Hyperelliptic Isogeny Based Cryptography

Isogeny based cryptography is a cryptosystem that uses isogenies between Abelian varieties.  While the cryptosystems that use isogenies among elliptic curves have become part of the folklore, we focus on isogenies of genus two curves and hyperelliptic genus 3 curves.

Since most of our activities are related to grants and involvement in industry, we publish relatively little on the topic and only with permission from our partners.  If you are a mathematician or computer scientist interested in getting involved, please fill the following form 

Activities

• Isogeny based post-quantum cryptography, NATO-Science for Peace and Security, Advanced Research Workshops, Hebrew University of Jerusalem, July 29-31, 2024

Literature and some recent preprints

• Curves, Jacobians, and cryptography, G. Frey, T. Shaska, Contemp. Math, vol. 724,   AMS,  (2019), pg. 279--344

• Machine learning for moduli space of genus two curves and an application to post-quantum cryptograph, (submitted)

• Isogenies of genus 2 Jacobians and cryptography, NATO D-series, (to appear)

Timeline on (n,n)-split Jacobians

• Jacobi, Anzeige von Legendre, Theorie des fonetions elliptiques. Troi- sieme Supplement. 1832. 

• Review of  Legendre, Th\'eorie des fonctions elliptiques. Troiseme suppl\'em ent. 1832. J. reine angew. Math. 8,  413-417.

• Jacobi; Ges. Werke Bd. 1. Berlin 1881, pag. 373; 

• Kotänyi, Zur Reduction hypereUiptischer Integrale. Wiener Sitzb. Bd. 88. 1883, Abth. IT, pag. 401.

• Brioschi, Sur la reduction de l'integrale hyperelliptique ä l'elliptique par une transformation du troisieme degre. Ann. de TEc. norm. sup. (3) Bd. 8. 1891, pag. 227.

• Bolza, Zur Reduction hyperelliptischer Integrale erster Ordnung auf elliptische mittelst einer Transformation dritten Grades. Math. Ann. Bd. 50. 1898, pag. 314

• Bolza,  Zur Reduction hyperelliptischer Integrale erster Ordnung auf elliptische mittelst einer Transformation dritten Grades. Nachtrag. Math. Ann. Bd. 51. 1899, pag. 478.

  • • Hayashida, Tsuyoshi; Nishi, Mieo; Existence of curves of genus two on a product of two elliptic curves. J. Math. Soc. Japan   17 (1965), 1–16.

• Kuhn M. R.: Curves of genus 2 with split Jacobian. Trans. Amer. Math. Soc. 307 (1988), 41--49

  • • G. Frey and E. Kani, Curves of genus 2 covering elliptic curves and an arithmetic application. Arithmetic algebraic geometry (Texel, 1989), 153-176, Progr. Math., 89, Birkh¨auser Boston, MA, 1991.

    • G. Frey, On elliptic curves with isomorphic torsion structures and corresponding curves of genus 2. Elliptic curves, modular forms, and Fermat’s last theorem (Hong Kong, 1993), 79-98, Ser. Number Theory, I, Internat. Press, Cambridge, MA, 1995.

• Shaska, T. Curves of genus 2 with (N,N) decomposable Jacobians. J. Symbolic Comput. 31 (2001), no. 5, 603--617.

• Shaska, T.:   Curves of genus two covering elliptic curves. Thesis (Ph.D.)--University of Florida. 2001. 72 pp. ISBN: 978-0493-20012-5, ProQuest LLC

• Shaska, Tanush; Völklein, Helmut Elliptic subfields and automorphisms of genus 2 function fields. Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), 703--723, Springer, Berlin, 2004.

• Shaska, T. Genus 2 fields with degree 3 elliptic subfields. Forum Math. 16 (2004), no. 2, 263--280.

• Magaard, Shaska, V\"olklein; Genus 2 curves that admit a degree 5 map to an elliptic curve. Forum Math. 21 (2009), no. 3, 547--566.

• Shaska, T. Genus two curves covering elliptic curves: a computational approach. Computational aspects of algebraic curves, 206--231, Lecture Notes Ser. Comput., 13, World Sci. Publ., Hackensack, NJ, 2005.  

• Kumar, A.; Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields. Res. Math. Sci. 2 (2015), Art. 24, 46 pp.