Machine learning for moduli space of genus two curves and an application to isogeny based cryptography

E. Shaska and T. Shaska; Machine learning for moduli space of genus two curves and an application to isogeny based cryptography,   Journal of Algebraic Combinatorics

code/: deg3F.ipynb

data/: deg-3-height-4.sobj deg-3-w-heigh-4.sobj

References

[1] M. Beltrametti and L. Robbiano, Introduction to the theory of weighted projective spaces,  Expositiones Mathematicae 4 (1986), 111–162. ↑4

[2] L. Beshaj, A. Elezi, and T. Shaska, Isogenous components of Jacobian surfaces, Eur. J. Math. 6 (2020), no. 4, 1276–1302. MR4185170 ↑13

[3] L. Beshaj, J. Gutierrez, and T. Shaska, Weighted greatest common divisors and weighted heights, J. Number Theory 213 (2020), 319–346. MR4091944 ↑4, 8, 13

[4] L. Beshaj, R. Hidalgo, S. Kruk, A. Malmendier, S. Quispe, and T. Shaska, Rational points in the moduli space of genus two, Higher genus curves in mathematical physics and arithmetic geometry, 2018, pp. 83–115. MR3782461 ↑7, 10, 14

[5] A. Clingher, A. Malmendier, and T. Shaska, Six line configurations and string dualities, Comm. Math. Phys. 371 (2019), no. 1, 159–196. MR4015343 ↑2

[6] , Geometry of Prym varieties for certain bielliptic curves of genus three and five, Pure Appl. Math. Q. 17 (2021), no. 5, 1739–1784. MR4376094 ↑2

[7] , On isogenies among certain Abelian surfaces, Michigan Math. J. 71 (2022), no. 2,  227–269. MR4484238 ↑2

[8] Elira Curri, On the stability of binary forms and their weighted heights, Albanian J. Math. 16 (2022), no. 1, 3–23. MR4448533 ↑14

[9] E. V. Flynn and Yan Bo Ti, Genus two isogeny cryptography, Post-quantum cryptography, 2019, pp. 286–306. MR3989010 ↑11

[10] Gerhard Frey and Tony Shaska, Curves, Jacobians, and cryptography, Algebraic curves and their applications, 2019, pp. 279–344. MR3916746 ↑5

[11] Jun-ichi Igusa, Arithmetic variety of moduli for genus two, Ann. of Math. (2) 72 (1960), 612–649. MR0114819 ↑8

[12] , On Siegel modular forms of genus two, Amer. J. Math. 84 (1962), 175–200. MR0141643 ↑9

[13] , On Siegel modular forms genus two. II, Amer. J. Math. 86 (1964), 392–412. MR0168805 ↑9

[14] Robert M. Kuhn, Curves of genus 2 with split Jacobian, Trans. Amer. Math. Soc. 307 (1988), no. 1, 41–49. MR936803 ↑7, 10

[15] Abhinav Kumar, Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields, Res. Math. Sci. 2 (2015), Art. 24, 46. MR3427148 ↑2

[16] K. Magaard, T. Shaska, and H. V¨olklein, Genus 2 curves that admit a degree 5 map to an elliptic curve, Forum Math. 21 (2009), no. 3, 547–566. MR2526800 ↑2, 7, 10, 11, 12, 13

[17] A. Malmendier and T. Shaska, The Satake sextic in F-theory, J. Geom. Phys. 120 (2017), 290–305. MR3712162 ↑2

[18] , A universal genus-two curve from Siegel modular forms, SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), Paper No. 089, 17. MR3731039 ↑8, 9, 19

[19] , From hyperelliptic to superelliptic curves, Albanian J. Math. 13 (2019), no. 1, 107–200. MR3978315 ↑8, 9, 14, 18

[20] Andrew Obus and Tanush Shaska, Superelliptic curves with many automorphisms and CM Jacobians, Math. Comp. 90 (2021), no. 332, 2951–2975. MR4305376 ↑2

[21] Stuart Russell and Peter Norvig, Artificial intelligence. A modern approach, Englewood Cliffs, NJ: Prentice-Hall International, 1995 (English). ↑20, 21

[22] Sajad Salami and Tony Shaska, Local and global heights on weighted projective varieties, Houston J. Math. 49 (2023), no. 3, 603–636 (English). ↑2, 4, 15

[23] Maria Corte-Real Santos, Craig Costello, and Sam Frengley, An algorithm for efficient detection of (n,n)-splittings and its application to the isogeny problem in dimension 2, 2022. https://eprint.iacr.org/2022/1736. ↑2, 20

[24] E. Shaska and T. Shaska, Geometric learning for weighted spaces, 2025. in progress. ↑27

[25] E. Shaska and T. Shaska, Isogeny based cryptography on abelian surfaces, Quantum computation and quantum security, 2025, pp. xii+375. ↑20

[26] , Weighted projective clustering, 2025. in preparation. ↑27

[27] T. Shaska, Curves of genus 2 with (N,N) decomposable Jacobians, J. Symbolic Comput. 31 (2001), no. 5, 603–617. MR1828706 ↑2, 7, 9, 10, 12

[28] T. Shaska, Curves of genus two covering elliptic curves, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)–University of Florida. MR2701993 ↑2, 7, 10

[29] T. Shaska, Genus 2 fields with degree 3 elliptic subfields, Forum Math. 16 (2004), no. 2, 263–280. MR2039100 ↑2, 7, 10, 11, 13, 17

[30] T. Shaska, Genus two curves with many elliptic subcovers, Comm. Algebra 44 (2016), no. 10, 4450–4466. MR3508311 ↑2, 9, 10, 11, 13

[31]  T. Shaska, Artificial neural networks on graded vector spaces (2025), available at 2407.19031. ↑27

[32] T. Shaska and L. Beshaj, The arithmetic of genus two curves, Information security, coding theory and related combinatorics, 2011, pp. 59–98. MR2963126 ↑8

[33] T. Shaska and H. V¨olklein, Elliptic subfields and automorphisms of genus 2 function fields, Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), 2004, pp. 703–723. MR2037120 ↑2, 7, 9, 10, 17