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Pierre de Fermat, an amateur mathematician, set upon the mathematical world a challenge so difficult it took 357 years to prove. This challenge, known as Fermat's Last Theorem, has many different ways of being expressed, but it simply states that for $n > 2$, the equation $x^n + y^n = z^n$ has no nontrivial solution. The first set of attempts of proofs came from mathematicians using the essentially elementary tools provided by number theory: the notable mathematicians were Leonhard Euler, Sophie Germain and Ernst Kummer. Kummer was the final mathematician to try to use essentially elementary number theory as the basis for his proof and even exclaimed that Fermat's Last Theorem could not be solved using number theory alone; Kummer claimed that greater mathematics would have to be developed in order to prove this ever-growing mystery. The 20th century arrives and two Japanese mathematicians, Goro Shimura and Yutaka Taniyama, shock the world by claiming two highly distinct branches of mathematics, elliptic curves and modular forms, were in fact one and the same. Gerhard Frey then took this claim to the extreme by stating that this claim, the Taniyama-Shimura conjecture, was the necessary link to finally prove Fermat's Last Theorem was true. Frey's statement was then validated by Kenneth Ribet by proving that the Frey Curve could not indeed be a modular form. The final piece of the puzzle placed, the English mathematician Andrew Wiles embarked on a 7 year journey to prove Fermat's Last Theorem as now the the proof of the theorem rested in his area of expertise, that being elliptic curves. In 1994, Wiles published his complete proof of Fermat's Last Theorem, putting an end to one of mathematics' greatest mysteries.
ContributorsBoyadjian, Hoveeg Krikor (Author) / Bremner, Andrew (Thesis director) / Jones, John (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2016-12
Description
In the modern world with the ever growing importance of technology, the challenge of information security is of increasing importance. Cryptographic algorithms used to encode information stored and transmitted over the internet must be constantly improving as methodology and technology for cyber attacks improve. RSA and Elliptic Curve cryptosystems such as El Gamal or Diffie-Hellman key exchange are often used as secure asymmetric cryptographic algorithms. However, quantum computing threatens the security of these algorithms. A relatively new algorithm that is based on isogenies between elliptic curves has been proposed in response to this threat. The new algorithm is thought to be quantum resistant as it uses isogeny walks instead of point addition to generate a shared secret key. In this paper we will analyze this algorithm in an attempt to understand the theory behind it. A main goal is to create isogeny graphs to visualize degree 2 and 3 isogeny walks that can be taken between supersingular elliptic curves over small fields to get a better understanding of the workings and security of the algorithm.
ContributorsLoucks, Sara J (Author) / Jones, John (Thesis director) / Bremner, Andrew (Committee member) / Computer Science and Engineering Program (Contributor) / School of Film, Dance and Theatre (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
In this paper we outline a method of producing reduced elliptic curves with many integral points and provide the results of the outlined computations, including several curves with hundreds of integral points. The first three sections give back-ground and describe our work with integral points on elliptic curves. The last section is unrelated to elliptic curves and provides a complete classification of self-descriptive numbers.
ContributorsJones, Benjamin Redington (Author) / Bremner, Andrew (Thesis director) / Sprung, Florian (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2020-12