Given 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 $2/5$. We give a Las Vegas algorithm, analyzed without a smoothness heuristic, with expected time and memory \[ p^{1/3}\exp\bigl(O(\sqrt{\log p\,\log\log p})\bigr) = p^{1/3+o(1)}. \] The algorithm fixes in advance a family of degrees that are products of small primes. Known counting results provide many isogenies of these degrees from curves to their Frobenius conjugates, and a collision estimate shows that the isogenies occur on sufficiently many distinct curves for a random walk to reach one of them. From such a curve, the algorithm splits a degree into two parts, enumerates two lists of shorter isogenies, and matches their targets to obtain an isogeny to the conjugate, whose composition with Frobenius gives the required endomorphism.
Let $M=\widetilde H^*(MO(2);\mathbb F_2)$ and let $\mathcal A(1)$ be the subalgebra of the mod $2$ Steenrod algebra generated by $\operatorname{Sq}^1$ and $\operatorname{Sq}^2$. In this work, we determine the quotient $\mathbb F_2\otimes_{\mathcal A(1)}M$ in every internal degree $h\geq0$, where internal degree $h$ cor...
In this paper, we consider the problem of refuting $\mathbb{F}_2$-linear equations with random right-hand sides. Formally, we give a sub-exponential $2^{O(n/\log n)}$-time randomized algorithm that takes as input an arbitrary $m \times n$ matrix $A$ and a uniformly random vector $b \in \mathbb{F}_2^m$, and outputs a wi...
Pravesh K. Kothari, Andrew D. Lin, P. Manohar· 0 citations
The search for supersingular elliptic curves is a fundamental computational problem in isogeny-based cryptography. A recent metaheuristic formulation over $\mathbb{F}_{p^2}$ introduced the NonMultiplicity Distance (NMD) objective, measuring the deviation of the Frobenius trace from a multiple of $p$, and showed that un...
Let $X=\mathbb{P}^3$ and let \[ \alpha_n=(0,1,-\tfrac12,\tfrac16-n)\in H^{\mathrm{even}}(X,\mathbb{Q}) \] with respect to the basis $1,H,H^2,H^3$, where $H=c_1(\mathcal{O}_X(1))$. We prove that every Gieseker semistable sheaf on $X$ with Chern character $\alpha_n$ is stable and uniquely of the form $\iota_{P*}\mathcal{...
R. Anderson· Journal of Geometry and Phys...· 0 citations
The quantity $q(A\circ A^{-1})$, the minimum modulus of the eigenvalues of $A\circ A^{-1}$, arises naturally in connection with positive diagonal symmetrizability. For an invertible $\mathbf{M}$-matrix $A$ of order $n$, the classical bounds $\frac{2}{n}\leq q(A\circ A^{-1})\leq 1$ are known. We discuss the sharpness of...
Let $A\in\mathbb{R}^{m\times n}$ have columns of Euclidean norm at most one. We prove that $\operatorname{disc}(A)\le2395\left(1+\log_+\frac n9\right)^{1/4}+2\sqrt2$. Here $\log_+t=\max\{0,\log t\}$. Building on Bansal and Jiang's affine spectral independence framework, we remove the $(\log\log n)^{7/4}$ factor from th...
E. Ercan· 3 citations
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