# Chloe Martindale

I am a number theorist currently working as a Postdoc at Technical University of Eindhoven in the group of Prof. dr. Tanja Lange. I am currently working on pairing-based cryptography and the discrete logarithm problem for elliptic curves as part of the USEIT project. My other research interests include Hilbert modular forms, genus 2 and 3 curves with complex multiplication, and isogeny graphs.

## Research

• PhD thesis: Isogeny Graphs, Modular Polynomials, and Applications (currently under revision), supervised by Marco Streng
• Isogenies for point counting on genus two hyperelliptic curves with maximal real multiplication, Sean Ballentine, Aurore Guillevic, Elisa Lorenzo Garcia, Chloe Martindale, Maike Massierer, Benjamin Smith, and Jaap Top
• ### Contact information:

 Office: 6.097a MetaForum, Technische Universiteit Eindhoven Telephone: +31-40247 2541 E-mail: chloemartindale (at) gmail (dot) com Address: MetaForum building, 5612 AZ, Eindhoven, The Netherlands

I am giving some lectures in 2MMC10 Cryptology - Fall 2017 at the Technische Universiteit Eindhoven. The main lecturer for this course is Tanja Lange. Videos, blackboard pictures, and lecture notes for the lectures from 19/09/2017-12/10/2017 will be posted here:

• Exercise sheet 3 (Due 28 Sep 2017 for 2MMC10 and 26 Oct 2017 for Mastermath). Make sure to check the lecture notes for help with the exercise sheet! I'm afraid we didn't have time to cover every you need in the lecture itself.
• 28 Sep 2017: Crypto protocols relying on hardness of DLP (Diffie-Hellman, ElGamal encryption, ElGamal signatures), and attacks on DLP (Pohlig-Hellman attack, Baby-step giant-step, Pollard's rho method). Lecture notes, blackboard pictures, videos.
• 03 Oct 2017: Constructed F32, Rabin's irreducibility test, CDH vs DDH, Baby-step-giant-step example, Pollard rho example, statement index calculus. Lecture notes, blackboard pictures, videos.
• 10 Oct 2017: Quick overview of arithmetic in finite fields following questions from students. Defined Edwards curves, proved that the $$\mathbb{F}_{p}$$ points of an Edwards curve form a group, looked at complexity of arithmetic on Edwards curves. Lecture notes, blackboard pictures, videos.
• 12 Oct 2017: Elliptic curves in Weierstrass form: group law in pictures and in formulae. Montgomery curves: definition and definition of group law. Transformation from Edwards form to Montgomery form. Lecture notes, blackboard pictures, videos.
• Co-supervised Bachelor Thesis of Ivo Kok, 'Rings in which every ideal has 2 generators', Spring 2014, with Marco Streng.
• Here is a growing list of Hilbert modular polynomials. The theory behind this and the algorithm to compute them is in Chapter 2 of my PhD thesis. The code to compute these polynomials is here.
Examples for totally real field $$\mathbb{Q}(\sqrt{5})$$:
Examples for totally real field $$\mathbb{Q}(\sqrt{2})$$:
• Isogeny graphs of abelian varieties and applications to the discrete logarithm problem, (pdf), talk in the Heilbronn Seminar in Bristol, UK (December 2017).
• User empowerment for security and privacy in IoT, (pdf), talk in the dcypher Symposium in Utrecht, the Netherlands (October 2017).
• Isogeny graphs, modular polynomials, and point counting for higher genus curves, (pdf), talk in the Number theory seminar at INRIA in Nancy, France (July 2017).
• Constructing genus 2 curves over finite fields with a prescribed number of points, (pdf), talk in the DIAMANT Symposium in Breukelen, the Netherlands (June 2017).
• Counting points on genus 2 curves over finite fields, (pdf), talk in the Number Theory Seminar at l'Insitut Fourier, Grenoble (May 2017).
• Counting points on genus 2 curves over finite fields, (pdf), talk in the Number Theory Seminar at EPFL, Lausanne (November 2016).
• Studying genus 2 and 3 curves using isogeny graphs and modular polynomials, (pdf), talk in the Algebra Seminar at Universiteit Leiden, Leiden (November 2016).
• From conic sections to isogeny graphs, (slides), Colloquium talk at University College Dublin (June 2016).
• Modularity of Elliptic Curves over $$\mathbb{Q}$$, (pdf), talk in the Elliptic Curves Seminar at Universiteit Leiden, Leiden (April 2016).
• Counting points of Jacobians of Genus 2 curves over large finite fields, progress report on a joint research project at AGC2016, led by Ben Smith and Jaap Top, AGC2016, UCLA (February 2016).
• Isogeny Graphs, (pdf), talk in the PhD Colloquium at Universiteit Leiden, Leiden (October 2015).
• The theory of canonical lifts, (pdf), Universiteit Leiden (July 2015).
• The Galois representation associated to modular forms (Part I), (pdf), Universiteit Leiden (May 2015).
• An algorithm for computing Hilbert modular varieties (pdf), LFANT Seminar, IMB, Université de Bordeaux (September 2014).
• Elliptic curves and jacobians of curves of genus 2, talk on master thesis supervised by Prof. E. Victor Flynn, University of Oxford (April 2013).