Reseach Interests & Publications

Research Interests

• • Algebraic number theory: Ideal lattices, well-rounded ideal lattices, unit groups of number fields, the size function of a number field, and elliptic curve isogenies.

  • Applying ideal lattices and isogenies in cryptography and coding theory.

Publications​​​ 

1. Principal well-rounded ideals of quadratic fields and Pell equations, with Morgan Smith, in preparation.

2. Rigorous Certification of Fundamental Units in Number Fields, with Randy Yee, Felix Fontein, and Michael J. Jacobson, Jr., in preparation.

3. Fast arithmetic in the divisor class group, with Jens Dietrich Bauch and Sumin Leem, in preparation.

4. Orientations and cycles in supersingular isogeny graphs, with  Sarah Arpin, Mingjie Chen, Kristin E. Lauter, Renate Scheidler, Katherine E. Stange, In: Bucur, A., Ho, W., Scheidler, R. (eds) Research Directions in Number Theory. Association for Women in Mathematics Series, vol 33. Springer, Cham. 

5. Reduced ideals from the reduction algorithm, with Peng Tian,  Results in Mathematics, Volume 79, article number 50, (2024).

6. The size function for imaginary cyclic sextic fields, with Peng Tian and Amy Feaver, Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 3, pp. 841-866.

7. Computing mod $\ell$- Galois representations associated to modular forms for small primes, with Peng Tian and  Dung Duong, AIMS Mathematics 2023, Volume 8, Issue 12: 28766-28779.

8. Well-rounded ideal lattices of cyclic cubic and quartic fields, with Nam H. Le, Dat T. Tran,  Communications in Mathematics, October 18, 2023, Volume 31 (2023), Issue 2 (Special issue: Euclidean lattices: theory and applications). 

9. Orienteering with one endomorphism, with  Sarah Arpin, Mingjie Chen, Kristin E. Lauter, Renate Scheidler, Katherine E. Stange, La Matematica (2023).  

10. Well-rounded twists of ideal lattices from imaginary quadratic fields, with Nam H. Le, Dat T. Tran, Journal of Algebra and Its Applications, Vol. 21, No. 07, 2250133 (2022). pdf 

11. An efficient multivariate threshold ring signature scheme, with Dung Hoang Duong, Willy Susilo and Le Van Luyen, Computer Standards & Interfaces 74 (2021): 103489.    pdf

12. Lattice Blind Signatures with Forward Security, with Huy Quoc Le, Dung Hoang Duong, Willy Susilo, Viet Cuong Trinh, Josef Pieprzyk, and Thomas Plantard, In Australasian Conference on Information Security and Privacy 2020, Nov 30 (pp. 3--22). Springer, Cham.     pdf

13. A multivariate blind ring signature scheme, with Dung H. Duong and Willy Susilo, The Computer Journal, Volume 63, Issue 8, August 2020, Pages 1194–1202.  pdf

14. Choosing subfields for LUOV and LRainbow Signature Scheme, with Dung Hoang Duong and Le Van Luyen, IET Information Security, 2019.    pdf     (errata)

15. Computation of triangular integral bases, with Jens Dietrich Bauch, THE OPEN BOOK SERIES 2 (2019), Thirteenth Algorithmic Number Theory Symposium.

16. The size function for cyclic cubic fields, with Peng Tian,  Int. J. Number Theory 14, 399 (2018).   pdf  (errata)

17. The size function of quadratic extensions of complex quadratic fields, Journal de théorie des nombres de Bordeaux, 29 no. 1 (2017), p. 243-259.      pdf 

18. Computing dimensions of spaces of Arakelov divisors of number fields, Int. J. Number Theory, March 2017, Vol. 13, No. 02:  pp. 487-512  pdf

19. Well-Rounded Lattices for Coset Coding in MIMO Wiretap Channels, with Oliver W. Gnilke, Amaro Barreal, Alex Karrila, David A. Karpuk and Camilla Hollanti, 26th International Telecommunication Networks and Applications Conference, ITNAC 2016., p. 289-294. pdf 

20. Well-Rounded Lattices for Reliability and Security in Rayleigh Fading SISO Channels, with O. W. Gnilke,  A. Karrila and C. Hollanti, Information Theory Workshop (ITW), 2016 IEEE, 359-363. pdf

21.  On reduced Arakelov divisors of real quadratic fields, Acta Arith. 3483 (2016), 297-315. pdf 

22. A generalization of reduced Arakelov divisors of a number field,  Journal of Number Theory 167 (2016) 104 - 117.        pdf   (errata)