### Jump to:

- New
- To Appear
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- Older preprints
- Pedagogical Articles
- Expositional Publications
- Ph.D. Thesis
- Unpublish(ed/able) Notes

The following articles are colour-coded to help you find certain topics:

**Arithmetic of thin and Kleinian groups** **Arithmetic of Abelian varieties** **Cryptography**

## New Preprints

**Sesquilinear pairings on elliptic curves**

17 pages.

[ arXiv:2405.14167 ]

**Prime and Thickened Prime Components in Apollonian Circle Packings**

With Elena Fuchs, Holley Friedlander, Piper Harris, Catherine Hsu, James Rickards, Katherine Sanden and Damaris Schindler.

41 pages.

[ arXiv:coming soon; email me ]

**Reciprocity obstructions in semigroup orbits in $\text{SL}(2, \mathbb{Z})$**

With James Rickards.

27 pages.

[ arXiv:2401.01860 | github code ]

## To appear

**Extending class group action attacks via sesquilinear pairings**

With Joseph Macula

20 pages. To appear at AsiaCrypt 2024.

[ eprint:2024/880 ]**The Local-Global Conjecture for Apollonian circle packings is false**

With Summer Haag, Clyde Kertzer and James Rickards.

15 pages, to appear in Annals of Mathematics.

[ arXiv:2307.02749 | github code | github data ]

## Published

**Orientations and cycles in supersingular isogeny graphs**

With Sarah Arpin, Mingjie Chen, Kristin E. Lauter, Renate Scheidler, and Ha T. N. Tran.

Proceedings of Women in Number Theory 5 (2024), pp. 25-86.

[ eprint:2022/562 | arXiv:2205.03976 | published ]**Failing to hash into supersingular isogeny graphs**

With Jeremy Booher, Ross Bowden, Javad Doliskani, Tako Boris Fouotsa, Steven D. Galbraith, Sabrina Kunzweiler, Simon-Philipp Merz, Christophe Petit, Benjamin Smith, Yan Bo Ti, Christelle Vincent, José Felipe Voloch, Charlotte Weitkämper and Lukas Zobernig.

The Computer Journal (CFAIL Issue), Volume 67, Issue 8, August 2024, pp. 2702–2719 (extended abstract accepted to CFAIL 2022).

[ eprint:2022/518 | arXiv:2205.00135 | CFAIL presentation | published | free access ]

**Factoring using multiplicative relations modulo n: a subexponential algorithm inspired by the index calculus**

Mathematical Cryptology, 3(2), 2–10 (issue for MathCrypt 2023).

[ published | eprint:2022/1588 | arXiv:2211.06821 | github ]

**Orienteering with one endomorphism**

With Sarah Arpin, Mingjie Chen, Kristin E. Lauter, Renate Scheidler, and Ha T. N. Tran.

La Matematica (currently early online access)

[ published | arXiv:2201.11079 | github ]

**Algebraic Number Starscapes**

With Edmund Harriss and Steve Trettel.

Experimental Mathematics, 31:4, 1098-1149

[ arXiv:2008.07655 | published | accompanying website ]**Monogenic fields arising from trinomials**

With Ryan Ibarra, Henry Lembeck, Mohammad Ozaslan, and Hanson Smith.

Accepted to Involve – A Journal of Mathematics.

[ arXiv:1908.09793 | published ]**Improved torsion point attacks on SIDH variants**

With Victoria de Quehen, Péter Kutas, Chris Leonardi, Chloe Martindale, Lorenz Panny, and Christophe Petit.

Advances in Cryptology — CRYPTO 2021, Springer LNCS 12827 (2021), 432-470.

[ published | arXiv:2005.14681 | IACR eprint 2020/633 | video | code ]**Algebraic aspects of solving Ring-LWE, including ring-based improvements in the Blum-Kalai-Wasserman algorithm**

SIAM Journal on Applied Algebra and Geometry, 5(2) (2021): pp. 366–387.

[ published | arXiv:1902.07140 | IACR eprint 2019/183 | local pdf | code ]**Local-global principles in circle packings**

with Elena Fuchs and Xin Zhang

Compositio Mathematica, 155:6 (2019): pp. 1118-1170.

[ arXiv:1707.06708 | published ]

**The dynamics of super-Apollonian continued fractions**

with Sneha Chaubey, Elena Fuchs and Robert Hines

Transactions of the American Mathematical Society, 372 (2019), pp. 2287-2334.

[ arXiv:1703.08616 | published ]

**A family of monogenic S**_{4}quartic fields arising from elliptic curves

with T. Alden Gassert and Hanson Smith

Journal of Number Theory, 197 (2019): pp. 361-382.

[ arXiv:1708.03953 | published ]**The Apollonian structure of Bianchi groups**

Transactions of the American Mathematical Society, 370 (2018), pp. 6169-6219.

[ arXiv:1505.03121 | published ]**Attacks on the Search RLWE Problem with Small Errors**

with Hao Chen and Kristin Lauter

SIAM Journal on Applied Algebra and Geometry, 1-1 (2017): pp. 665-682.

[ IACR eprint 2015/971 | published ]**Visualizing the arithmetic of quadratic imaginary fields**

International Mathematics Research Notices 2018:12 (2018), 3908–3938.

[ arXiv:1410.0417 | published | free access link ]**Security Considerations for Galois Non-dual RLWE Families**

with Hao Chen and Kristin Lauter

*Selected Areas in Cryptography 2016 — SAC 2016*, LNCS vol 10532, pp. 443-462.

[ IACR eprint 2016/193 | published ]**Index divisibility in dynamical sequences and cyclic orbits modulo p**

with Annie S. Chen and T. Alden Gassert

New York Journal of Mathematics, 23 (2017), pp. 1045-1063.

[ arXiv:1608.02177 | published ]**Arithmetic properties of the Frobenius traces defined by a rational abelian variety**with Alina Carmen Cojocaru, Rachel Davis and Alice Silverberg, and with two appendices by J-P. Serre

International Mathematics Research Notices, 12 (2017), 3557-3602.

[ arXiv:1504.00902 | published | free access link ]**The sensual Apollonian circle packing**

Expositiones Mathematicae, 34-4 (2016), 364-395.

[ arXiv:1208.4836 | published ]**RLWE Cryptography for the Number Theorist**with Yara Elias, Kristin E. Lauter, and Ekin Ozman

Women in Numbers 3: Research Directions in Number Theory Springer AWM Series, Vol. 3 (2016), 271-290.

[ IACR eprint 2015/758 | arXiv:1508.01375 | published ]**Integral points on elliptic curves and explicit valuations of division polynomials**

Canadian Journal of Mathematics, 68 (2016), 1120-1158.

[ arXiv:1108.3051 | published ]**Provably weak instances of Ring-LWE**

with Yara Elias, Kristin E. Lauter, and Ekin Ozman.

Advances in Cryptology — CRYPTO 2015, Springer LNCS 9215 (2015), 63-92.

[ arXiv:1502.03708 | published ]**A duality principle for selection games**with Lionel Levine and Scott Sheffield

Proceedings of the American Mathematical Society, 141 (2013), 4349-4356.

[ arXiv:1110.2712 | published ]**How to make the most of a shared meal: plan the last bite first**

with Lionel Levine

American Mathematical Monthly, 119-7 (2012), 550-565.

[ arXiv:1104.0961 | published | computer scripts ]**Algebraic divisibility sequences over function fields**

with Patrick Ingram, Valery Mahe, Joseph H. Silverman, and Marco Streng

In memory of Alf van der Poorten, mathematician, colleague, friend.

Journal of the Australian Mathematical Society (special issue dedicated to Alf van der Poorten), 92-1 (2012), 99-126.

[ arXiv:1105.5633 | published ]**Character sums with division polynomials**

with Igor E. Shparlinski

Canadian Mathematical Bulletin, 55 (2012), 850-857.

[ arXiv:0912.5246 | published ]**Amicable pairs and aliquot cycles for elliptic curves**

with Joseph H. Silverman.

Experimental Mathematics, 20-3 (2011), 329-357.

[ arXiv:0912.1831| published | related website ]**Terms in elliptic divisibility sequences divisible by their indices**

with Joseph H. Silverman

Acta Arithmetica, 146 (2011), 355-378.

[ arXiv:1001.5303 | published ]**Pairings on hyperelliptic curves**

with Jennifer Balakrishnan, Juliana Belding, Sarah Chisholm, Kirsten Eisenträger and Edlyn Teske.

Dedicated to the memory of Isabelle Déchène (1974-2009)

WIN — Women in Numbers: Research Directions in Number Theory, Fields Institute Communications, 60 (2011), 87-120.

[ arXiv:0908.3731 | published ]**Elliptic nets and elliptic curves**

Algebra and Number Theory, 5-2 (2011), 197-229.

[ arXiv:0710.1316 | published | errata ]**The elliptic curve discrete logarithm problem and equivalent hard problems for elliptic divisibility sequences**

with Kristin E. Lauter

Selected Areas in Cryptography 2008, Springer LNCS, 5381 (2009), 309-327.

[ IACR eprint 2008/099 | arXiv:0803.0728 | published ]**The Tate pairing via elliptic nets**

Pairing-Based Cryptography — PAIRING 2007, Springer LNCS, 4575 (2007), 329-348.

[ IACR eprint 2006/392 | published ]

## Older Preprints

**An arborist’s guide to the rationals**

Under revision, 7 pages.

[ arXiv:1403.2928 ]

**General Zelevinsky Algebras**with Peter Hoffman and Chris WooffUnder revision, 23 pages.

[ pdf ]

## Pedagogical Articles

**Standards Based Grading in an Introduction to Abstract Mathematics**

To appear in PRIMUS: Problems, Resources and Issues in Mathematics Undergraduate Studies, 27 pages plus appendices.

[ pdf | published ]

## Expositional Publications

**On the importance of illustration in mathematical research**

With Rémi Coulon, Gabriel Dorfsman-Hopkins, Edmund Harriss, Martin Skrodzki, and Glen Whitney

*Notices of the American Mathematical Society*, January 2024.

[ arXiv:2307.04636 | published ]**The ingenious physical factoring devices of D. N. Lehmer**

*Math Horizons*, 30:2 (2022), pp. 8-11.

[ published ]**The Farey structure of the Gaussian integers**

*Asia Pacific Math Newsletter*, (6)2, 2016, pp. 10-14.

[ published ]

**Visualizing imaginary quadratic fields**

*Canadian Mathematical Society Notes*September 2016, 2 pages.

[ published ]

**An illustration in number theory (2019 Lecture Sampler)**

*Notices of the American Mathematical Society,*66(03), 2019, pp. 411-413.

[ published ]

## Ph.D. Thesis

**Elliptic nets and elliptic curves**

Ph.D. Thesis, Brown University, 2008, 297 pages including computer code and errors.

[ pdf | Brown library ]