## Involve: A Journal of Mathematics

• Involve
• Volume 12, Number 5 (2019), 755-765.

### The number of rational points of hyperelliptic curves over subsets of finite fields

#### Abstract

We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of quadratic residues and nonresidues in the image of such subsets over uniformly random hyperelliptic curves of given degrees. We find a constant probability of such a high difference and show the existence of sets with an exceptionally large discrepancy.

#### Article information

Source
Involve, Volume 12, Number 5 (2019), 755-765.

Dates
Received: 19 January 2018
Revised: 21 June 2018
Accepted: 28 July 2018
First available in Project Euclid: 29 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.involve/1559095403

Digital Object Identifier
doi:10.2140/involve.2019.12.755

Mathematical Reviews number (MathSciNet)
MR3954294

Zentralblatt MATH identifier
07072552

#### Citation

Nelson, Kristina; Solymosi, József; Tom, Foster; Wong, Ching. The number of rational points of hyperelliptic curves over subsets of finite fields. Involve 12 (2019), no. 5, 755--765. doi:10.2140/involve.2019.12.755. https://projecteuclid.org/euclid.involve/1559095403

#### References

• L. Carlitz, “The arithmetic of polynomials in a Galois field”, Amer. J. Math. 54:1 (1932), 39–50.
• H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, and F. Vercauteren (editors), Handbook of elliptic and hyperelliptic curve cryptography, Chapman & Hall, Boca Raton, FL, 2006.
• H. Hasse, “Zur Theorie der abstrakten elliptischen Funktionenkörper, III: Die Struktur des Meromorphismenrings, die Riemannsche Vermutung”, J. Reine Angew. Math. 175 (1936), 193–208.
• F. Hess, G. Seroussi, and N. P. Smart, “Two topics in hyperelliptic cryptography”, pp. 181–189 in Selected areas in cryptography (Toronto, 2001), edited by S. Vaudenay and A. M. Youssef, Lecture Notes in Comput. Sci. 2259, Springer, 2001.
• C. Pelekis and J. Ramon, “Hoeffding's inequality for sums of dependent random variables”, Mediterr. J. Math. 14:6 (2017), art. id. 243.
• J. M. Pollard, “A Monte Carlo method for factorization”, Nordisk Tidskr. Informationsbehandling 15:3 (1975), 331–334.
• R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems”, Comm. ACM 21:2 (1978), 120–126.
• T. Satoh, “Generating genus two hyperelliptic curves over large characteristic finite fields”, pp. 536–553 in Advances in cryptology: EUROCRYPT 2009 (Cologne, 2009), edited by A. Joux, Lecture Notes in Comput. Sci. 5479, Springer, 2009.
• J. P. Schmidt, A. Siegel, and A. Srinivasan, “Chernoff–Hoeffding bounds for applications with limited independence”, SIAM J. Discrete Math. 8:2 (1995), 223–250.
• I. A. Semaev, “Evaluation of discrete logarithms in a group of $p$-torsion points of an elliptic curve in characteristic $p$”, Math. Comp. 67:221 (1998), 353–356.