Francisco Rodríguez‐Henríquez

Gora Adj, Jesús-Javier Chi-Domínguez, Víctor Mateu et al.

In SIDH and SIKE protocols, public keys are defined over quadratic extensions of prime fields. We present in this work a projective invariant property characterizing affine Montgomery curves defined over prime fields. We then force a secret 3-isogeny chain to repeatedly pass through a curve defined over a prime field in order to exploit the new property and inject zeros in the A-coefficient of an intermediate curve to successfully recover the isogeny chain one step at a time. Our results introduce a new kind of fault attacks applicable to SIDH and SIKE.

Jesús-Javier Chi-Domínguez, Francisco Rodríguez-Henríquez, Benjamin Smith

Let $q = 2^n$, and let $E / \mathbb{F}_{q^{\ell}}$ be a generalized Galbraith--Lin--Scott (GLS) binary curve, with $\ell \ge 2$ and $(\ell, n) = 1$.We show that the GLS endomorphism on $E / \mathbb{F}_{q^{\ell}}$ induces an efficient endomorphism on the Jacobian $J_H(\mathbb{F}_q)$ of the genus-$g$ hyperelliptic curve $H$ corresponding to the image of the GHS Weil-descent attack applied to $E/\mathbb{F}_{q^\ell}$, and that this endomorphism yields a factor-$n$ speedup when using standard index-calculus procedures for solving the Discrete Logarithm Problem (DLP) on $J_H(\mathbb{F}_q)$. Our analysis is backed up by the explicit computation of a discrete logarithm defined on a prime-order subgroup of a GLS elliptic curve over the field $\mathbb{F}_{2^{5\cdot 31}}$. A Magma implementation of our algorithm finds the aforementioned discrete logarithm in about $1,035$ CPU-days.