Improved Algorithms for Finding Fixed-Degree Isogenies Between Supersingular Elliptic Curves

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 E1, E2 be supersingular elliptic curves over F p 2 . We present improved classical and quantum algorithms that compute an isogeny of degree d between E1 and E2 if it exists. Let d ≈ p 1/2+ϵ for some ϵ > 0. Our essentially memory-free algorithms have better time complexity than meetin-the-middle algorithms, which require exponential memory storage, in the range 1/2 ≤ ϵ ≤ 3/4 on a classical computer. For quantum computers, we improve the time complexity in the range 0 < ϵ < 5/2. Our strategy is to compute the endomorphism rings of both curves, compute the reduced norm form associated to Hom(E1, E2) 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 [3] does not reach NIST level I security.