Finite elds of the form F2m play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these elds can have a signi cant impact on the resource requirements for quantum arithmetic. In particular, we show how the Gaussian normal basis representations and \ghost-bit basis" representations can be used to implement inverters with a quantum circuit of depth O(mlog(m)). To the best of our knowledge, this is the rst construction with subquadratic depth reported in the literature. Our quantum circuit for the computation of multiplicative inverses is based on the Itoh-Tsujii algorithm which exploits the property that, in a normal basis representation, squaring corresponds to a permutation of the coe cients. We give resource estimates for the resulting quantum circuit for inversion over binary elds F2m based on an elementary gate set that is useful for fault-tolerant implementation. Elliptic curves over nite elds F2m play a prominent role in modern cryptography. Published quantum algorithms dealing with such curves build on a short Weierstrass form in combination with a ne or projective coordinates. In this thesis we show that changing the curve representation allows a substantial reduction in the number of T-gates needed to implement the curve arithmetic. As a tool, we present a quantum circuit for computing multiplicative inverses in F2m in depth O(mlogm) using a polynomial basis representation, which may be of independent interest. Finally, we change our focus from the design of circuits which aim at attacking computational assumptions on asymmetric cryptographic algorithms to the design of a circuit attacking a symmetric cryptographic algorithm. We consider a block cipher, SERPENT, and our design of a quantum circuit implementing this cipher to be used for a key attack using Grover's algorithm as in [18]. This quantum circuit is essential for understanding the complexity of Grover's algorithm. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2016. / FAU Electronic Theses and Dissertations Collection

"This thesis describes how an elliptic curve cryptosystem can be implemented on low cost microprocessors without coprocessors with reasonable performance. We focus in this paper on the Intel 8051 family of microcontrollers popular in smart cards and other cost-sensitive devices, and on the Motorola Dragonball, found in the Palm Computing Platform. The implementation is based on the use of the Optimal Extension Fields GF((2^8-17)^17) for low end 8-bit processors, and GF((2^13-1)^13) for 16-bit processors. Two advantages of our method are that subfield modular reduction can be performed infrequently, and that an adaption of Itoh and Tsujii's inversion algorithm may be used for the group operation. We show that an elliptic curve scalar multiplication with a fixed point, which is the core operation for a signature generation, can be performed in a group of order approximately 2^134 in less than 2 seconds on an 8-bit smart card. On a 16-bit microcontroller, signature generation in a group of order approximately 2^169 can be performed in under 700 milliseconds. Unlike other implementations, we do not make use of curve parameters defined over a subfield such as Koblitz curves."

Graduation date: 2003

Security issues have started to play an important role in the wireless communication and computer networks due to the migration of commerce practices to the electronic medium. The deployment of security procedures requires the implementation of cryptographic algorithms. Performance has always been one of the most critical issues of a cryptographic function, which determines its effectiveness. Among those cryptographic algorithms are the elliptic curve cryptosystems which use the arithmetic of finite fields. Furthermore, fields of characteristic two are preferred since they provide carry-free arithmetic and at the same time a simple way to represent field elements on current processor architectures. Multiplication is a very crucial operation in finite field computations. In this contribution, we compare most of the multiplier architectures found in the literature to clarify the issue of choosing a suitable architecture for a specific application. The importance of the measuring the energy consumption in addition to the conventional measures for energy-critical applications is also emphasized. A new parallel-in serial-out multiplier based on all-one polynomials (AOP) using the shifted polynomial basis of representation is presented. The proposed multiplier is area efficient for hardware realization. Low hardware complexity is advantageous for implementation in constrained environments such as smart cards. Architecture of an elliptic curve coprocessor has been developed using the proposed multiplier. The instruction set architecture has been also designed. The coprocessor has been simulated using VHDL to very the functionality. The coprocessor is capable of performing the scalar multiplication operation over elliptic curves. Point doubling and addition procedures are hardwired inside the coprocessor to allow for faster operation.

Security issues have started to play an important role in the wireless communication and computer networks due to the migration of commerce practices to the electronic medium. The deployment of security procedures requires the implementation of cryptographic algorithms. Performance has always been one of the most critical issues of a cryptographic function, which determines its effectiveness. Among those cryptographic algorithms are the elliptic curve cryptosystems which use the arithmetic of finite fields. Furthermore, fields of characteristic two are preferred since they provide carry-free arithmetic and at the same time a simple way to represent field elements on current processor architectures. Multiplication is a very crucial operation in finite field computations. In this contribution, we compare most of the multiplier architectures found in the literature to clarify the issue of choosing a suitable architecture for a specific application. The importance of the measuring the energy consumption in addition to the conventional measures for energy-critical applications is also emphasized. A new parallel-in serial-out multiplier based on all-one polynomials (AOP) using the shifted polynomial basis of representation is presented. The proposed multiplier is area efficient for hardware realization. Low hardware complexity is advantageous for implementation in constrained environments such as smart cards. Architecture of an elliptic curve coprocessor has been developed using the proposed multiplier. The instruction set architecture has been also designed. The coprocessor has been simulated using VHDL to very the functionality. The coprocessor is capable of performing the scalar multiplication operation over elliptic curves. Point doubling and addition procedures are hardwired inside the coprocessor to allow for faster operation.

In the thesis we study the elliptic curves and its use in cryptography. Elliptic curvesencompasses a vast area of mathematics. Elliptic curves have basics in group theory andnumber theory. The points on elliptic curve forms a group under the operation of addition.We study the structure of this group. We describe Hasse’s theorem to estimate the numberof points on the curve. We also discuss that the elliptic curve group may or may not becyclic over finite fields. Elliptic curves have applications in cryptography, we describe theapplication of elliptic curves for discrete logarithm problem and ElGamal cryptosystem.

In the thesis we study the elliptic curves and its use in cryptography. Elliptic curvesencompasses a vast area of mathematics. Elliptic curves have basics in group theory andnumber theory. The points on elliptic curve forms a group under the operation of addition.We study the structure of this group. We describe Hasse’s theorem to estimate the numberof points on the curve. We also discuss that the elliptic curve group may or may not becyclic over finite fields. Elliptic curves have applications in cryptography, we describe theapplication of elliptic curves for discrete logarithm problem and ElGamal cryptosystem.

In this thesis we study a class of authentication codes with secrecy. We obtain the maximum success probability of the impersonation attack and the maximum success probability of the substitution attack on these authentication codes with secrecy. Moreover we determine the level of secrecy provided by these authentication codes. Our methods are based on the theory of algebraic function fields over finite fields. We study a certain class of algebraic function fields over finite fields related to this class of authentication codes. We also determine the number of rational places of this class of algebraic function fields.

This work is composed of two independent parts, both addressing problems related to algebraic curves over finite fields. In the first part, we characterize all irreducible plane curves defined over Fq which are Frobenius non-classical for different powers of q. Such characterization gives rise to many previously unknown curves which turn out to have some interesting properties. For instance, for n [greater-than or equal to] 3 a curve which is both q- and qn-Frobenius non-classical will have its number of Fqn-rational points attaining the Stöhr-Voloch bound. In the second part, we study the arc property of several plane curves and present new complete (N, d)-arcs in PG(2, q). Some of these arcs (viewed as linear (N, 3,N - d)-codes) are just a small constant away from the Griesmer bound and for some small values of q the bound is achieved. In addition, this part also answers a question of Voloch about the arc property of a certain family of curves with many rational points, and another question of Giulietti et al about the arc property of q-Frobenius non-classical plane curves. / text

The goal of this thesis is to apply an approach due to Elkies to study the distribution of rational point counts for certain families of curves and surfaces over finite fields. A vector space of polynomials over a fixed finite field gives rise to a linear code, and the weight enumerator of this code gives information about point count distributions. The MacWilliams theorem gives a relation between the weight enumerator of a linear code and the weight enumerator of its dual code. For certain codes C coming from families of varieties where it is not known how to determine the distribution of point counts directly, we analyze low-weight codewords of the dual code and apply the MacWilliams theorem and its generalizations to gain information about the weight enumerator of C. These low-weight dual codes can be described in terms of point sets that fail to impose independent conditions on this family of varieties. Our main results concern rational point count distributions for del Pezzo surfaces of degree 2, and for certain families of genus 1 curves. These weight enumerators have interesting geometric and coding theoretic applications for small q. / Mathematics