International Association for Cryptologic Research

International Association
for Cryptologic Research


Ezekiel J. Kachisa


Generating more Kawazoe-Takahashi Genus 2 Pairing-friendly Hyperelliptic Curves
Ezekiel J Kachisa
Constructing pairing-friendly hyperelliptic curves with small $\rho$-values is one of challenges for practicability of pairing-friendly hyperelliptic curves. In this paper, we describe a method that extends the Kawazoe-Takahashi method of generating families of genus $2$ ordinary pairing-friendly hyperelliptic curves by parameterizing the parameters as polynomials. With this approach we construct genus $2$ ordinary pairing-friendly hyperelliptic curves with $2 <\rho \le 3$.
Implementing cryptographic pairings: a magma tutorial
In this paper we show an efficient implementation if the Tate, ate, and R-ate pairings in magma. This will be demostrated by using the KSS curves with embedding degree k=18
Fast hashing to G2 on pairing friendly curves
When using pairing-friendly ordinary elliptic curves over prime fields to implement identity-based protocols, there is often a need to hash identities to points on one or both of the two elliptic curve groups of prime order $r$ involved in the pairing. Of these $G_1$ is a group of points on the base field $E(\F_p)$ and $G_2$ is instantiated as a group of points with coordinates on some extension field, over a twisted curve $E'(\F_{p^d})$, where $d$ divides the embedding degree $k$. While hashing to $G_1$ is relatively easy, hashing to $G_2$ has been less considered, and is regarded as likely to be more expensive as it appears to require a multiplication by a large cofactor. In this paper we introduce a fast method for this cofactor multiplication on $G_2$ which exploits an efficiently computable homomorphism.
Constructing Brezing-Weng pairing friendly elliptic curves using elements in the cyclotomic field
Ezekiel J. Kachisa Edward F. Schaefer Michael Scott
We describe a new method for constructing Brezing-Weng-like pairing-friendly elliptic curves. The new construction uses the minimal polynomials of elements in a cyclotomic field. Using this new construction we present new ``record breaking'' families of pairing-friendly curves with embedding degrees of $k \in \{16,18,36,40\}$, and some interesting new constructions for the cases $k \in \{8,32\}$