }$ for varying ${mathbf A}, {mathbf B}$ and ${mathbf Z}$ in $mathbb Z_2^n$, where $BJoin in { BIGodot,BIGboxplus }$. By using some of these properties, we present impossible differentials for 1-round IDEA and Pseudo-Hadamard Transform.