"The object of this paper is to present a detailed, illustrated discussion of a portion of the theory of bilinear forms as found in Volume 2 of Lectures in Abstract Algebra by Nathan Jacobson. This theory involves the fundamental properties of bilinear forms on finite dimensional vector spaces over arbitrary division rings and includes such things as a discussion of the relationship between matrices and bilinear forms as well as the definition of the transpose of a transformation relative to a pair of bilinear forms; a definition independent of the matrix representations of the forms"--Introduction. / Typescript. / "May, 1955." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: Nickolas Heerema, Professor Directing Paper. / Includes bibliographical references (leaf 30).

Inaug.-diss.--Breslau. / Lebenslauf.

No description available.

Thesis--Bonn. Extra t.p. with thesis statement inserted. / Includes bibliographical references (p. 45).

Thesis--Bonn. Extra t. p. with thesis statement inserted. / Bibliography: p. 45.

Thesis (Ph. D.)--University of Missouri-Columbia, 2005. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (May 23, 2006) Vita. Includes bibliographical references.

Thesis (M.S.)--University of North Carolina at Wilmington, 2004. / Includes bibliographical references (leaf : 32).

No description available.

Orientador: Ricardo Dahab / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-14T04:06:33Z (GMT). No. of bitstreams: 1 Morais_EduardoMoraesde_M.pdf: 1060073 bytes, checksum: 349e64f9f2a5456f3af935607dfa5e3d (MD5) Previous issue date: 2009 / Resumo: A criptografia baseada em identidades representa uma alternativa ao modelo de certificação digital, exigindo menor esforço para solucionar o problema de autenticidade da chave pública, mas perdendo a custódia da chave privada, que será gerada por uma autoridade de confiança. O modelo de criptografia sem certificados soluciona o problema da custódia da chave privada sem a utilização de certificados digitais. Neste modelo, o usuário tem a posse de uma parte da chave privada e com isso a chave pública passa a ser constituída de uma parte gerada pela autoridade de confiança e uma parte gerada pelo usuário. A cifrassinatura é uma primitiva criptográfica que reúne as vantagens do ciframento e da assinatura em uma única operação, permitindo maior eficiência e segurança. A literatura possui diversas propostas de ciframento sem certificados e assinatura sem certificados, mas não tem uma proposta genérica de cifrassinatura sem certificados. Este trabalho propõe um protocolo de cifrassinatura sem certificados eficiente, que pode ser implementado usando dois emparelhamentos bilineares. Considerando a importância de emparelhamentos bilineares para a construção do protocolo proposto, este trabalho apresenta os conceitos matemáticos necessários para a obtenção de emparelhamentos bilineares eficientes e resistentes a ataques ao problema do logaritmo discreto sobre a curva elíptica e sobre o corpo de extensão resultante do cálculo do emparelhamento bilinear. São apresentados também algoritmos eficientes para aritmética de precisão arbitrária, aritmética de curvas elípticas e cálculo de emparelhamentos. Além disso, são discutidos modelos formais de segurança, como por exemplo o modelo do oráculo aleatório. Finalmente, o modelo de criptografia baseada em identidades e o modelo de criptografia sem certificados são discutidos e com isso é possível apresentar a proposta de cifrassinatura sem certificados e argumentar que esta proposta _e segura e eficiente / Abstract: Identity based cryptography is an alternative to digital certification, which requires less computational effort to solve the problem of public key authenticity. On the other hand, identity based cryptography has the problem of key escrow, because the private key is generated by a trust authority. The certificateless cryptography model solves the key escrow problem without digital certificates. In this model, the user computes a parcial private key that is used to compose the entire private key. In the same way, the public key has two parts: one generated by the user and the other generated by the trust authority. Signcryption is a cryptographic primitive that has the advantages of encryption and signature together in a single operation, allowing the construction of secure and efficient protocols. The literature has many certificateless encryption and certificateless signature protocols, but there is no generic and efficient certificateless signcryption scheme. This work proposes an efficient certificateless signcryption protocol, that can be implemented with just two bilinear pairings. Considering the importance of bilinear pairings for the construction of the proposed protocol, this work presents the mathematical concepts for efficient bilinear pairings, that can resist against discrete logarithm atacks on the elliptic curve and on the extension field. This works also presents efficient algorithms for big number arithmetic, elliptic curve arithmetic and the Miller algorithm for pairings. It also presents formal security models, such as the random oracle model. Finally, identity based cryptography and certificateless cryptography models are defined and the proposed certificateless signcryption scheme is presented and we argue that it is secure and eficient, although no formal proof is given / Mestrado / Criptografia / Mestre em Ciência da Computação

In 1883 Leopold Kronecker published a paper containing “a few explanatory remarks” to an earlier paper of his from 1866. His work loosely connected the theory of integral binary bilinear forms to the theory of integral binary quadratic forms. In this dissertation we discover the statements within Kronecker's paper and offer detailed arithmetic proofs. We begin by developing the theory of binary bilinear forms and their automorphs, providing a classification of integral binary bilinear forms up to equivalence, proper equivalence and complete equivalence. In the second chapter we introduce the class number, proper class number and complete class number as well as two refinements, which facilitate the development of a connection with binary quadratic forms. Our third chapter is devoted to deriving several class number formulas in terms of divisors of the determinant. This chapter also contains lower bounds on the class number for bilinear forms and classifies when these bounds are attained. Lastly, we use the class number formulas to rigorously develop Kronecker's connection between binary bilinear forms and binary quadratic forms. We supply purely arithmetic proofs of five results stated but not proven in the original paper. We conclude by giving an application of this material to the number of representations of an integer as a sum of three squares and show the resulting formula is equivalent to the well-known result due to Gauss.