AIMS - Hyperelliptic Isogeny Based Cryptography

Hyperelliptic Isogeny Based Cryptography

Isogeny-based cryptography is one of the leading directions in post-quantum cryptography. It relies on isogenies—special morphisms between abelian varieties that preserve their algebraic structure. Early cryptosystems focused on elliptic curves, giving rise to well-known constructions such as SIDH and CSIDH. In recent years, however, attention has turned to higher-genus analogues, especially Jacobians of hyperelliptic curves of genus two and three, which provide richer moduli spaces, larger endomorphism rings, and new avenues for secure cryptographic protocols.

A particularly active area of research involves (n,n)-isogenies between Jacobians of genus-two curves. A Jacobian J(C) is said to be (n,n)-split if it is isogenous to a product of two elliptic curves via an isogeny of type (n,n). The moduli of such Jacobians form an irreducible, two-dimensional subvariety inside the moduli space of genus-two curves. Geometrically, this surface corresponds to a Humbert surface of discriminant n^2 in the Siegel modular threefold—one of the most classical and beautiful objects in algebraic geometry.

The first explicit computation of these loci for n>2 was carried out in Tony Shaska’s doctoral thesis, Genus 2 Curves Covering Elliptic Curves (University of Florida, 2001). This work provided explicit equations for the corresponding Humbert surfaces, describing all genus-two curves whose Jacobians are (n,n)-split. These results opened the way for modern studies of higher-genus isogenies and remain a foundational reference in the field.

Today, this geometric framework underpins the development of hyperelliptic isogeny-based cryptosystems, extending the ideas of elliptic isogeny cryptography to more complex moduli spaces. The structure of the (n,n)-split locus determines the shape of isogeny graphs, the endomorphism rings of the underlying abelian varieties, and the computational assumptions that define the security of next-generation protocols. From both a mathematical and cryptographic perspective, this line of research continues to bridge deep geometry with practical post-quantum cryptography.

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.

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