Paper 2026/2283 Original: eprint.iacr.org/2026/2283 ↗

A Kleptographic Attack on CSIDH

Abstract

This note reports an attack scenario for CSIDH that does not appear to have been considered in the literature. A subverted device of Alice can leak a shared curve between Alice and Bob through Alice's public curves in later CSIDH sessions. The attack adapts the Young--Yung kleptographic attack from classical Diffie--Hellman to CSIDH using the Goldreich--Levin theorem and rejection sampling. We prove that a CSIDH public curve $[\mathfrak a]\star E_0$ remains computationally indistinguishable from an honest public curve even when its secret class $[\mathfrak a]$ is rejection-sampled according to a hard-core bit of a hidden parallelization curve $[\mathfrak a][\mathfrak c]\star E_0$ formed by the same class $[\mathfrak a]$ together with an independent curve $[\mathfrak c]\star E_0$. The proofs rely only on the computational assumptions underlying CSIDH itself. We hope that this note provides a useful starting point for the study of algorithm-substitution attacks in isogeny-based cryptography, complementing existing work on side-channel and fault attacks.