There has often been a connection between music and mathematics. The world of musical composition is full of combinations of orderings of different musical notes, each of which has different sound quality, length, and em phasis. One of the more intricate composition styles is twelve-tone music, where twelve unique notes (up to octave isomorphism) must be used before they can be repeated. In this thesis, we aim to show multiple ways in which mathematics can be used directly to compose twelve-tone musical scores.

Hash-based signature schemes are a class of post-quantum algorithms that usually consist of hash-trees built upon OTS solutions. These schemes have small key sizes, eficient processing and are simple to implement, while their security properties rely basically on the pre-image or collision resistance of the their underlying hash function. Despite such advantages, however, they have relatively large signature sizes compared to traditional signature algorithms. One way of tackling this issue is to reduce the sizes of their underlying OTS algorithms. Besides that, in applications where signature verifications are done much more frequently than signature generation (e.g. trusted software distribution), it is desirable that signature verification has less overhead than signature generation. In this work, two contributions are presented to tackle these issues. The first one is a probabilistic technique that, with negligible processing overhead, allows reductions in the underlying OTS signature sizes; namely, up to 12.5% average size reduction can be achieved depending on the w parameter chosen for the signature (however, for w = 4 the reduction is only 0.2%). The second contribution is a novel OTS scheme which has all advantages of W-OTS and W-OTS+ and yet has much faster signature verification times at the cost of slightly slower signature generation times. / Esquemas de assinaturas baseadas em hash são uma classe de algoritmos pós-quânticos que basicamente consistem em árvores de hash construídas em cima de soluções de assinaturas unitárias (OTS). Tais esquemas possuem tamanhos pequenos de chaves, processamento eficiente e são simples de se implementar, enquanto que a segurança desses esquemas baseia-se na resistência à pré-imagem ou à colisão das funções de hash utilizadas. Apesar dessas vantagens, eles possuem tamanhos de assinaturas relativamente grandes comparados aos algoritmos tradicionais de assinatura. Nesse caso, uma forma de lidar com essa questão é reduzir os tamanhos de assinatura das OTS utilizadas. Além disso, em aplicações em que se faz muito mais verificações de assinatura do que gerações de assinatura, é desejável que a vericação seja significativamente mais rápida do que a geração. Nesse trabalho, duas contribuições são apresentadas para mitigar os problemas acima mencionados. A primeira é uma técnica probabilística que permite a redução do tamanho das assinaturas nas OTS utilizadas com custo adicional de processamento desprezível, isto é, pode-se alcançar uma redução média de até 12.5% dependendo do valor de w escolhido para a assinatura (no entanto, para w = 4 a redução é de apenas 0.2%). A segunda contribuição é um esquema inovador de assinatura digital que possui todas as vantagens do W-OTS e do W-OTS+, além de possuir verificação bem mais rápida do que estes em troca de uma geração de assinatura um pouco mais lenta.

Cette version du mémoire a été tronquée d'un élément protégé par le droit d'auteur, l'''Annexe 5a audio, K'viria de Riho''. Une version plus complète est disponible en ligne pour les membres de la communauté de l'Université de Montréal et peut aussi être consultée à la Bibliothèque de la Faculté de Musique de l'UdeM. / Ce mémoire porte sur la notion de fusion musicale à la multiplicité d'influences dans ma démarche en composition musicale. Je commencerai par définir le terme de fusion, les différents types de fusion musicale ainsi que leur relation à la tradition musicale d'un côté, et leur relation à une démarche contemporaine en composition musicale. Puis, j'analyserai quantité des styles et traditions qui forment la base de la littérature musicale dont je me sers dans la construction de mon langage personnel, autant du côté des démarches déjà existantes en fusion que du côtés des musiques relevant de traditions non-fusion. Celles qui m'ont le plus influencées sont la musique classique européenne et la musique classique du Moyen-Orient, puis, d'autres influences phares se trouvent dans les musiques classiques d'Inde, la musique vocale de Géorgie, la musique de gamelan Balinais, et dans le Hard Rock ainsi que le Rock Progressif. J'explorerai ensuite de manière plus concrète les modes musicaux, la mélodie et son enrichissement grâce à l'ornementation mélodique, et le rythme, dans lequel la métrique apparaît comme essentielle, et la polyrythmie comme toute aussi enrichissante. Ces trois paramètres – mode, mélodie et rythme - forment les trois grands piliers de mon langage musical. Je ferai des aller-retours entre l'apport des traditions, leur réinterprétation créative et l'expansion de leurs paramètres jusqu'à en dépasser les frontières, ce qui débouchera sur la possibilité et la volonté de la création d'un nouveau style musical en soi, qui dépasserait, peut-être, l'esthétique de la fusion. / This memoir discusses the concept of musical fusion with multiple influences in my compositional works. I will start by defining the term fusion, the different types of fusion as well as their relationship to musical tradition, on the one hand, and their relationship to a contemporary musical compositional endeavor. I will then analyse diverse musical styles and traditions that form the basis of the musical literature that I use in constructing my personal musical language, both in the branch of already existing fusion musics, and in the branch of non-fusion musics. The musics that most influenced me are european classical music and middle-eastern classical music, and then, other important ones include classical indian music, vocal georgian music, balinese gamelan music, Hard Rock and Progressive Rock. Afterwards I will explore in more concrete detail the musical modes, the melody and its complexification with the mechanisms of ornamentation, rhythm, within which time signatures appear to be an essential aspect, and polyrhythm to be it's important mechanism of complexification. These three parameters – mode, melody and rhythm – form the three great pillars of my musical language. I will go back-and-forth between the intakes from tradition, their creative reinterpretation and the expansion of their parameters, until we reach a point where we go beyond their frontier, at which point we will envision the possibility and the will to create a new musical style in itself, which, perhaps, goes beyond the esthetic of fusion.