[ David Gruenewald ]
[ Research | Contact | Teaching ]

Research

Interests

In 2012 I completed a postdoc working with John Boxall in the Laboratoire de Mathématiques Nicolas Oresme at the Université de Caen, supported by the ANR's PACE project (Pairings and Advances in Cryptology for E-cash). Before that, I spent 3 months in Department of Mathematics at Radboud Universiteit Nijmegen, supported by the DIAMANT cluster working with Wieb Bosma on complex continued fractions. Before that I was a postdoc at eRISCS at the Université d'Aix-Marseille (in Marseille). Before that, I was a doctoral student in the Number Theory group of the School of Mathematics and Statistics at the University of Sydney, where my supervisor was David R. Kohel.

I was interested in computing with modular forms and their associated moduli spaces. My attention was focused on genus 2 with practical applications to hyperelliptic curve cryptography.

I was awarded my PhD in December 2009. In my thesis entitled "Explicit Algorithms for Humbert Surfaces", I find explicit practical models for moduli spaces of Abelian surfaces, in particular Humbert surfaces and Shimura curves. The equations can be found here.

In July 2008 I went to Microsoft Research for the northern summer, working as a research intern under the guidance of Kristin Lauter and Reinier Bröker. We implemented an improved version of the CRT algorithm in Magma which makes use of (3,3)-isogeny relations I had previously computed.

I enjoy all computational aspects of number theory. At the beginning of 2006 I did some computations for Alf van der Poorten on width 6 Somos sequences arising from continued fraction expansions of genus 2 curves. I primarily use Magma for my computations, but have also worked with Sage and Mathematica.

Publications

• Heuristics on pairing-friendly abelian varieties (with John Boxall), LMS Journal of Computation and Mathematics, Volume 18, Issue 01, June 2015, pp 419-443. DOI: 10.1112/S1461157015000091 (preprint version)

Abstract: We discuss heuristic asymptotic formulae for the number of isogeny classes of pairing-friendly abelian varieties of fixed dimension g ≥ 2 over prime finite fields. In each formula, the embedding degree k ≥ 2 is fixed and the rho-value is bounded above by a fixed real ρ₀ > 1. The first formula involves families of ordinary abelian varieties whose endomorphism ring contains an order in a fixed CM-field K of degree g and generalizes previous work of the first author when g=1. It suggests that, when ρ₀ < g, there are only finitely many such isogeny classes. On the other hand, there should be infinitely many such isogeny classes when ρ₀ > g. The second formula involves families whose endomorphism ring contains an order in a fixed totally real field K₀⁺of degree g. It suggests that, when ρ₀ > 2g/(g+2) (and in particular when ρ₀ > 1 if g = 2), there are infinitely many isogeny classes of g-dimensional abelian varieties over prime fields whose endomorphism ring contains an order of K₀⁺. We also discuss the impact that polynomial families of pairing-friendly abelian varieties has on our heuristics, and review the known cases where they are expected to provide more isogeny classes than predicted by our heuristic formulae.

• Complex numbers with bounded partial quotients (with Wieb Bosma), Journal of the Australian Mathematical Society, Volume 93, Issue 1-2 (dedicated to Alf van der Poorten), October 2012, pp. 9-20. DOI: 10.1017/S1446788712000638 (preprint version)

Abstract: Conjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction expansion are rationals and quadratic irrationals. We show that the corresponding statement is not true for complex algebraic numbers in a very strong sense, by constructing for every even degree d algebraic numbers of degree d that have bounded complex partial quotients in their Hurwitz continued fraction expansion. The Hurwitz expansion is the generalization of the nearest integer continued fraction expansion for complex numbers. In the case of real numbers the boundedness of regular and nearest integer partial quotients is equivalent.

• Explicit CM theory for level 2-structures on abelian surfaces (with Reinier Bröker and Kristin Lauter), Algebra & Number Theory 5-4 (2011), 495--528. DOI: 10.2140/ant.2011.5.495 (preprint version).

Abstract: For a complex abelian surface A with endomorphism ring isomorphic to the maximal order in a quartic CM field K, the Igusa invariants j₁(A), j₂(A), j₃(A) generate an unramified abelian extension of the reflex field of K. In this paper we give an explicit geometric description of the Galois action of the class group of this reflex field on j₁(A), j₂(A), j₃(A). Our description can be expressed by maps between various Siegel modular varieties, and we can explicitly compute the action for ideals of small norm. We use the Galois action to modify the CRT method for computing Igusa class polynomials, and our run time analysis shows that this yields a significant improvement. Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby implying that the "isogeny volcano" algorithm to compute endomorphism rings of ordinary elliptic curves over finite fields does not have a straightforward generalization to computing endomorphism rings of abelian surfaces over finite fields.

• Computing Humbert surfaces and applications, in Arithmetic, Geometry, Cryptography and Coding Theory 2009, Contemporary Mathematics, vol. 521, Amer. Math. Soc., Providence, RI, 2010, pp. 59-69 (preprint version).

• Complex numbers with bounded partial quotients: AustMS Conference, The University of Sydney, 1^st October 2013.
• Heuristics on pairing-friendly abelian varieties: Computational Algebra Seminar, The University of Sydney, 18^th April 2013.
• Computing isogeny graphs in genus 2 using CM lattices, ECC 2012, Querétaro, Mexico, 31^st October 2012 (pdf)
• Computing "isogeny graphs" using CM lattices: LACAL seminar, EPFL, 9^th March 2012
• Computing "isogeny graphs" using CM lattices: Workshop on Algorithms for Curves, Moduli, and Isogenies, Laboratoire d'informatique (LIX), École Polytechnique, Palaiseau, 7^th July 2011
• Computing "isogeny graphs" using CM lattices: Geocrypt 2011, Corsica, 22^nd June 2011 (pdf)
• Hyperelliptic Curves, Cryptography and Factorization Algorithms: Séminarie de Cryptographie, Université de Caen, 27^th January 2011
• Hyperelliptic Curves, Cryptography and Factorization Algorithms: Algemeen Wiskundecolloquium, Radboud Universiteit Nijmegen, 8^th December 2010 (pdf)
• Humbert Surfaces and Applications: DIAMANT Symposium, Lunteren, 27^th November 2010
• Explicit CM in Genus 2: Intercity Number Theory Seminar, Radboud Universiteit Nijmegen, 1^st October 2010
• Humbert Surfaces and Isogeny Relations: Séminaire de Cryptographie, Institut de Recherche en Mathématiques de Rennes, 15^th January 2010
• Humbert Surfaces and Applications: Réunion CHIC, Institut Henri Poincaré, Paris, 6^th October 2009 (pdf)
• An Introduction to Hyperelliptic Curves: Crypto'Puces, île de Porquerolles, 4^th June 2009 (pdf)
• Humbert Surfaces and Isogeny Relations: AGCT-12, CIRM at Luminy, Marseille, 3^rd April 2009 (pdf)
• Explicit CM in Genus 2: Computational Algebra Seminar, The University of Sydney, 19^th March 2009 (pdf)
• Explicit CM in Genus 2: Number Theory Seminar, Institut de Mathématiques de Luminy, 9^th October 2008
• Computing Humbert Surfaces: AustMS Conference, La Trobe University, 25^th September 2007 (pdf)
• Computing Humbert Surfaces: Journées Arithmétiques, University of Edinburgh, Scotland, July 2007
• Introduction to the Weil Conjectures: Algebraic Geometry seminar, UNSW, 13^th June 2007
• Néron Models: Complex Multiplication lecture series, Number Theory Seminar, Sydney, semester 1, 2006
• Polarized Abelian Varieties: lecture series following Shimura's Complex Multiplication of Abelian Varieties, Number Theory Seminar, Sydney, 15^th September 2005
• Humbert Surfaces: Number Theory Seminar, Sydney, 25^th May 2005
• The Moduli of Abelian Varieties: Number Theory Seminar, Sydney, 16^th March 2005
• Endomorphisms of Complex Abelian Varieties: Abelian Varieties lecture series, Number Theory Seminar, Sydney, 29^th October 2004
• Principally Polarized Complex Abelian Varieties and their Moduli Space: Abelian Varieties lecture series, Number Theory Seminar, Sydney, 15^th October 2004
• Line Bundles on Complex Tori: Abelian Varieties lecture series, Number Theory Seminar, Sydney, August - September 2004
• N-dimensional Spheres, Cubes and the Tower of Hanoi: talk given at MANSW's (Mathematical Association of NSW) Mathematical Enrichment Day (formerly known as their Talented Students' Day),
• 23^rd July, 2004 at University of Technology Sydney
• 22^nd July, 2005 at Macquarie University
• 21^st July, 2006 at University of New South Wales
• 20^th July, 2007 at The University of Sydney
• 7^th June, 2016 at The University of Sydney
• 13^th June, 2017 at Sydney University
• 25^th June, 2018 at Macquarie University
• 24^th June, 2019 at University of Technology Sydney
• Hecke Operators: third talk in the Modular Forms lecture series, Number Theory Seminar, Sydney, April 2004
• Introduction to Modular Forms: talk given to the Sydney University Mathematics Society, October 2003
• Introduction to Elliptic Curves: talk given to the Sydney University Mathematics Society, May 2003

Conference and Workshop participation

• Australian Mathematical Society Annual Meeting, The University of Sydney, 30^th September — 3^rd October 2013
• ECC 2012, Querétaro, Mexico, 28^th — 31^st October 2012
• ECC 2011, INRIA, Nancy, France, 19^th — 21^st September 2011
• Geocrypt 2011, near Bastia, Corsica, 20^th — 24^th June 2011
• AGCT-13, CIRM Luminy, Marseille, 14^th — 18^th April 2011
• DIAMANT Symposium, Lunteren, 26^th — 27^th November 2010
• ANTS-IX , INRIA Nancy, France, 19^th — 23^rd July 2010
• Réunion CHIC, Institut Henri Poincaré, Paris, 5^th — 6^th October 2009
• Crypto'Puces, Île de Porquerolles, 2^nd — 6^thJune 2009
• AGCT-12, CIRM Luminy, Marseille, 30^th March — 3^rd April 2009
• ESF Workshop on Codes, Cryptography and Coding Theory, Institut de Mathématiques de Luminy, 25^th — 29^th March 2009
• Australian Mathematical Society Annual Meeting, La Trobe University, 25^th — 28^th September 2007
• Journées Arithmétiques, University of Edinburgh, Scotland, 2^nd — 6^th July 2007
• Australian Mathematical Society Annual Meeting, Macquarie University, 25^th — 29^th September 2006
• Computing with Modular forms, MSRI, Berkeley, 31^st July — 11^th August 2006
• Explicit Arithmetic Geometry, Institut Henri Poincaré, Paris, 6^th — 10^th December 2004

Contact Details

Teaching

All MATHyyy courses (after 2015) correspond to lecture courses given at ACU Strathfield.

All MATHzzz courses correspond to lecture courses given at Macquarie University.

= lectures and tutorials given at ACU.

= small group teaching activities (SGTA's are board tutorials) at Macquarie University.

• MATH123 Mathematics 123 tutorials
• MATH1002 Linear Algebra tutorials
• MATH1014 Introduction to Linear Algebra tutorials

• Tutor at National Mathematics Summer School, Australian National University, Canberra, 7^th — 20^th January
• MATH311 Linear Programming and Applications
• MATH1014 Introduction to Linear Algebra tutorials

• Tutor at National Mathematics Summer School, Australian National University, Canberra, 8^th — 21^st January
• MATH1002 Linear Algebra tutorials
• MATH1014 Introduction to Linear Algebra tutorials

• Tutor at National Mathematics Summer School, Australian National University, Canberra, 3^rd — 16^th January
• MATH1002 Linear Algebra tutorials
• MATH1014 Introduction to Linear Algebra tutorials
• MATH311 Linear Programming and Applications

• MATH1002 Linear Algebra tutorials
• MATH1003 Integral Calculus and Modelling tutorials
• MATH1014 Introduction to Linear Algebra tutorials

• Tutor for 2-Unit maths bridging course, Sydney University, 11^th — 26^th February
• MATH1901 Differential Calculus (Advanced) tutorials
• MATH1002 Linear Algebra tutorials
• MATH1903 Integral Calculus and Modelling (Advanced) tutorials

• Tutor at National Mathematics Summer School, Australian National University, Canberra, 4^th — 17^th January
• Tutor for 2-Unit maths bridging course, Sydney University, 9^th — 24^th February

• Tutor at National Mathematics Summer School, Australian National University, Canberra, 6^th — 19^th January
• Tutor for 2-Unit maths bridging course, Sydney University, 11^th — 26^th February
• MATH1011 Life Sciences Calculus tutorials

• Tutor at National Mathematics Summer School, Australian National University, Canberra, 7^th — 20^th January
• Tutor for 2-Unit maths bridging course, Sydney University, 12^th — 27^th February
• MATH1011 Life Sciences Calculus tutorials

• Tutor at National Mathematics Summer School, Australian National University, Canberra, 8^th — 21^st January
• Tutor for 2-Unit maths bridging course, Sydney University, 13^th — 28^th February
• MATH1002 Linear Algebra tutorials

• Tutor at National Mathematics Summer School, Australian National University, Canberra, 2^nd — 15^th January
• Tutor for 2-Unit maths bridging course, Sydney University, 14^th February - 1^st March
• MATH1901 Differential Calculus (Advanced) tutorials
• MATH1012 Life Sciences Algebra tutorials

• MATH1001 Differential Calculus tutorials
• MATH1003 Integral Calculus tutorials

Does this web page look familiar? I borrowed the template from Ben Smith (I'll return it or the favour one day!)