Skip to main content
×
×
Home

Constructing abelian surfaces for cryptography via Rosenhain invariants

  • Craig Costello (a1), Alyson Deines-Schartz (a2), Kristin Lauter (a3) and Tonghai Yang (a4)
Abstract

This paper presents an algorithm to construct cryptographically strong genus $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$ curves and their Kummer surfaces via Rosenhain invariants and related Kummer parameters. The most common version of the complex multiplication (CM) algorithm for constructing cryptographic curves in genus 2 relies on the well-studied Igusa invariants and Mestre’s algorithm for reconstructing the curve. On the other hand, the Rosenhain invariants typically have much smaller height, so computing them requires less precision, and in addition, the Rosenhain model for the curve can be written down directly given the Rosenhain invariants. Similarly, the parameters for a Kummer surface can be expressed directly in terms of rational functions of theta constants. CM-values of these functions are algebraic numbers, and when computed to high enough precision, LLL can recognize their minimal polynomials. Motivated by fast cryptography on Kummer surfaces, we investigate a variant of the CM method for computing cryptographically strong Rosenhain models of curves (as well as their associated Kummer surfaces) and use it to generate several example curves at different security levels that are suitable for use in cryptography.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Constructing abelian surfaces for cryptography via Rosenhain invariants
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Constructing abelian surfaces for cryptography via Rosenhain invariants
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Constructing abelian surfaces for cryptography via Rosenhain invariants
      Available formats
      ×
Copyright
References
Hide All
1.Atkin, A. O. L. and Morain, F., ‘Elliptic curves and primality proving’, Math. Comp. 61 (1993) no. 203, 2968.
2.Bernstein, D. J., ‘A software implementation of NIST P-224’, Talk at ECC, October 2001.
3.Bernstein, D. J., ‘Elliptic vs. hyperelliptic, part I’, Talk at ECC, September 2006.
4.Bernstein, D. J., Chuengsatiansup, C., Lange, T. and Schwabe, P., ‘Kummer strikes back: new DH speed records’, http://cr.yp.to/papers.html#kummer.
5.Bernstein, D. J. and Lange, T., ‘Faster addition and doubling on elliptic curves’, Advances in cryptology – ASIACRYPT 2007, Lecture Notes in Computer Science 4833 (ed. Kurosawa, K.; Springer, 2007) 2950.
6.Bos, J. W., Costello, C., Hisil, H. and Lauter, K., ‘Fast cryptography in genus 2’, Advances in cryptology – EUROCRYPT 2013, Lecture Notes in Computer Science 7881 (eds Johansson, T. and Nguyen, P. Q.; Springer, 2013) 194210. Full version: http://eprint.iacr.org/2012/670.
7.Broker, R., Gruenewald, D. and Lauter, K., ‘Explicit CM-theory for level 2-structures on abelian surfaces’, Algebra Number Theory 5 (2011) no. 4, 495528.
8.Bruinier, J. H., Kudla, S. S. and Yang, T., ‘Special values of Green functions at big CM points’, Int. Math. Res. Not. IMRN 2012 (2012) no. 9, 19171967.
9.Cardona, G. and Quer, J., ‘Field of moduli and field of definition for curves of genus 2’, Computational aspects of algebraic curves, Lecture Notes Series on Computing 13 (World Scientific, Hackensack, NJ, 2005) 7183.
10.Chudnovsky, D. and Chudnovsky, G. V., ‘Sequences of numbers generated by addition in formal groups and new primality and factorization tests’, Adv. Appl. Math. 7 (1986) no. 4, 385434.
11.Cosset, R., ‘Factorization with genus 2 curves’, Math. Comp. 79 (2010) no. 270, 11911208.
12.Edwards, H., ‘A normal form for elliptic curves’, Bull. Amer. Math. Soc. 44 (2007) no. 3, 393422.
13.Eisentraeger, K. and Lauter, K., ‘A CRT algorithm for constructing genus 2 curves over finite fields’, Arithmetic, geometry, and coding theory (AGCT 2005), Séminaires et Congrès 21 (eds Rodier, F. and Vladut, S.; Societe Mathematique de France, 2011) 161176.
14.Enge, A. and Thomé, E., ‘Computing class polynomials for abelian surfaces’, Experiment. Math., to appear, http://eprint.iacr.org/2013/299.
15.Gaudry, P., ‘Fast genus 2 arithmetic based on theta functions’, J. Math. Cryptology 1 (2007) no. 3, 243265.
16.Gaudry, P., Houtmann, T., Kohel, D. R., Ritzenthaler, C. and Weng, A., ‘The 2-adic CM method for genus 2 curves with application to cryptography’, Advances in cryptology – ASIACRYPT 2006, Lecture Notes in Computer Science 4284 (eds Lai, X. and Chen, K.; Springer, 2006) 114129.
17.Gaudry, P. and Schost, E., ‘Genus 2 point counting over prime fields’, J. Symbolic Comput. 47 (2012) no. 4, 368400.
18.Goren, E. Z. and Lauter, K. E., ‘Genus 2 curves with complex multiplication’, Int. Math. Res. Not. IMRN 2012 (2012) no. 5, 10681142.
19.Gruenewald, D., ‘Computing Humbert surfaces and applications’, Arithmetic, geometry, cryptography and coding theory 2009 (American Mathematical Society, Providence, RI, 2010) 5969.
20.Igusa, J., ‘On Siegel modular forms of genus two’, Amer. J. Math. (1962) 175200.
21.Lauter, K. and Viray, B., ‘An arithmetic intersection formula for denominators of Igusa class polynomials’, Preprint, 2012, arXiv:1210.7841.
22.Lenstra, A. K., Lenstra, H. W. and Lovász, L., ‘Factoring polynomials with rational coefficients’, Math. Ann. 261 (1982) no. 4, 515534.
23.Lubicz, D. and Robert, D., ‘A generalisation of Miller’s algorithm and applications to pairing computations on abelian varieties’, Cryptology ePrint Archive, Report 2013/192, 2013, http://eprint.iacr.org/.
24.Mestre, J., ‘Construction de courbes de genre 2 à partir de leurs modules’, Effective methods in algebraic geometry, Progress in Mathematics 94 (Birkhäuser, Boston, MA, 1991) 313334.
25.Milne, J. S., ‘Class field theory (v4.02)’, 2013, available at www.jmilne.org/math/.
26.Montgomery, P. L., ‘Speeding the Pollard and elliptic curve methods of factorization’, Math. Comp. 48 (1987) no. 177, 243264.
27.Pila, J., ‘Frobenius maps of abelian varieties and finding roots of unity in finite fields’, Math. Comp. 55 (1990) no. 192, 745763.
28.Schoof, R., ‘Elliptic curves over finite fields and the computation of square roots mod p’, Math. Comp. 44 (1985) no. 170, 483494.
29.Shimura, G., Introduction to the arithmetic theory of automorphic functions, vol. 1 (Princeton University Press, Princeton, NJ, 1971).
30.Shimura, G., Abelian varieties with complex multiplication and modular functions, vol. 46 (Princeton University Press, Princeton, NJ, 1998).
31.Spallek, A., ‘Kurven vom Geschlecht 2 und ihre Anwendung in public-key-Kryptosystemen’, PhD Thesis, Inst. für Experimentelle Mathematik, 1994.
32.Streng, M., ‘Complex multiplication of abelian surfaces’, PhD Thesis, Leiden University, June 2010,https://openaccess.leidenuniv.nl/handle/1887/15572.
33.Streng, M., ‘An explicit version of Shimura’s reciprocity law for Siegel modular functions’, CoRR, 2012, arXiv:abs/1201.0020.
34.van Wamelen, P., ‘Equations for the Jacobian of a hyperelliptic curve’, Trans. Amer. Math. Soc. 350 (1998) no. 8, 30833106.
35.van Wamelen, P., ‘Examples of genus two CM curves defined over the rationals’, Math. Comp. 68 (1999) no. 225, 307320.
36.Weng, A., ‘Constructing hyperelliptic curves of genus 2 suitable for cryptography’, Math. Comp. 72 (2003) no. 241, 435458.
37.Yang, T., ‘Arithmetic intersection on a Hilbert modular surface and the Faltings height’, Asian J. Math. 17 (2013) no. 2, 335382.
38.Yang, T., ‘Rational structure of over and explicit Galois action on CM points’, Preprint, 2014.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed