Acknowledgements and Bibliography
Thank Yous and references.
Acknowledgements
First and foremost let me say thank you to Dr. Jelena Schmaltz for all of her support, guidance, energy, focus, ideas, and her continuing passion for her subject and her students. I could not have done this without your help and guidance. Our immediate connection on the subject matter was only the beginning of our journey. Your openness to ideas, your gentle navigation of advice, your sense of both the importance and the beauty of what we study, and you practical and straightforward approach to the subject matter were a joy to work with.
This work completed as part of a course (SCI395) with University of New England - under the supervision of Dr. Jelena Schmaltz.
I would like to acknowledge Wix for helping with their platform, to build this site.
And a special thanks to Desmos (especially their 2D and 3D graphing calculator) for helping me to build the visualisations and animations for this work.
Desmos
To see the Desmos graph used to create the tangent lines in a finite field animations: https://www.desmos.com/calculator/ak7ycoyblp
This graph, while a working example, requires work to make it more readable, and more easily reusable by others. It is the hope of the author to find time to link to more of the graphs used to build this site (for others' use), and to further extend this work to a more shareable state. Please contact: Nicholas Gledhill - if the extension to the Desmos work would be of practical help. See contact details, below.
Bibliography
Austin, M. B., David, & Schlicker, S. (2022, July 13). Limits, Continuity, and Differentiability. Grand Valley State University. https://math.libretexts.org/@go/page/5286
Buterin, V. (2017, January 16). Exploring Elliptic Curve Pairings. Medium. https://medium.com/@VitalikButerin/exploring-elliptic-curve-pairings-c73c1864e627
Chasteen-Boyd, D. (2015). The Unexpected Connection Between Internet Security and the Riemann Hypothesis. Inquiro (Volume 19). UAB'S Undergraduate Research Journal. https://www.uab.edu/inquiro/issues/past-issues/volume-9/the-unexpected-connection-between-internet-security-and-the-riemann-hypothesis
Duis, N. (2011, May 2). Transforming a general cubic elliptic curve equation to Weierstrass form, a Sage implementation. Technische Universiteit Eindhoven. https://people.tamu.edu/~rojas//cubic2weierstrass.pdf
Friedl, S. (2017). An elementary proof of the group law for elliptic curves. Groups Complexity Cryptology, 9, 117 - 123.
Hayden, H, H., IV. (2021). Elliptic curves and their Practical Applications. [MSU Graduate Theses]. 3656. https://bearworks.missouristate.edu/theses/3656
Johansen, F. (2021). Computing with real numbers. Fredrik Johansson's website. 3656. https://bearworks.missouristate.edu/theses/3656
Koblitz, A.H., Koblitz, N., & Menezes, A. (2011). Elliptic Curve Cryptography: The Serpentine Course of a Paradigm Shift. IACR Cryptol. ePrint Arch., 2008 , 390.
Leandro J. Created Playlists [YouTube Channel]. YouTube. Retrieved April 2, 2024. from https://www.youtube.com/watch?v=n41Z0c9Jm4Y&list=PL1xkDS1G9As7E_fPaLaFchq1a27I9a5tO
Lemmermeyer, F. (2004, February 25). Lecture 7. Bilkent University, Faculty of Science. http://www.fen.bilkent.edu.tr/~franz/ta/ta07.pdf
Onete, Cristina. (2008). Visualisation of Modern Key Exchange Schemes for more than two Parties in CrypTool and their Security Analysis.
Shevchuk, O. (2020). Introduction to Elliptic Curve Cryptography. Institute for Applied Information Processing and Communication, The University of Chicago Mathematics. math.uchicago.edu/~may/REU2020/REUPapers/Shevchuk.pdf
Southerland, A. (2019). Finite fields and integer arithmetic. 18.783 - Elliptic Curves. https://math.mit.edu/classes/18.783/2019/LectureNotes3.pdf
Sudan, M. (2005, September 28). Lecture 6, Algebra and Computation. Harvard John A. Paulson, School of Engineering and Applied Science, Harvard University.
Vasundhara, S. (2017). The Advantages of Elliptic Curve Cryptography for Security. Global Journal of Pure and Applied Mathematics. 13 (9), 4995-5011
Washington, L. C. (2008). Elliptic Curves Number Theory and Cryptography (Second Edition).
Wikipedia contributors. (2024, May 15). Modular arithmetic. Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/w/index.php?title=Modular_arithmetic&oldid=1224040388
Wikipedia contributors. (2024, January 26). Modular multiplicative inverse. Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/w/index.php?title=Modular_multiplicative_inverse&oldid=1199167146
Wenberg, S. L. (2013). Elliptic curves and their cryptographic applications. [EWU Masters Thesis Collection]. 160. https://dc.ewu.edu/theses/160
Zwick, D. (2013). Math 2280 - Lecture 6: Substitution Methods for First-Order ODEs and Exact Equations. University of Utah, Department of Mathematics, College of Science.
Other Web Resources
A list of other web resources that were read and used to support the information on this site - resource which proved somewhat difficult to site properly, or attribute to individual authors.
Hobson, Nick; Nichol, Reid; and Weisstein, Eric W. (date unknown). Modular Inverse. MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ModularInverse.html
Landesman, A. (date unknown). Notes on Finite Fields. Harvard Mathematics Department Home page. https://people.math.harvard.edu/~landesman/assets/finite-fields.pdf
Lynn, B. (date unknown). Elliptic Curves, Notes. PBC Library (The Pairing-Based Cryptography Library), Applied Cryptography Group, Stanford University. https://crypto.stanford.edu/pbc/notes/elliptic/weier.html
Norton, A. (date unknown). (IR)REVERSABILITY [sic] IN MATHEMATICS. Institute of Education Services. https://files.eric.ed.gov/fulltext/ED583787.pdf
Reid, I. (date unknown). Lecture 2, Vanishing points and horizons. Applications of projective transformations. University of Adelaide, School of Computer Science, and Australian Institute for Machine Learning. https://cs.adelaide.edu.au/~ianr/Teaching/CompGeom/lec2.pdf
Sullivan, N. (2013, October 24). A (Relatively Easy To Understand) Primer on Elliptic Curve Cryptography. Cloudflare. https://blog.cloudflare.com/a-relatively-easy-to-understand-primer-on-elliptic-curve-cryptography
Various. (date unknown). Homogeneous First Order Differential Equations. Calculus, Core Mathematics, Maths Resources, Online Resources, Academic Skills Kit, Newcastle University. https://www.ncl.ac.uk/webtemplate/ask-assets/external/maths-resources/core-mathematics/calculus/homogeneous-first-order-differential-equations.html
Weisstein, Eric W. (date unknown). Elliptic Curve. MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipticCurve.html
Weisstein, Eric W. (date unknown). Elliptic Curve Group Law. MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipticCurveGroupLaw.html
Weisstein, Eric W. (date unknown). Weierstrass Elliptic Function. From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeierstrassEllipticFunction.html
Contact
For further details regarding this this site, the information contained within, the material covered for the course (SCI395), or anything else related, please contact:
Nicholas Gledhill
Text: 0419 609 025
(PLEASE text first, if introducing yourself... I get far too many spam calls, and text first to say "hi" - with a reason - is a great way to filter the hot from the cold - thank you!)
