Papers

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

Arithmetic of Kleinian groups      Arithmetic of Abelian varieties      Cryptography

 

New Preprints

  • Algebraic aspects of solving Ring-LWE, including ring-based improvements in the Blum-Kalai-Wasserman algorithm
    Submitted, 26 pages.
    [ arXiv:1902.07140 | IACR eprint 2019/183 | local pdf | code ]

To appear

  • Local-global principles in circle packings
    with Elena Fuchs and Xin Zhang
    To appear in Compositio Mathematica, 54 pages.
    [ arXiv:1707.06708 ]
  • The dynamics of super-Apollonian continued fractions
    with Sneha Chaubey, Elena Fuchs and Robert Hines
    To appear in Transactions of the American Mathematical Society, 41 pages.
    [ arXiv:1703.08616 | early view publication ]

Published

  • A family of monogenic S4 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 ]
  • Attacks on the Search RLWE Problem with Small Errors
    with Hao Chen and Kristin Lauter
    SIAM Journal on Applied Algebra and Geometry, 1-1 (2019): pp. 665-682.
    [ IACR eprint 2015/971 | published ]
  • The Apollonian structure of Bianchi groups
    Transactions of the American Mathematical Society, 370 (2018), pp. 6169-6219.
    [ arXiv:1505.03121 | published ]
  • Visualizing the arithmetic of quadratic imaginary fields
    International Mathematics Research Notices (2017) (currently advance access).
    [ 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

  • 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 ]

Unpublish(ed/able) Notes

  • Formulary for elliptic divisibility sequences and elliptic nets
    [ pdf ]
  • Notes on the spin homomorphism with Yongqi Feng
    [ pdf ]
  • Notes on Bhargava’s composition laws
    [ pdf ]