In many areas of engineering and applied mathematics, spectral methods provide very powerful tools for solving and analyzing problems. For instance, large to extremely large sizes of numbers can efficiently be multiplied by using discrete Fourier transform and convolution property. Such computations are needed when computing π to millions of digits of precision, factoring and also big prime search projects. When it comes to the utilization of spectral techniques for modular operations in public key cryptosystems two difficulties arise; the first one is the reduction needed after the multiplication step and the second is the cryptographic sizes which are much shorter than the optimal asymptotic crossovers of spectral methods. In this dissertation, a new modular reduction technique is proposed. Moreover, modular multiplication is given based on this reduction. These methods work fully in the frequency domain with some exceptions such as the initial, final and partial transformations steps. Fortunately, the new technique addresses the reduction problem however, because of the extra complexity coming from the overhead of the forward and backward transformation computations, the second goal is not easily achieved when single operations such as modular multiplication or reduction are considered. On the contrary, if operations that need several modular multiplications with respect to the same modulus are considered, this goal is more tractable. An obvious example of such an operation is the modular exponentiation i.e., the computation of c=m[superscript e] mod n where c, m, e, n are large integers. Therefore following the spectral modular multiplication operation a new modular exponentiation method is presented. Since forward and backward transformation calculations do not need to be performed for every multiplication carried during the exponentiation, the asymptotic crossover for modular exponentiation is decreased to cryptographic sizes. The method yields an efficient and highly parallel architecture for hardware implementations of public-key cryptosystems. / Graduation date: 2006

Thesis (Ph.D.) -- University of Adelaide, Dept. of Electrical Engineering, 1979. / Typescript (photocopy).

Thesis (M.S.)--Oregon State University, 2006. / Printout. Includes bibliographical references (leaves 43-44). Also available on the World Wide Web.

In this thesis the last two digits of m^k, for different cases of the positive integers m and k, in the base of 10 has been determined. Moreover, using fundamental theory from elementary number theory and abstract algebra, results most helpful in finding the last two digits in any base b has been regarded and developed, such as how to reduce large m and k to more manageable numbers.

Typescript (photocopy) / 175, 66 leaves : ill., (part col.), charts ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Electrical Engineering, 1979

In this article we will deal with the prime numbers and its current use in encryption algorithms. Encryption algorithms make possible the exchange of sensible data in internet, such as bank transactions, email correspondence and other internet transactions where privacy is important.

Not available / text

Thesis (M.S.)--Oregon State University, 2004. / Printout. Includes bibliographical references (leaves 63-64). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Texas at Austin, 2005. / Vita. Includes bibliographical references.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-01-06T13:23:39Z No. of bitstreams: 1 renatodacruzavelar.pdf: 1970202 bytes, checksum: e954f9890b80be2f892f0d33a4f38974 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-01-25T16:18:59Z (GMT) No. of bitstreams: 1 renatodacruzavelar.pdf: 1970202 bytes, checksum: e954f9890b80be2f892f0d33a4f38974 (MD5) / Made available in DSpace on 2016-01-25T16:18:59Z (GMT). No. of bitstreams: 1 renatodacruzavelar.pdf: 1970202 bytes, checksum: e954f9890b80be2f892f0d33a4f38974 (MD5) Previous issue date: 2015-04-11 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Este trabalho tem como principal objetivo apresentar um abordagem da aritmética modular direcionada para o aluno do 1o ano do ensino médio regular, baseado na experiência do autor nessa modalidade de ensino, fazendo uma breve revisão de alguns requisitos básicos para compreensão do conteúdo. A teoria é apresentada utilizando uma linguagem simples, sempre seguida de exemplos, sendo alguns deles retirados de provas de nível nacional, além de propor atividades para fixação, seguidas das respectivas soluções e atividades de aplicação, que permitem a verificação e percepção da importância do conteúdo. / This work aims to present a modular arithmetic approach directed to the student on his first year of regular high school, based on the experience of author in this type of education, making a brief review of some basic requirements to understand the content. The theory is presented using simple language, always followed by examples, some of which are drawn from national tests, and to propose activities for fixation, followed by their solutions and activities application that allow the verification and perceived importance of the content.