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The Advanced Encryption Standard (AES) is the current standard for symmetric key ciphers and is algorithmically secure. Side channel attacks that target power consumption can reveal the secret key in AES implementations. Masking data with random variables is one of the main methods used to thwart power analysis attacks. Data can be masked with multiple random variables to prevent higher-order attacks at the cost of a large increase in area. This thesis tests the plausibility of using varied finite field construction to prevent power analysis attacks as an alternative to masking. Initially, a design using finite field architecture as the sole countermeasure was investigated. This was followed by varied field construction in conjunction with a low entropy masking scheme. Neither approach provided an acceptable trade off between security and area. Analysis then turned to a combined Boolean and multiplicative masking scheme. Varied construction provided little gain for multiplicative masking. However, varied constructions were found to greatly increase security when used in conjunction with a Boolean random mask. A novel masking scheme for AES resistant to second-order attacks is proposed. Instead of an additional mask, variation in finite field construction is exploited to increase resistance to second-order attacks in Boolean masked shares. As a result, the area requirement is substantially reduced. For an example AES encryption, the proposed design is 12% smaller compared to the previous best design, with a small drop in achievable security level.

Cryptographic and coding theory algorithms use arithmetic operations over nite elds.Finite eld multiplications over GF(2m) are critical components for these systems. Wellknown irreducible polynomials are all-one-polynomials (AOP), equally spaced polynomial(ESP), trinomial and pentanomial. Due to its simple structure, AOP based multiplica-tion is easy to implement and hence the AOP based multiplication of variable lengthcan be used as a standard computation core. In this thesis, rst of all, we employ lowregister complexity AOP based systolic multiplication core to propose multiplication overGF(2m) based on NIST recommended pentanomials. The proposed parallel and serial ar-chitectures use pre-computation (PC) modules to compute bits involve in multiplicationand re-combination (RC) modules to combine computed bits from PC to form vectorswhich will reduce the multiplication complexity. The corresponding architecture basedon the proposed algorithm is then synthesized by Xilinx ISE 14.1 on a Virtex 5 FPGAdevice and it is observed that the proposed structures has lower area-delay complexitythan the best of existing designs. Second, we propose a novel obfuscation mechanismto equip multiplication over dierent irreducible polynomials and addition operations.Desired functionality of the proposed obfuscated structure is achieved through correctinput sequence to controller (FSM). This is the rst architecture proposed which canimplement four types of polynomial multiplications and additions with obfuscated man-ner. The proposed architecture is synthesized and implemented in application specicintegration circuits (ASIC) platform and have achieved excellent area-time performance.

Capacity characterization of communication networks is the most fundamental problem in Information Theory, that underlies the design of various wireless and wired networks. The radical idea of "Interference alignment" has enabled Capacity or Degrees of Freedom characterization (DoF, a first order approximation) for many interference networks. Various alignment schemes developed have provided new and fundamental insights into the number of accessible signal dimensions in communication networks where the output signals are linear functions of the input signals. Most of the prior art deal with generic channels wherein the channel coefficients are assumed to be independent and drawn from a continuous distribution, continuous alphabet with infinite diversity, and the network is often single-hop. These assumptions are challenged due to the following reasons : 1) In MIMO systems, poor scattering environment and network topology lead to spatial dependencies that are manifested as rank deficient channels, 2) Multi-hop dependencies arise due to the presence of relays, and 3) Linear network coding applications (as in wired networks) act as finite field counterparts of wireless networks, with limited diversity.In this thesis, Capacity / DoF of linear communication networks are characterized for "Non-generic channels". One of the significant problems considered is the DoF of the K-user MIMO rank deficient interference channel, with different ranks for the direct and the cross channels. For this rank deficient interference channel, it is shown that the rank deficiency of direct channels does not help DoF and the rank-deficiency of cross-channels does not hurt DoF. The main challenge is to account for the spatial dependencies introduced by rank deficiencies in the interference alignment schemes that typically rely on the independence of channel coefficients. Another interesting problem is the DoF of Two-hop MIMO rank deficient interference channel with different channel ranks in the first and the second hops, for which a rank-matching principle is identified reminiscent of impedance matching in circuit theory. For this channel, the DoF loss is shown to be the rank-mismatch between the two hops. Finally, capacity results for the finite field counterparts of wireless networks are presented, exploring the implications of channels being from a finite alphabet with limiteddiversity. By characterizing the capacity of constant finite field channels over F_{p^n} for 2-user X channel and 3-user interference channel, interesting parallels are drawn between p and SNR, and n and Channel Diversity.

Traditionally number theorists have studied, both theoretically and computationally, elliptic curves and their L-functions over number fields, in particular over the rational numbers. Much less work has been done over function fields, especially computationally, where the underlying geometry of the function field plays an intimate role in the arithmetic of elliptic curves. We make use of this underlying geometry to develop a method to compute the L-function of an elliptic curve and its twists over the function field of the projective line over a finite field. This method requires computing the number of points on an elliptic curve over a finite field, for which we present a novel algorithm. If the j-invariant of an elliptic curve over a function field is non-constant, its L-function is a polynomial, hence its analytic rank and value at a given point can be computed exactly. We present data in this direction for a family of quadratic twists of four fixed elliptic curves over a few function fields of differing characteristic. First we present analytic rank data that confirms a conjecture of Goldfeld, in stark contrast to the corresponding data in the number field setting. Second, we present data on the integral moments of the value of the L-function at the symmetry point, which on the surface appears to refute random matrix theory conjectures.

In this paper, we provide a generalization of binary quadratic residue codes to the cases of higher power prime residues over the finite field of the same order, which we will call qth power residue codes. We find generating polynomials for such codes, define a new notion corresponding to the binary concept of an idempotent, and use this to find square root lower bound for the codeword weight of the duals of such codes, which leads to a lower bound on the weight of the codewords themselves. In addition, we construct a family of asymptotically bad qth power residue codes.

Elliptic curve cryptography (ECC) offers a viable alternative to Rivest-Shamir-Adleman (RSA) by delivering equivalent security with a smaller key size. This has several advantages, including smaller bandwidth demands, faster key exchange, and lower latency encryption and decryption. The fundamental operation for ECC is scalar point multiplication, wherein a point P on an elliptic curve defined over a finite field is multiplied by a scalar k. The complexity of this operation requires a hardware implementation to achieve high performance. The algorithms involved in scalar point multiplication are constantly evolving, incorporating the latest developments in number theory to improve computation time. These competing needs, high performance and flexibility, have caused previous implementations to either limit their adaptability or to incur performance losses. This thesis explores the use of a hybrid-FPGA for scalar point multiplication. A hybrid- FPGA contains a general purpose processor (GPP) in addition to reconfigurable fabric. This allows for a software/hardware co-design with low latency communication between the GPP and custom hardware. The elliptic curve operations and finite field inversion are programmed in C code. All other finite field arithmetic is implemented in the FPGA hardware, providing higher performance while retaining flexibility. The resulting implementation achieves speedups ranging from 24 times to 55 times faster than an optimized software implementation executing on a Pentium II workstation. The scalability of the design is investigated in two directions: faster finite field multiplication and increased instruction level parallelism exploitation. Increasing the number of parallel arithmetic units beyond two is shown to be less efficient than increasing the speed of the finite field multiplier.

Linking the structure of a system with its dynamics is an important problem in the theory of finite dynamical systems. For monomial dynamical systems, that is, a system that can be described by monomials, information about the limit cycles can be obtained from the monomials themselves. In particular, this work contains sufficient and necessary conditions for a monomial dynamical system to have only fixed points as limit cycles.