Publications

Articles

• A special configuration of 12 conics and generalized Kummer surfaces, with Xavier Roulleau and Alessandra Sarti, Manuscripta Mathematica, 2021. [ArXiV, HAL, DOI]
• On Sato-Tate distributions, extremal traces, and real multiplication in genus 2, with Yih-Dar Shieh, in Arithmetic, Geometry, Cryptography and Coding Theory, AMS Contemporary Mathematics, 770, 2021. [ArXiv, HAL, DOI]
• SQISign: compact post-quantum signatures from quaternions and isogenies, with Luca de Feo, Antonin Leroux, Christophe Petit, Benjamin Wesolowski, Asiacrypt 2020 (South Korea), Lecture Notes in Computer Science, 12491, 64-93, 2020. [ePrint, HAL, DOI]
• Orienting supersingular isogeny graphs, with Leonardo Colò, Journal of Mathematical Cryptology, 14, 414-437, 2020. [ArXiv, ePrint, HAL, DOI]
• Arithmetic statistics of Galois groups, Proceedings of the Thirteenth Algorithmic Number Theory Symposium, The Open Book Series, Vol. 2, 353-374, 2019. [ArXiv, HAL, DOI]
• Twisted μ₄-normal form for elliptic curves, in Advances in Cryptology — Eurocrypt 2017 (Paris), Lecture Notes in Computer Science, 10210, 659-678, 2017. [ArXiv, ePrint, HAL, DOI]
• Twisted Hessian curves, with Daniel J. Bernstein, Tanja Lange, and Chitchanok Chuengsatiansup, in Progress in Cryptology — Latincrypt 2015 (Guadalajara), Lecture Notes in Computer Science, 9230, 269-294, 2015. [ePrint, HAL, DOI]
• The geometry of efficient arithmetic on elliptic curves, in Algorithmic Arithmetic, Geometry and Coding Theory, AMS Contemporary Mathematics, 637, 95-110, 2015. [ArXiv, HAL, DOI]
• On the quaternion l-isogeny path problem, with K. Lauter, C. Petit, and J.-P. Tignol, LMS Journal of Computational Mathematics, 17, 418-432, 2014. [ArXiv, ePrint, HAL]
• Théorie des nombres et cryptographie, with Igor Shparlinski, Arithmétique et dynamique, Chaires Jean Morlet 2014, SMF Journée Annuelle, 27, 2014. [HAL]
• Efficient arithmetic on elliptic curves in characteristic 2, in Progress in Cryptology — Indocrypt 2012 (Kolkata), Lecture Notes in Computer Science, 7668, 378-398, 2012. [ArXiv, HAL, DOI]
• Complete addition laws on abelian varieties, with C. Arene and C. Ritzenthaler, LMS Journal of Computational Mathematics, 15, 308-316, 2012. [ArXiv, HAL]
• Counting points on genus 2 curves with real multiplication, with P. Gaudry and B. Smith, in Progress in Cryptology — Asiacrypt 2011 (Seoul), Lecture Notes in Computer Science, 7073, 504-519, 2011. [ArXiv, HAL, DOI]
• Addition law structure of elliptic curves, Journal of Number Theory, 131, Issue 5 (special volume: Elliptic Curve Cryptography), 894-919, 2011. [ArXiv, HAL, DOI]
• Arithmetic of split Kummer surfaces: Montgomery endomorphism of Edwards products, in Coding and Cryptology — Third International Workshop (IWCC 2011, Qingdao), Lecture Notes in Computer Science, 6639, 238-245, 2011. [ArXiv, HAL, DOI]
• Double-base number system for multi-scalar multiplications, with C. Doche and F. Sica, in Advances in Cryptology — Eurocrypt 2009 (Cologne), Lecture Notes in Computer Science, 5479, 502-519, 2009. [ePrint, DOI]
• Complex multiplication and canonical lifts, in Algebraic Geometry and Its Applications, Series on Number Theory and Its Applications, 5, 67-83, 2008.
• Higher dimensional 3-adic CM construction, with R. Carls and D. Lubicz, Journal of Algebra, 319, no. 3, 971-1006, 2008. [ArXiv, HAL, DOI]
• The Weierstrass subgroup of a curve has maximal rank, with M. Girard and C. Ritzenthaler, Bulletin of the London Mathematical Society, 38, Issue 06, 925-931, 2006. [ArXiv, HAL, DOI]
• The 2-adic CM method for genus 2 curves with application to cryptography, with P. Gaudry, T. Houtmann, C. Ritzenthaler, and A. Weng, in Progress in Cryptology — Asiacrypt 2006, (Shanghai), Lecture Notes in Computer Science, 4284, 114-129, 2006. [HAL, DOI]
• Classification of genus 3 curves in special strata of the moduli space, with M. Girard, in Algorithmic Number Theory — 7th International Symposium (ANTS VII, Berlin), Lecture Notes in Computer Science, 4076, 346-360, 2006. [ArXiv, DOI]
• Efficiently computable endomorphisms for hyperelliptic curves, with B. Smith, in Algorithmic Number Theory — 7th International Symposium (ANTS VII, Berlin), Lecture Notes in Computer Science, 4076, 495-509, 2006. [ArXiv, HAL, DOI]
• Efficient scalar multiplication by isogeny decompositions, with C. Doche and T. Icart, in Public Key Cryptography — PKC 2006 (New York), Lecture Notes in Computer Science, 3958, 191-206, 2006. [ePrint, DOI] For comparative analysis (and improvements to our formulae), see the Explicit Formulas Database web page of Bernstein and Lange on hyperelliptic.org.
• The AGM-X₀(N) Heegner point lifting algorithm and elliptic curve point counting, in Advances in Cryptology — Asiacrypt 2003 (Taipei), Lecture Notes in Computer Science, 2894, 124-136, 2003. [DOI]
• Fundamental domains for Shimura curves, with Helena Verrill, Journal of Number Theory of Bordeaux, 15, no. 1, 205-222, 2003.
• Rational groups of elliptic curves suitable for cryptography, Cryptography and Computational Number Theory (Singapore, 1999), Progress in Computer Science and Applied Logic, 20, 69-80, 2001. [DOI]
• Counting the number of points on affine diagonal curves, with Cunsheng Ding and San Ling, in Cryptography and Computational Number Theory (Singapore, 1999), Progress in Computer Science and Applied Logic, 20, 15-24, 2001. [DOI]
• Component groups of quotients of J₀(N), with William Stein, in Algorithmic Number Theory — 4th International Symposium (ANTS IV, Leiden), Lecture Notes in Computer Science, 1838, 405-412, 2000. [DOI]
• On exponential sums and group generators for elliptic curves over finite fields, with Igor Shparlinski, in Algorithmic Number Theory — 4th International Symposium (ANTS IV, Leiden), Lecture Notes in Computer Science, 1838, 395-404, 2000. [DOI]
• Hecke module structure of quaternions, in Class Field Theory — Its Centenary and Prospect (Tokyo, 1998), Advanced Studies in Pure Mathematics, 30, 177-196, 2000. [Euclid]
• Split group codes, with Cunsheng Ding and San Ling, IEEE Transactions on Information Theory, 46, no. 2, 485-495, 2000. [DOI]
• Elementary 2-group character codes, with Cunsheng Ding and San Ling, IEEE Transactions on Information Theory, 46, no. 1, 280-283, 2000. [DOI]
• Explicit sequence expansions, with San Ling and Chaoping Xing, in Sequences and their Applications — Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999. [DOI]
• Endomorphism rings of elliptic curves over finite fields. Thesis, University of California, Berkeley, December, 1996.

Edited Books

• Arithmetic, Geometry, Cryptography and Coding Theory, Proceedings of AGCT 2015 (CIRM, Luminy), edited with Alp Bassa and Alain Couvreur, AMS Contemporary Mathematics, 686, 2017. [DOI]
• Frobenius Distributions: Lang-Trotter and Sato-Tate Conjectures, Proceedings of a CIRM workshop, edited with Igor Shparlinski, AMS Contemporary Mathematics, 663, 2016. [DOI]
• Arithmetic, Geometry, Cryptography and Coding Theory 2009, Proceedings of AGCT 2009 (CIRM, Luminy) and Geocrypt 2009 (Pointe-à-Pitre, Guadeloupe), edited with Robert Rolland, AMS Contemporary Mathematics, 521, 2010. [HAL, DOI]
• Algorithmic Number Theory — 5th International Symposium, (ANTS V, Sydney), edited with C. Fieker, Lecture Notes in Computer Science, 2369, 2002. [DOI]

Books

• Cryptography, introductory text on cryptography based on teaching (see also MATH3024 and MATH3925), with exercises in Sage, 2007.
  

Talks et al.

• Arithmetic statistics of Galois groups. Slides from the Algorithmic Number Theory Symposium, Madison, 16 July 2018.
• Elliptic curves in μ₄-normal form. Slides from Eurocrypt 2017, Paris, 1 May 2017.
• The mathematics of Sudoku. Introductory slides of a stage hippocampe on Sudoku (for high school students), 17 October 2016.
• The geometry of efficient arithmetic on elliptic curves. 1. Slides of talk at the Cryptography seminar at the Mathematical Institute, Oxford University, 26 October 2016, 2. slides of talk at the Séminaire CCA (Codage, Cryptographie et Algorithmes), INRIA Paris-Rocquencourt, 14 October 2016, and 3. handwritten scanned notes of a talk at the Mathematical Structures for Cryptography, Lorentz Center, Universiteit Leiden, 25 August 2016.
• Théorie des nombres et cryptographie. Slides from a presentation at the Journée Annuelle de la SMF, IHP, Paris, 20 Juin 2014.
• The geometry of efficient arithmetic on elliptic curves. Slides from Geocrypt 2013, Tahiti, 7 October 2013.
• Frobenius distributions. Slides from the ANR PEACE meeting, IHP, Paris, 12 April 2013.
• Elliptic curves in characteristic 2. Slides from the Rump session of ECC 2011, Nancy, 19 September 2011.
• Sato-Tate and notions of generality. Slides from a talk at Explicit Methods in Number Theory, Oberwolfach, 19 July 2011.
• Arithmetic of split Kummer surfaces: Montgomery endomorphism of Edwards products. Slides from a talk at the International Workshop on Coding and Cryptography 2011, Qingdao, 2 June 2011.
• A normal form for elliptic curves in characteristic 2. Slides from a talk at the CIRM workshop AGCT-13 (Arithmetic, Geometry, Cryptography and Coding Theory), Luminy, 15 March 2011.
• Structure of elliptic curves and addition laws. Slides from a talk at the final GTEM conference, Barcelona, 9 September 2010.
• Sage dans l'enseignement: Cryptographie. Sage worksheet pour un exposé en cryptographie dans le colloque Journées Sage dans l'enseignement, Luminy, 24 February 2010.
• Sage: Open source software for mathematics. Sage worksheet for an introduction to Sage at Sage Days 20, Luminy, 22-26 February 2010.
• On CM and RM for abelian surfaces. Slides from CIRM workshop Number theory and applications, Luminy, 3 December 2009 and the BMS-LMS Conference 2009 special session on Algebraic Geometry and Cryptography, Leuven, 4-5 December, 2009.
• ECHIDNA: Open source extensions to Magma in Sage (printed pdf) and Sage worksheet. Presentation at Sage Days 16, Barcelona, June 2009.
• Le logiciel Sage. Sage worksheet for a talk in the ATI seminar in Luminy, 16 October 2008.
• Lower dimensional complex multiplication. Slides from the Computational Number Theory workshop at FoCM'08, Hong Kong, 24-26 June 2008.
• Complex multiplication and canonical lifts. Slides from SAGA 2007, Tahiti, 2007.
• Construction of CM moduli by p-adic lifting. Notes from a talk at the 50th annual meeting of the Australian Mathematical Society, Sydney, 2006.
• Classification of genus 3 curves in special strata of the moduli space. Slides from talk at ANTS VII in Berlin, 24 July 2006.
• Construction p-adique des invariant CM des courbes de genre 2. Slides from talk at Université Toulouse Le Mirail, 12 December 2005.
• Tores algébriques et jacobiennes generalisées en cryptographie. Slides from talk at the Institut de Mathématiques de Luminy, 8 December 2005.
• Constructive p-adic CM for genus 2 curves. Slides from talk at Center for Communications Research, San Diego, 6 October 2005.
• An l-adic CM method for genus 2. Slides from talk at the Journées Arithmétiques, Marseille, July 2005.
• Introduction to magma and applications. Slides from talk at the African Institute for the Mathematical Sciences, Capetown, 2005.
• The p-adic CM-method for genus 2, with P. Gaudry, T. Houtmann, C. Ritzenthaler, and A. Weng; extended preprint version, 2005.
• p-adic construction of CM curves, with C. Ritzenthaler; unpublished precursor to joint article with Gaudry, Houtmann and Weng, 2004.
• Igusa class invariants and the AGM. Slides from Banff, 13-18 November, and Magma Workshop, Göttingen, 11-15 December, 2004.
• Constructive and destructive facets of torus-based cryptography. Unpublished preprint, 2004.
• The AGM-X₀(N) Algorithm. Slides from presentation at Asiacrypt 2003, Taipei.
• On invariants of Shimura curves. Slides from a talk on Shimura curves at the Arithmetic Geometry workshop at MSRI, Berkeley, December 2000.
• Part I (Magma language) and Part II (Number theoretic applications). Slides from a presentation on the magma computer language and selected applications in number theory, MSRI, Berkeley, October 2000.
• Computing modular curves via quaternions. Unpublished note based on talk at Computational Algebraic Number Theory, Sydney, December 1997.
• Coding theory: algebraic geometry of linear algebra. Rough notes of talk on algebraic geometric codes, National University of Singapore, April, 1997.

David Kohel (David.Kohel@univ-amu.fr) Institut de Mathématiques de Marseille