The Supersingular Isogeny Problem in Time and Memory $p^{1/3+o(1)}$, Unconditionally
An unconditional algorithm solves the supersingular isogeny problem in p to the 1/3-plus-little-o time and memory.
The paper gives an unconditional Las Vegas algorithm for the supersingular OneEnd problem: finding a non-scalar endomorphism of a supersingular elliptic curve over the field with p squared elements. Known reductions mean the algorithm also solves the supersingular endomorphism-ring and isogeny problems. Expected time and memory are p to the 1/3 times a subexponential factor, equivalently p to the 1/3 plus little-o of 1, improving the previous unconditional exponent of 2/5 without a smoothness heuristic. It uses predetermined smooth degrees, random-walk collisions, split-degree list matching, and composition with Frobenius.
- Solving OneEnd also solves endomorphism-ring and supersingular isogeny problems.
- Expected time and memory are p to the 1/3 plus little-o of 1.
- The result improves the prior unconditional exponent of 2/5.
- No smoothness heuristic is required for the Las Vegas analysis.
From the original article
1 min read at arxiv.orgGiven a supersingular elliptic curve $E/\mathbb{F}_{p^2}$, the $\mathsf{OneEnd}$ problem asks for a non-scalar endomorphism of $E$. By known reductions, solving this problem also solves the supersingular endomorphism ring and isogeny problems. Wesolowski obtained exponent $1/3$ under an assumption on the factorization of a small degree, whereas the previous unconditional exponent was $2/5$.
By José Luis Delgado · arxiv.org
A short excerpt; the article and its images belong to the publisher. Summary, key facts and assessment above are AI-written from the article and can be wrong.