We investigate the origins of mathematical societies and journals. We argue that the origins of today’s professional societies and journals have their roots in the informal gatherings of mathematicians in 17th century Italy, France, and England. The small gatherings in these nations began as academies and after gaining government recognition and support, they became the ancestors of the professional societies that exist today. We provide a brief background on the influences of the Renaissance and Reformation before discussing the formation of mathematical academies in each country.

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A new class of moduli called the low-weight polynomial form integers (LWPFIs) is introduced. LWPFIs are expressed in a low-weight, monic polynomial form, <em>p</em> = <em>f</em>(<em>t</em>). While the generalized Mersenne numbers (GMNs) proposed by Solinas allow only powers of two for <em>t</em>, LWPFIs allow any positive integers. In our first proposal of LWPFIs, we limit the coefficients of <em>f</em>(<em>t</em>) to be 0 and ±1, but later we extend LWPFIs to allow any integer of less than <em>t</em> for the coefficients of <em>f</em>(<em>t</em>). Modular multiplication using LWPFIs is performed in two phases: 1) polynomial multiplication in Z[<em>t</em>]/<em>f</em>(<em>t</em>) and 2) coefficient reduction. We present an efficient coefficient reduction algorithm based on a division algorithm derived from the Barrett reduction algorithm. We also show a coefficient reduction algorithm based on the Montgomery reduction algorithm. We give analysis and experimental results on modular multiplication using LWPFIs. <br /><br /> New three, four and five-way squaring formulae based on the Toom-Cook multiplication algorithm are presented. All previously known squaring algorithms are symmetric in the sense that the point-wise multiplication step involves only squarings. However, our squaring algorithms are asymmetric and use at least one multiplication in the point-wise multiplication step. Since squaring can be performed faster than multiplication, our asymmetric squaring algorithms are not expected to be faster than other symmetric squaring algorithms for large operand sizes. However, our algorithms have much less overhead and do not require any nontrivial divisions. Hence, for moderately small and medium size operands, our algorithms can potentially be faster than other squaring algorithms. Experimental results confirm that one of our three-way squaring algorithms outperforms the squaring function in GNU multiprecision library (GMP) v4. 2. 1 for certain range of input size. Moreover, for degree-two squaring in Z[<em>x</em>], our algorithms are much faster than any other squaring algorithms for small operands. <br /><br /> We present a side channel attack on XTR cryptosystems. We analyze the statistical behavior of simultaneous XTR double exponentiation algorithm and determine what information to gather to reconstruct the two input exponents. Our analysis and experimental results show that it takes <em>U</em>^{1. 25} tries, where <em>U</em> = max(<em>a</em>,<em>b</em>) on average to find the correct exponent pair (<em>a</em>,<em>b</em>). Using this result, we conclude that an adversary is expected to make <em>U</em>^{0. 625} tries on average until he/she finds the correct secret key used in XTR single exponentiation algorithm, which is based on the simultaneous XTR double exponentiation algorithm.

A new class of moduli called the low-weight polynomial form integers (LWPFIs) is introduced. LWPFIs are expressed in a low-weight, monic polynomial form, <em>p</em> = <em>f</em>(<em>t</em>). While the generalized Mersenne numbers (GMNs) proposed by Solinas allow only powers of two for <em>t</em>, LWPFIs allow any positive integers. In our first proposal of LWPFIs, we limit the coefficients of <em>f</em>(<em>t</em>) to be 0 and ±1, but later we extend LWPFIs to allow any integer of less than <em>t</em> for the coefficients of <em>f</em>(<em>t</em>). Modular multiplication using LWPFIs is performed in two phases: 1) polynomial multiplication in Z[<em>t</em>]/<em>f</em>(<em>t</em>) and 2) coefficient reduction. We present an efficient coefficient reduction algorithm based on a division algorithm derived from the Barrett reduction algorithm. We also show a coefficient reduction algorithm based on the Montgomery reduction algorithm. We give analysis and experimental results on modular multiplication using LWPFIs. <br /><br /> New three, four and five-way squaring formulae based on the Toom-Cook multiplication algorithm are presented. All previously known squaring algorithms are symmetric in the sense that the point-wise multiplication step involves only squarings. However, our squaring algorithms are asymmetric and use at least one multiplication in the point-wise multiplication step. Since squaring can be performed faster than multiplication, our asymmetric squaring algorithms are not expected to be faster than other symmetric squaring algorithms for large operand sizes. However, our algorithms have much less overhead and do not require any nontrivial divisions. Hence, for moderately small and medium size operands, our algorithms can potentially be faster than other squaring algorithms. Experimental results confirm that one of our three-way squaring algorithms outperforms the squaring function in GNU multiprecision library (GMP) v4. 2. 1 for certain range of input size. Moreover, for degree-two squaring in Z[<em>x</em>], our algorithms are much faster than any other squaring algorithms for small operands. <br /><br /> We present a side channel attack on XTR cryptosystems. We analyze the statistical behavior of simultaneous XTR double exponentiation algorithm and determine what information to gather to reconstruct the two input exponents. Our analysis and experimental results show that it takes <em>U</em>^{1. 25} tries, where <em>U</em> = max(<em>a</em>,<em>b</em>) on average to find the correct exponent pair (<em>a</em>,<em>b</em>). Using this result, we conclude that an adversary is expected to make <em>U</em>^{0. 625} tries on average until he/she finds the correct secret key used in XTR single exponentiation algorithm, which is based on the simultaneous XTR double exponentiation algorithm.

It has been said that Pythagoras discovered the perfect musical intervals by chance when he heard sounds of hammers striking an anvil at a nearby smithy. The sounds corresponded to the same intervals Pythagoras had been studying. He experimented with various instruments and apparatus to confirm what he heard. Math, and in particular, numbers are connected to music, he concluded. The discovery of musical intervals and the icon of the musical blacksmith have been familiar tropes in history, referenced in literary, musical, and visual arts. Countless authors since Antiquity have written about the story of the discovery, most often found in theoretical texts about music. However, modern scholarship has judged the narrative as a myth and a fabrication. Its refutation of the story is peculiar because modern scholarship has failed to disprove the nature of Pythagoras’s discovery with valid physical explanations. This report examines the structural elements of the story and traces its evolution since Antiquity to the early modern period to explain how an author interprets the narrative and why modern scholarship has deemed it a legend. The case studies of Nicomachus of Gerasa, Claudius Ptolemy, Boethius, and Marin Mersenne reveal not only how the story about Pythagoras’s discovery functions for each author, but also how the alterations in each version uncover an author’s views on music. / text

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2014-10-23T12:34:52Z No. of bitstreams: 2 Dissertação - Glauber Cristo Alves de Carvalho - 2013.pdf: 2320575 bytes, checksum: 5671a75a3a3b2b110d7431a79726479c (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2014-10-23T12:38:41Z (GMT) No. of bitstreams: 2 Dissertação - Glauber Cristo Alves de Carvalho - 2013.pdf: 2320575 bytes, checksum: 5671a75a3a3b2b110d7431a79726479c (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-10-23T12:38:41Z (GMT). No. of bitstreams: 2 Dissertação - Glauber Cristo Alves de Carvalho - 2013.pdf: 2320575 bytes, checksum: 5671a75a3a3b2b110d7431a79726479c (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-03-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This paper presents a brief history about the numbers. After some important definitions to understand the texts. Following, we encounter the world of prime numbers. This part is presented some important properties, findings and open problems. The study of these figures have managed to find some formulas to generate them, which are presented throughout the text. It presents some numbers especias such as Fermat primes, Mersene, Shopie German and others. Finally, we have an application that uses many properties presented. / Neste trabalho é apresentado um breve histórico sobre os números. Após, algumas definições importantes para compreensão dos textos. Seguindo, nos deparamos com o universo dos números primos. Nesta parte é apresentado algumas propriedades importantes, descobertas e problemas em aberto. O estudo sobre estes números já conseguiu encontrar algumas fórmulas para gerá-los, que são apresentadas no decorrer do texto. Apresenta-se alguns números especias, como os primos de Fermat, Mersene, Shopie German e outros. Por fim, temos uma aplicação que utiliza muitas propriedades apresentadas.

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