Isogenies

Elliptic curves were introduced to the ancient science of cryptography in the mid 1980s, and Elliptic Curve Cryptography (ECC) has since been growing rapidly. However, owing to the incompleteness of the Weierstrass addition law, elliptic curve cryptosystems based on the Weierstrass model are vulnerable to side-channel attacks. New addition algorithms and elliptic curve models have been proposed to take elliptic curve cryptosystems resistant to side-channel attacks. A promising model in this regard is the Edwards model introduced in 2007. The Edwards addition law is both complete and has the fastest known implementations for elliptic curve operations like addition and doubling. As a part of this work we study the Edwards model in relation to ECC with an emphasis on its computational aspects. We also study two encoding schemes, Elligator and Elligator Square, for representing elliptic curve points as bit strings indistinguishable from uniform random bit strings, both of which have formulations over Edwards curves. We also study isogenies and their computation using analogues of V ́elu’s and Kohel’s formulas for the Edwards model, which turn out to be simpler and more efficient than those for the Weierstrass model. We implement an hitherto unavailable library for Edwards curves, and two ECC algorithms using the implemented Edwards curves, in the mathematical software Sage.

In 2005, Jao, Miller, and Venkatesan proved that the DLP of elliptic curves with the same endomorhism ring is random reducible under the GRH. In this talk, we discuss a possible generalization of this result to hyperelliptic curves of genus 2 (and 3) defined over a finite field and show the difficulties involved. First, we explain the role of the endomorphism rings of the Jacobian and the polarization. Following the work of Jao, Miller and Venkatesan, we construct isogeny graphs for genus 2 curves. Specifically, we discuss the connection between isogenies and ideal classes in the Jacobian of these curves. This project is research in progress and we describe the current status of this research.

Abstract. The Gallant-Lambert-Vanstone (GLV) algorithm uses efficiently computable endomorphisms to accelerate the computation of scalar multiplication of points on an abelian variety. Freeman and Satoh proposed for cryptographic use two families of genus 2 curves defined over Fp which have the property that the corresponding Jacobians are (2, 2)isogenous over an extension field to a product of elliptic curves defined over F p 2. We exploit the relationship between the endomorphism rings of isogenous abelian varieties to exhibit efficiently computable endomorphisms on both the genus 2 Jacobian and the elliptic curve. This leads to a four-dimensional GLV method on Freeman and Satoh’s Jacobians and on two new families of elliptic curves defined over F p 2.