[ Benjamin Smith ]

Contact Details

Benjamin Smith
Projet TANC - INRIA Saclay–Île-de-France
Laboratoire d'informatique (LIX)
École polytechnique
91128 Palaiseau cedex
France

Email: smith@lix... (the rest of the address is left as an exercise.)

My PGP public key

Research

Interests

• Explicit constructions of isogenies in higher genus
• Effective constructions of real and complex multiplications on arithmetic Jacobians
• Applications of algebraic correspondences in computational number theory and cryptology

I have been a chargé de récherche at INRIA, working at the Laboratoire d'Informatique (LIX) at the École polytechnique in France, since November 2007. Before that, I was a postdoctoral research assistant in the Mathematics department and Information Security Group at Royal Holloway, University of London, working for Steven Galbraith. Before that, I was a student in the number theory research group of the School of Mathematics and Statistics at the University of Sydney, where my supervisor was David R. Kohel.

I was a developer and programmer with the Magma project between 2000 and 2003, contributing to the General Number Field Sieve, linear programming and algebraic geometry packages, as well as documentation and bug squashing. I completed honours at Sydney in 2001, in the Categories and Combinatorics research group.

Publications/Preprints

• Families of Explicit Isogenies of Hyperelliptic Jacobians. In the Proceedings of AGCT 12. Preprint available at http://fr.arxiv.org/abs/0909.5406 .

  Abstract: We describe three-dimensional families of pairs of hyperelliptic curves of genus 6, 12, and 14, two-dimensional families of hyperelliptic curves of genus 3, 6, and 7, and one-dimensional families of hyperelliptic curves of genus 5, 10 and 15, all of which are equipped with an an explicit isogeny from their Jacobian to another hyperelliptic Jacobian. We show that the Jacobians are generically simple, and determine the types of the isogenies. The families are derived from Cassou-Noguès and Couveignes' explicit classification of pairs (f,g) of polynomials such that f(x₁) - g(x₂) is reducible.

• Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves (Extended version). Journal of Cryptology 22 #4 (2009). Preprint available at http://arxiv.org/abs/0806.2995.
• Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves (Condensed version). Advances in Cryptology: EUROCRYPT 2008, Istanbul (Springer LNCS 4965).

  Abstract: We describe the use of explicit isogenies to reduce Discrete Logarithm Problems (DLPs) on Jacobians of hyperelliptic curves of genus three to Jacobians of non-hyperelliptic curves of genus three, which are vulnerable to faster index calculus attacks. We provide algorithms which compute an isogeny with kernel isomorphic to $(Z/2Z)^3$ for any hyperelliptic genus three curve. These algorithms provide a rational isogeny for a positive fraction of all hyperelliptic genus three curves defined over a finite field of characteristic p > 3. Subject to reasonable assumptions, our algorithms provide an explicit and efficient reduction from hyperelliptic DLPs to non-hyperelliptic DLPs for around $18.57\%$ of all hyperelliptic genus three curves over a given finite field.

• Distortion Maps for Genus 2 Curves (with Steven D. Galbraith, Jordi Pujolas, and Christophe Ritzenthaler). Journal of Mathematical Cryptology 3 #1 (2009). Preprint available from eprint and arXiv.

  Abstract: Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more complicated since the full torsion subgroup has rank 2g. In this paper we prove that distortion maps always exist for supersingular curves of genus g>1 and we give several examples in genus 2.

• Efficiently Computable Endomorphisms for Hyperelliptic Curves (with David R. Kohel), in Algorithmic Number Theory: ANTS-VII, Berlin (Springer LNCS 4076). Preprint available at http://arxiv.org/abs/math.NT/0603505.

  Abstract: Elliptic curves have a well-known and explicit theory for the construction and application of endomorphisms, which can be applied to improve performance in scalar multiplication. Recent work has extended these techniques to hyperelliptic Jacobians, but one obstruction is the lack of explicit models of curves together with an efficiently computable endomorphism. In the case of hyperelliptic curves there are limited examples, most methods focusing on special CM curves or curves defined over a small field. In this article we describe three infinite families of curves which admit an efficiently computable endomorphism, and give algorithms for their efficient application.

• Discrete Logarithms in Generalized Jacobians (with Steven D. Galbraith), preprint available at http://arxiv.org/abs/math.NT/0610073.

  Abstract: Déchène has proposed generalized Jacobians as a source of groups for public-key cryptosystems based on the hardness of the Discrete Logarithm Problem (DLP). Her specific proposal gives rise to a group isomorphic to the semidirect product of an elliptic curve and a multiplicative group of a finite field. We explain why her proposal has no advantages over simply taking the direct product of groups. We then argue that generalized Jacobians offer poorer security and efficiency than standard Jacobians.

• Explicit Endomorphisms and Correspondences: Ph.D. Thesis.
• Computations in Algebraic Geometry with Magma: poster presented at the 5th Algorithmic Number Theory Symposium (ANTS-V), Sydney 2002

Selected talks

• Isogenies and the DLP in genus 3: Eurocrypt 2008, Istanbul, March 2008; ECC, Dublin, September 2007; AGCT, Luminy, November 2007
• Explicit isogenies of hyperelliptic Jacobians: London Number Theory Seminar, King's College London, UK, May 2006; Oxford Number Theory Seminar, January 2007; LIX, Paris, February 2007; LORIA, Nancy, March 2007
• Efficiently computable endomorphisms for hyperelliptic curves: Number Theory Seminar, University of Sydney, Australia, January 2006 Pure Mathematics Seminar, Royal Holloway, University of London, February 2006; ANTS-VII, Berlin, July 2006;
• Richelot correspondences and explicit modular isogeny graphs: Number Theory Seminar, Sydney, Australia, June 2005; XXIVeme Journees Arithmetiques, Marseille, France, July 2005
• Realising Endomorphisms of Jacobians with Correspondences: Workshop in Computational Arithmetic Geometry, PIMS/Simon Fraser University, Vancouver, Canada, July 2004; ECHIDNA-II Workshop in Arithmetic Geometry and Applications, Sydney, Australia, January 2005
• The Elliptic Curve Method for integer factorisation: Sydney University Mathematics Society, Sydney, Australia, August 2003

Alibis: Conferences and Workshops

• Explicit Methods in Number Theory, Oberwolfach, July 2009
• AGCT 12, Luminy, March-April 2009
• ESF Exploratory Workshop: Curves, Coding Theory and Cryptography, Luminy, March 2009
• ASIACRYPT 2008, Melbourne, December 2008
• ECC 2008, Utrecht, September 2008
• C4: Computations on Curves for Crypto and Coding, Paris, June 2008
• ANTS-VIII, Banff, May 2008
• Eurocrypt 2008, Istanbul, March 2008
• AMS/MAA Joint meetings 2008 (Special session on low-genus curves and applications), San Diego, January 2008
• AGCT 11, Luminy, November 2007
• ECC 2007, Dublin, September 2007
• 25th Journées arithmetiques, Edinburgh, July 2007
• Workshop on Computational challenges arising in algorithmic number theory and Cryptography , Fields Institute, Toronto, Canada, 30 October -- 3 November 2006
• Thematic Program in Cryptography , Fields Institute, Toronto, Canada, 23 October -- 3 November 2006
• Magma 2006, TU-Berlin, Germany, 30 July -- 2 August 2006
• ECRYPT workshop on Cryptanalysis, TU-Berlin, Germany, 29-30 July 2006
• ANTS-VII: 7th Algorithmic Number Theory Symposium, TU-Berlin, Germany, 23-28 July 2006
• Curves, Isogenies and Cryptologic Applications, Ecole Polytechnique, Paris, France, 18 July 2006
• PQCrypto 2006: International workshop on postquantum cryptography, Katholieke Universiteit Leuven, Belgium, 24-26 June 2006
• XXIViemes Journees Arithmetiques, Marseille, France, 4-8 July 2005
• ECHIDNA-II Workshop in Arithmetic Geometry and Applications, Sydney, Australia, 12-14 January 2005
• Explicit Methods in Number Theory, Institut Henri Poincare, Paris, France, November-December 2004
• Workshop on Computational Arithmetic Geometry, PIMS/Simon Fraser University, Vancouver, Canada, July 2004
• 8th Canadian Number Theory Association meeting, Fields Institute/University of Toronto, Canada, June 2004
• ANTS-VI: 6th Algorithmic Number Theory Symposium, University of Vermont, June 2004
• Special activity on minimal models, Mathematical Sciences Institute, Australian National University, Canberra, Australia, August 2003
• Workshop on computational arithmetic geometry, University of Sydney, Australia, June 2003
• Fields institute conference in number theory in Honour of Professor H.C. Williams, Banff, Canada, May 2003
• ECHIDNA: Elliptic Curves and Higher Dimensional Analogues, University of Sydney, Australia, July 2002
• ANTS-V: 5th Algorithmic Number Theory Symposium, University of Sydney, Australia, July 2002

(Reproduced with permission of Chris Onstad, Achewood.com.)