## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 22, Number 4 (2015), 589-602.

### Bisection for genus 2 curves with a real model

Josep M. Miret, Jordi Pujolàs, and Nicolas Thériault

#### Abstract

Integer multiplication in Jacobians of genus $2$ curves over a finite field $\mathbb{F}_q$ is a fundamental operation for hyperelliptic curve cryptography. Algorithmically, the result of this operation is given by the very well known algorithms of Cantor. One method to reverse duplication in these cases consists in associating, to every preimage of the desired doubled divisor defined over $\mathbb{F}_q$, a root in $\mathbb{F}_q$ of the so called bisection polynomial. We generalize this approach to genus $2$ curves with two points at infinity, both in even and odd characteristics. We attach a bisection polynomial to each possible type of Mumford coordinate, we show the factorization of these in terms of Galois orbits in the set of bisections, and we compare the efficiency of our approach versus brute-force adaptations of the existing methods to our setting.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 22, Number 4 (2015), 589-602.

**Dates**

First available in Project Euclid: 18 November 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1447856061

**Digital Object Identifier**

doi:10.36045/bbms/1447856061

**Mathematical Reviews number (MathSciNet)**

MR3429173

**Zentralblatt MATH identifier**

1345.14033

**Subjects**

Primary: 11Y40: Algebraic number theory computations 11G20: Curves over finite and local fields [See also 14H25] 14H45: Special curves and curves of low genus 11T71: Algebraic coding theory; cryptography 14G50: Applications to coding theory and cryptography [See also 94A60, 94B27, 94B40]

**Keywords**

Halving algorithm genus 2 curves real model finite fields

#### Citation

Miret, Josep M.; Pujolàs, Jordi; Thériault, Nicolas. Bisection for genus 2 curves with a real model. Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 4, 589--602. doi:10.36045/bbms/1447856061. https://projecteuclid.org/euclid.bbms/1447856061