With the advent of quantum computers, the security of number-theoretic cryptography has been compromised. Consequently, new cryptosystems have been suggested in the field of non-commutative group theory. In this thesis, we provide all the necessary background to understand and work with the Artin groups. We then show that Artin groups of finite type and Artin groups of large type possess an easily-computable normal form by explicitly writing the algorithms. This solution to the word problem makes these groups candidates to be cryptographic platforms. Finally, we present some combinatorial problems that can be used in group-based cryptography and we conjecture, through empirical evidence, that the conjugacy problem in Artin groups of large type is not a hard problem.

With the advent of quantum computers, the security of number-theoretic cryptography has been compromised. Consequently, new cryptosystems have been suggested in the field of non-commutative group theory. In this thesis, we provide all the necessary background to understand and work with the Artin groups. We then show that Artin groups of finite type and Artin groups of large type possess an easily-computable normal form by explicitly writing the algorithms. This solution to the word problem makes these groups candidates to be cryptographic platforms. Finally, we present some combinatorial problems that can be used in group-based cryptography and we conjecture, through empirical evidence, that the conjugacy problem in Artin groups of large type is not a hard problem.

With the advent of quantum computers, the security of number-theoretic cryptography has been compromised. Consequently, new cryptosystems have been suggested in the field of non-commutative group theory. In this thesis, we provide all the necessary background to understand and work with the Artin groups. We then show that Artin groups of finite type and Artin groups of large type possess an easily-computable normal form by explicitly writing the algorithms. This solution to the word problem makes these groups candidates to be cryptographic platforms. Finally, we present some combinatorial problems that can be used in group-based cryptography and we conjecture, through empirical evidence, that the conjugacy problem in Artin groups of large type is not a hard problem.

Originally presented as the author's Thesis (Ph. D.)--Scuola normale superiore Pisa. / Includes bibliographical references (p. [125]-131) and index.

Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2018. / Submitted by Raquel Viana (raquelviana@bce.unb.br) on 2018-07-04T19:56:19Z No. of bitstreams: 1 2018_BrunodePaulaMiranda.pdf: 934554 bytes, checksum: eee7a917cdecb7aa3b6c58ad0476d279 (MD5) / Approved for entry into archive by Raquel Viana (raquelviana@bce.unb.br) on 2018-07-09T17:43:26Z (GMT) No. of bitstreams: 1 2018_BrunodePaulaMiranda.pdf: 934554 bytes, checksum: eee7a917cdecb7aa3b6c58ad0476d279 (MD5) / Made available in DSpace on 2018-07-09T17:43:26Z (GMT). No. of bitstreams: 1 2018_BrunodePaulaMiranda.pdf: 934554 bytes, checksum: eee7a917cdecb7aa3b6c58ad0476d279 (MD5) Previous issue date: 2018-07-04 / Em 1963, e Lewis provaram que se a forma diagonal F(x) = a1xd1 +...+ aNxdN com coeficientes em Qp, o corpo dos números p-ádicos, satisfazer N > d2, então existe solução não trivial para F(x) = 0. Muito estudo tem sido realizado afim de generalizar esse resultado para extensões finitas de Qp. Aqui, estudamos o caso F(x) 2 K[x] com K sendo a extensão quadrática não ramificada de Q2 e provamos dois resultados: Se d não _e potência de 2, então N > d2 garante a existência de solucão não trivial para F(x) = 0. Além disso, se d = 6, N = 29 garante existência de solucão não trivial para F(x) = 0. / In 1963, Davenport and Lewis proved that if the diagonal form F(x) = a1xd1 +...+ aNxdN with coeficients in Qp, the field of p-adic numbers, satisfies N > d2, then there exists non-trivial solution for F(x) = 0. Since then, there has been a lot of study in order to generalize this result to finite extensions of Qp. Here, we study the case F(x) 2 K[x] where K is the quadratic unramified extension of Q2 and we prove two results: if d is not a power of 2 , then N > d2 guarantees non-trivial solution for F(x) = 0. Furthermore, if d = 6, N = 29 guarantees non-trivial solution for F(x) = 0.

Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2013. / Submitted by Albânia Cézar de Melo (albania@bce.unb.br) on 2013-09-09T14:24:13Z No. of bitstreams: 1 2013_LucianaLimaVentura.pdf: 572550 bytes, checksum: 0ce7cf628a3d83b89a7518122378820d (MD5) / Approved for entry into archive by Guimaraes Jacqueline(jacqueline.guimaraes@bce.unb.br) on 2013-09-09T15:46:43Z (GMT) No. of bitstreams: 1 2013_LucianaLimaVentura.pdf: 572550 bytes, checksum: 0ce7cf628a3d83b89a7518122378820d (MD5) / Made available in DSpace on 2013-09-09T15:46:43Z (GMT). No. of bitstreams: 1 2013_LucianaLimaVentura.pdf: 572550 bytes, checksum: 0ce7cf628a3d83b89a7518122378820d (MD5) / Uma versão da Conjectura de Artin afirma que para um sistema homogêneo com duas equações diagonais de grau k, cujos coeficientes são inteiros, ter solução p-ádica não trivial é suficiente que o número de variáveis seja maior que 2 k2. Nesse trabalho, vamos mostrar que a conjectura é verdadeira quando o grau é 2T . 3 ou 3T . 2, para T≥ 2. ______________________________________________________________________________ ABSTRACT / One version of Artin's Conjecture states that for a homogeneous system with two diagonal equations of degree k, whose coe cients are integers, exists a nontrivial p-adic solution provided the number of variables is greater than 2 k2. In this paper, we show that the conjecture is true when the degree is 2T . 3 or 3T . 2, for T≥ 2.

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Fix R = k[[x]]. Let Q_n be the category whose objects are ((M₁,...,M_n),(f₁,...,f_n-1)) where each M_i is a free R-module and f_i:M_i⟶M_i+1 for each i=1,...,n-1, and in which the morphisms are the obvious ones. Let β_n be the full subcategory of Ω_n in which each map f_i is a monomorphism whose cokernel is a torsion module. It is shown that there is a full dense functor Ω_n⟶β_n. If X is an object of β_n, we say that X <u>diagonalizes</u> if X is isomorphic to a direct sum of objects ((M₁,...,M_n),(f₁,...,f_n-1)) in which each M_i is of rank one. We establish an algorithm which diagonalizes any diagonalizable object X of β_n, and which fails only in case X is not diagonalizable. Let Λ be an artin algebra of finite type. We prove that for a fixed C in mod(Λ) there are only finitely many modules A in mod(Λ) (up to isomorphism) for which a short exact sequence of the form 0⟶A⟶B⟶C⟶0 is indecomposable. / Ph. D. / incomplete_metadata