International Association
for Cryptologic Research

Finding isogenies between supersingular elliptic curves is a natural algorithmic problem which is known to be equivalent to computing the curves' endomorphism rings. When the isogeny is additionally required to have a specific known degree $d$, the problem appears to be somewhat different in nature, yet its hardness is also required in isogeny-based cryptography. Let $E_1,E_2$ be supersingular elliptic curves over $\mathbb{F}_{p^2}$. We present improved classical and quantum algorithms that compute an isogeny of degree $d$ between $E_1$ and $E_2$ if it exists. Let $d \approx p^{1/2+ \epsilon}$ for some $\epsilon>0$. Our essentially memory-free algorithms have better time complexity than meet-in-the-middle algorithms, which require exponential memory storage, in the range $1/2\leq\epsilon\leq 3/4$ on a classical computer. For quantum computers, we improve the time complexity in the range $0<\epsilon<5/2$. Our strategy is to compute the endomorphism rings of both curves, compute the reduced norm form associated to $\Hom(E_1,E_2)$ and try to represent the integer $d$ as a solution of this form. We present multiple approaches to solving this problem which combine guessing certain variables exhaustively (or use Grover's search in the quantum case) with methods for solving quadratic Diophantine equations such as Cornacchia's algorithm and multivariate variants of Coppersmith's method. For the different approaches, we provide implementations and experimental results. A solution to the norm form can then be efficiently translated to recover the sought-after isogeny using well-known techniques. As a consequence of our results we show that a recently introduced signature scheme from~\cite{BassoSIDHsign} does not reach NIST level I security.

Functional encryption is a new paradigm for encryption where decryption does not give the entire plaintext but only some function of it. Functional encryption has great potential in privacy-enhancing technologies but suffers from excessive computational overheads. We introduce the first hardware accelerator that supports functional encryption for quadratic functions. Our accelerator is implemented on a reprogrammable system-on-chip following the hardware/software codesign methogology. We benchmark our implementation for two privacy-preserving machine learning applications: (1) classification of handwritten digits from the MNIST database and (2) classification of clothes images from the Fashion MNIST database. In both cases, classification is performed with encrypted images. We show that our implementation offers speedups of over 200 times compared to a published software implementation and permits applications which are unfeasible with software-only solutions.