T. Shaska

My research interests are primarily on algebraic curves, weighted projective spaces and weighted heights, isogeny based cryptography and weighted homomorphic encryption, etc. In the last few years I have focused more on AI and machine learning. Below is a list of papers and preprints.

There are many versions of any of these papers online. I do NOT maintain nor update papers on http://arxiv.org/a/shaska_t_1 or other websites. Be aware that arxiv versions are not, for the most cases, the correct, published versions. Please check (and cite) the published versions.

If interested in collaborating or joining me in any research project, or find out about my recent work in homomorphic encryption, Machine learning, and post-quantum cryptography please fill the following form.  

2024 (work in progress) 

1. Machine learning for moduli space of genus two curves and an application to post-quantum cryptography

2. Automorphism groups  of rational functions

3. Machine learning and Julia reduction

4. Isogenies of Jacobian surfaces and post-quantum cryptography, NATO Series (late 2024)

5. Group actions and deep learning, Contemporary Math, 2024 (late 2024)

6. Weighted homomorphic encryption

Selected Journal articles

• S. Salami, T. Shaska: Vojta's conjecture on weighted projective varieties (submitted) 

• S. Salami, T. Shaska; Local and global heights on weighted projective varieties, Houston J. Math.(2023), Vol. 49, 3, pg. xx 

• A. Clingher, A. Malmendier, T. Shaska; Geometry of Prym varieties for special bielliptic curves of genus three and five,  Pure Appl. Math. Q. 17 (2021), no. 5, 1739–1784.

• A. Obus, T. Shaska;  Superelliptic curves with many automorphisms and CM Jacobians, Math. Comp. 90 (2021), no. 332, 2951–2975.

• A. Clingher, A. Malmendier, T. Shaska,  On isogenies among certain Abelian varieties, Michigan Mathematics Journal, 2021 (43 pages)

• A. Elezi, T. Shaska, Reduction of binary forms via the hyperbolic centroid, Lobachevskii J. Math. 42 (2021), no. 1, 84–95.

• L. Beshaj, J. Gutierrez, T. Shaska, Weighted greatest common divisors and weighted heights, J. Number Theory 213 (2020), 319–346.

• L. Beshaj, A. Elezi, T. Shaska,  Isogenous components of Jacobian surfaces , Eur. J. Math. 6 (2020), no. 4, 1276–1302.

• A. Clingher, A. Malmendier, T. Shaska, Configurations of 6 lines and string dualities, Comm. Math. Phys. 371 (2019), no. 1, 159–196.

• A. Malmendier and T. Shaska; From hyperelliptic to superelliptic curves, Albanian J. Math. Vol. 13. (2019), No. 1. 107-200.

• Shuichi Otake, Tony Shaska; Some remarks on the non-real roots of polynomials, Cubo 20, (2018), no. 2. 67-93.

• A. Malmendier and T. Shaska,  A universal genus-two curve from Siegel modular forms, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 13 (2017), 089, 17 pages

• A. Malmendier and T. Shaska,  The Satake sextic in F-theory, Journal of Geometry and Physics, vol. 120, 2017, 290-305.

• T. Shaska and C. Shor,  The q-Weierstrass points of genus 3 hyperelliptic curves with extra automorphisms, Comm. Algebra, 45 (2017), no. 5, 1879-1892.

• Genus two curves with many elliptic subcovers, Comm. Algebra 44 (2016), no. 10, 4450–4466.

• T. Shaska and C. Shor,  Theta functions and symmetric weight enumerators for codes over imaginary quadratic fields, Des. Codes Cryptogr., 76 (2015) no. 2, 217–235.

• T. Shaska,  Some remarks on the hyperelliptic moduli of genus 3, Comm. Algebra 42 (2014), no. 9, 4110–4130.

• T. Shaska and F. Thompson,  Bielliptic curves of genus 3 in the hyperelliptic moduli, Appl. Algebra Eng. Commun. Comput., 2013, 24 (5), 387-412.

• A. Elezi and T. Shaska,  Quantum codes from superelliptic curves, Albanian J. Math. vol 5, (2011), no. 4, 175-191.

• L. Beshaj, V. Hoxha, T. Shaska,  Superelliptic curves of level n and their invariants I , Albanian J. Math. Vol 5, Nr. 3, 2011.

• T. Shaska, C. Shor, S. Wijesiri,  Codes, modular lattices, and corresponding theta functions, Finite Fields Appl., 16 (2010), no. 2, 75 -- 87.

• K. Magaard, T. Shaska, H. Voelklein,  Genus 2 curves with degree 5 elliptic subcovers, Forum Math. 21 (2009), no. 3, 547–566.

• T. Shaska and V. Ustimenko,  Applications of liner algebra to the theory of algebraic graphs of large girth, Linear Algebra and Appl. 430, (2009), no. 7. 1826-1837.

• T. Shaska and V. Ustimenko,  On some applications of graphs to cryptography and turbocoding, Albanian J. Math. Vol 2, Nr. 3, 2008, 249-255.

• N. Pjero, M. Ramosaco, T. Shaska,  Degree even coverings of elliptic curves by genus two curves, Albanian J. Math, vol. 2. Nr. 3, 2008, 241-248.

• R. Sanjeeva, T. Shaska,  Determining equations of families of cyclic curves, Albanian J. Math. Vol 2, Nr. 3, 2008, 199-213.

• T. Shaska, S. Wijesiri, S. Wolf, L. Woodland,  Degree four coverings of elliptic curves by genus two curves, Albanian J. Math. vol. 2. Nr. 4. 2008, 307-315.

• T. Shaska, S. Wijesiri,  Codes over rings of size four, Hermitian lattices, and corresponding theta functions, Proc. Amer.Math. Soc., 136 (2008), no.3, 849-857.

• Some open problems in computational algebraic geometry, Albanian J. Math, vol I, Nr. 4, 2007, 297-319.

• E. Previato, T. Shaska, S. Wijesiri,  Thetanulls of cyclic curves of small genus, Albanian J. Math., vol. 1, Nr. 4, 2007, 265-282.

• T. Shaska, Q. Wang,  On the automorphism groups of AG-codes based on $C_{ab}$ curves, Serdica J. Computing, vol.1, Nr. 1, 2007, 193-206.

• D. Sevilla, T. Shaska,  Hyperelliptic curves with reduced automorphism group A_5, Appl. Algebra Engrg. Comm. Comput., vol. 18, Nr. 1-2, 2007, pg. 3-20.

• Subvarieties of the hyperelliptic moduli determined by prescribed group actions, Serdica Math. Journal, No. 4, 355-374, 2006.

• J. Gutierrez, T. Shaska,  Hyperelliptic curves with extra involutions, London Math. Soc. J. of Comp. Math., 8, (2005), 102-115.

• Some special families of hyperelliptic curves, J. Algebra Appl., 3 (2004), no. 1, 75--89.

• K. Magaard, T. Shaska, S. Shpectorov, H. Voelklein,
  The locus of curves with prescribed automorphism group, Communications in arithmetic fundamental groups (Kyoto, 1999/2001). Sūrikaisekikenkyūsho Kōkyūroku, No. 1267 (2002), 112–141.

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

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

Proceedings articles

1. T. Shaska, Reduction of superelliptic Riemann surfaces Contemporary Math, 2021 (to appear).

2. G. Frey and T. Shaska, Curves, Jacobians, and Cryptography Contemporary Math. vol. 724, AMS (2019), pg. 279-345.

3. A. Broughton, A. Wootton, T. Shaska; On automorphisms of algebraic curves Contemporary Math. vol. 724, AMS (2019), pg. 175-212.

4. Jorgo Mandili and Tony Shaska, Computing heights on weighted projective spaces Contemporary Math. vol. 724, AMS (2019), pg. 149-160.

5. Shuichi Otake, Tony Shaska, On the discriminant of a certain quadrinomials Contemporary Math. vol. 724, AMS (2019), pg. 55-72.

6. D. Joyner, T. Shaska, Self-inversive polynomials, curves, and codes Contemporary Math., AMS, (2018), vol. 703, pg. 189-208.

7. L. Beshaj, R. Hidalgo, A. Malmendier, S. Kruk, S. Quispe, T. Shaska, Rational points on the moduli space of genus two, Contemporary Math., AMS, (2018), vol. 703, pg. 83-115.

8. R. Hidalgo, T. Shaska, On the field of moduli of superelliptic curves Contemporary Math., AMS, (2018), vol. 703, pg. 47-62.

9. L. Beshaj, A. Elezi, T. Shaska, Theta functions of superelliptic curves , NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 24, 2015.

10. A. Elezi and T. Shaska, Weight distributions, zeta functions and Riemann hypothesis for linear and algebraic geometry codes,, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 24, 2015.

11. T. Shaska, C. Shor, Weierstrass points of superelliptic curves, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 24, 2015.

12. L. Beshaj, T. Shaska, E. Zhupa, The case for superelliptic curves NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 24, 2015.

13. L. Beshaj, T. Shaska, C. Shor, On Jacobians of curves with superelliptic components, Contemporary Math, Vol 629. pg. 3-15.

14. M. Izquierdo and T. Shaska, Cyclic curves and their automorphisms , NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 24, 2015. 

15. L. Beshaj and T. Shaska, Heights on algebraic curves , NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 24, 2015.

16. L. Beshaj and T. Shaska, Decomposition of some Jacobian varieties of dimension 3 , Artificial Intelligence and Symbolic Computation, LNCS vol. 8884, 193-204.

17. L. Beshaj and T. Shaska, The arithmetic of genus 2 curves , NATO ASI, Croatia 2010, ISO Press.

18. T. Shaska and G. Wijesiri, Theta functions and algebraic curves with automorphisms , New Challenges in digital communications, NATO Advanced Study Institute, 2009, pg. 193-237.

19. T. Shaska, S. Zheng, A Maple package for hyperelliptic curves , Maple Conference 2005, 399-408.

20. J. Gutierrez, T. Shaska, D. Sevilla, Hyperelliptic curves of genus 3 and their automorphisms , Lect. Notes Comp., vol 13. (2005), 109--123.

21. T. Shaska and C. Shor, Codes over $F_{p^2}$ and $F_p \times F_p$, Hermitian lattices, and corresponding theta functions Advances in Coding Theory and Cryptology, vol 3. (2007), pg. 70-80.

22. V. Krishnamoorthy, T. Shaska, H. Voelklein, Invariants of binary forms , Dev. in Math., vol 12, pg.101-122, Springer, 2004.

23. T. Shaska, Genus 2 curves covering elliptic curves, a computational approach , Lect. Notes in Comp, vol 13. (2005), 151-195.

24. A. Bialostocki, T. Shaska, Galois group of prime degree polynomials with non-real roots , Lect. Notes in Computing, 13, 2005, 243--255.

25. T. Shaska, Computational algebra and algebraic curves, ACM, SIGSAM Bulletin, Comm. Comp. Alg.,vol. 37, No. 4,117-124, 2003.

26. T. Shaska, J. Thompson, On the generic curve of genus 3 , Contemporary. Math., vol. 369, pg. 233-244, AMS, 2005.

27. T. Shaska, Determining the automorphism group of a hyperelliptic curve , ISSAC 05, 248--254, ACM, New York, 2003.

28. T. Shaska, Computational Aspects of Hyperelliptic Curves , Lecture Notes Ser. Comput., 10, 248--257, World Sci. Publishing, River Edge, NJ.

29. T. Shaska, Genus 2 curves with $(3,3)$-split Jacobian and large automorphism group. Algorithmic number theory (Sydney, 2002), 205--218, Lecture Notes in Comput. Sci., 2369, Springer, Berlin, 2002.

30. T. Shaska and H. Voelklein, Elliptic subfields and automorphisms of genus 2 function fields , Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000),703--723, Springer, Berlin, 2004.

Others

• Donagi, Ron; Shaska, Tony
  Integrable systems: a celebration of Emma Previato’s 65th birthday. Volume 1. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 458, 1-12 (2020).

• Donagi, Ron; Shaska, Tony
  Algebraic Geometry: a celebration of Emma Previato’s 65th birthday. Volume 2. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 459, 1-12 (2020).

• G. Hiss and T. Shaska
  Kay Magaard (1962--2018), Special issue in honor of Kay Magaard, Albanian J. Math. Vol. 12, (2018), no. 1, 33-35.

• B. Shaska, T. Shaska,
  Mësimdhënia e matematikës nëpërmjet problemeve klasike, Albanian J. Math., vol. 10, (2016), no. 1, 47-80.

• T. Shaska, Computational algebraic geometry J. Symbolic Comput. 57 (2013), 1–2.

• T. Shaska, Computational algebraic geometry and its applications Appl. Algebra Engrg. Comm. Comput. 24 (2013), no. 5, 309–311.

• T. Shaska, Quantum codes from algebraic curves with automorphisms. Condensed Matter Physics, 2008, Vol. 11, No 2 (54), 383-396.

• T. Shaska, M. Qarri Algebraic aspects of digital communications. Albanian J. Math. 2 (2008), no. 3, 141–144.

• A. Elezi, T. Shaska, Special issue on algebra and computational algebraic geometry Albanian J. Math. 1 (2007), no. 4, 175–177.

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

Highlights

• Computation of loci of genus 2 curves with (n,n)-split Jacobians. Thesis (University of Florida), 2001

• Discovering dihedral invariants for hyperelliptic curves with involutions (2001-2005)

• Discovering weighted heights (2015-2020)

• From hyperelliptic curves to superelliptic curves