Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: See the full text for the abstract / AFRIKAANSE OPSOMMING: Sien volteks vir die opsomming

In this dissertation the Bays-Lambossy theorem is generalized to GF(pn). The Bays-Lambossy theorem states that if two Cayley objects each based on GF(p) are isomorphic then they are isomorphic by a multiplier map. We use this characterization to show that under certain conditions two isomorphic Cayley objects over GF(pn) must be isomorphic by a function on GF(pn) of a particular type.

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

We study the cause of the signature over finite fields of integrability in two dimensional discrete dynamical systems by using theory from algebraic geometry. In particular the theory of elliptic curves is used to prove the major result of the thesis: that all birational maps that preserve an elliptic curve necessarily act on that elliptic curve as addition under the associated group law. Our result generalises special cases previously given in the literature. We apply this theorem to the specific cases when the ground fields are finite fields of prime order and the function field $mathbb{C}(t)$. In the former case, this yields an explanation of the aforementioned signature over finite fields of integrability. In the latter case we arrive at an analogue of the Arnol'd-Liouville theorem. Other results that are related to this approach to integrability are also proven and their consequences considered in examples. Of particular importance are two separate items: (i) we define a generalization of integrability called mixing and examine its relation to integrability; and (ii) we use the concept of rotation number to study differences and similarities between birational integrable maps that preserve the same foliation. The final chapter is given over to considering the existence of the signature of reversibility in higher (three and four) dimensional maps. A conjecture regarding the distribution of periodic orbits generated by such maps when considered over finite fields is given along with numerical evidence to support the conjecture.

Graduation date: 2003

We study the cause of the signature over finite fields of integrability in two dimensional discrete dynamical systems by using theory from algebraic geometry. In particular the theory of elliptic curves is used to prove the major result of the thesis: that all birational maps that preserve an elliptic curve necessarily act on that elliptic curve as addition under the associated group law. Our result generalises special cases previously given in the literature. We apply this theorem to the specific cases when the ground fields are finite fields of prime order and the function field $mathbb{C}(t)$. In the former case, this yields an explanation of the aforementioned signature over finite fields of integrability. In the latter case we arrive at an analogue of the Arnol'd-Liouville theorem. Other results that are related to this approach to integrability are also proven and their consequences considered in examples. Of particular importance are two separate items: (i) we define a generalization of integrability called mixing and examine its relation to integrability; and (ii) we use the concept of rotation number to study differences and similarities between birational integrable maps that preserve the same foliation. The final chapter is given over to considering the existence of the signature of reversibility in higher (three and four) dimensional maps. A conjecture regarding the distribution of periodic orbits generated by such maps when considered over finite fields is given along with numerical evidence to support the conjecture.

Orientador: Fernando Eduardo Torres Orihuela / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T15:12:25Z (GMT). No. of bitstreams: 1 Castilho_TiagoNunes_M.pdf: 813127 bytes, checksum: 313e9951b003dcd0e0876813659d7050 (MD5) Previous issue date: 2008 / Resumo: Nesta dissertacao estudamos cotas para o numero de pontos racionais de curvas definidas sobre corpos finitos tendo como ponto de partida a teoria de Stohr-Voloch / Abstract: In this work we study upper bounds on the number of rational points of curves over finite fields by using the Stohr-Voloch theory / Mestrado / Algebra Comutativa, Geometria Algebrica / Mestre em Matemática

This thesis introduces a new tower field representation, optimal tower fields (OTFs), that facilitates efficient finite field operations. The recursive direct inversion method presented for OTFs has significantly lower complexity than the known best method for inversion in optimal extension fields (OEFs), i.e., Itoh-Tsujii's inversion technique. The complexity of OTF inversion algorithm is shown to be O(m^2), significantly better than that of the Itoh-Tsujii algorithm, i.e. O(m^2(log_2 m)). This complexity is further improved to O(m^(log_2 3)) by utilizing the Karatsuba-Ofman algorithm. In addition, it is shown that OTFs are in fact a special class of OEFs and OTF elements may be converted to OEF representation via a simple permutation of the coefficients. Hence, OTF operations may be utilized to achieve the OEF arithmetic operations whenever a corresponding OTF representation exists. While the original OTF multiplication and squaring operations require slightly more additions than their OEF counterparts, due to the free conversion, both OTF operations may be achieved with the complexity of OEF operations. Furthermore, efficient finite field algorithms are introduced which significantly improve OTF multiplication and squaring operations. The OTF inversion algorithm was implemented on the ARM family of processors for a medium and a large sized field whose elements can be represented with 192 and 320 bits, respectively. In the implementation, the new OTF inversion algorithm ran at least six to eight times faster than the known best method for inversion in OEFs, i.e., Itoh-Tsujii inversion technique. According to the implementation results obtained, it is indicated that using the OTF inversion method an elliptic curve scalar point multiplication operation can be performed at least two to three times faster than the known best implementation for the selected fields.

Orientador: Paulo Roberto Brumatti / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T19:35:36Z (GMT). No. of bitstreams: 1 Negreiros_DiogoBrunoFernandes_M.pdf: 5674415 bytes, checksum: bdd28225d3cc5505f91fd61e797f2794 (MD5) Previous issue date: 2011 / Resumo: Este texto tem como objetivo o estudo de um tipo de código que possui relações com as teorias de curvas algébricas e de formas quadráticas. Começaremos introduzindo as definições e resultados sobre as três teorias que serão necessárias a este estudo. Depois apresentaremos os códigos a serem estudados bem como as relações entre seus sub-códigos e curvas algébricas e entre suas palavras e formas quadráticas. Observando que sub-códigos de peso mais baixo correspondem a curvas com mais pontos, nos dedicaremos a obter um processo para a descoberta de sub-códigos de peso mínimo dentro deste tipo de código. Tal processo será possível através de investigações sobre as formas quadráticas associadas a palavras. Finalizaremos com exemplos de aplicações do processo em alguns códigos, o que permite também calcular seus pesos de Hamming generalizados de ordem mais baixa / Abstract: This text's objective is the study of a kind of code wich has relations with the theories of algebraic curves and quadratic forms. We start by introducing definitions and results about the three theories we will need in such study. Later, we present the codes wich will be studied along with relations between its subcodes and algebraic curves and between its words and quadratic forms. Noting that lower weight subcodes correspond to curves with more points, we research a process to find minimum weight subcodes in this kind of code. This process will be possible through investigations on the quadratic forms related to words. Finally we set examples of applications of the process on some codes, and that gives us their lower order generalized Hamming weights / Mestrado / Matematica / Mestre em Matemática

Orientador: Plamen Emilov Kochloukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T11:39:36Z (GMT). No. of bitstreams: 1 Reis_JulioCesardos_D.pdf: 2452563 bytes, checksum: 63f8b1d463a36f74d57c1d71769dc9ae (MD5) Previous issue date: 2012 / Resumo: Na presente tese, fornecemos bases das identidades polinomiais graduadas de...Observação: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital / Abstract: In this PhD thesis we give bases of the graded polynomial identities of...Note: The complete abstract is available with the full electronic document / Doutorado / Matematica / Doutor em Matemática