Related papers: An $L (1/3 + \epsilon)$ Algorithm for the Discrete…
In this paper, we investigate the computational complexity of isomorphism testing for finite groups and quasigroups, given by their multiplication tables. We crucially take advantage of their various decompositions to show the following: -…
Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a Hidden Subgroup problem, in which an unknown subgroup H of a group G must be determined from a uniform superposition on a…
We continue our development of the invariant theory of genus one curves with the aim of computing certain twists of the universal family of elliptic curves parametrised by the modular curve X(n) for n = 2,3,4,5. Our construction makes use…
We consider the identity component of the Sato-Tate group of the Jacobian of curves of the form $$C_1\colon y^2=x^{2g+2}+c, C_2\colon y^2=x^{2g+1}+cx, C_3\colon y^2=x^{2g+1} +c,$$ where $g$ is the genus of the curve and $c\in\mathbb Q^*$ is…
We study the Jacobian scheme of a plane algebraic curve at an ordinary singularity, characterizing it through a geometric property. We compute the Tjurina number for a family of curves at an ordinary singularity showing that it reaches the…
This paper presents a control framework on Lie groups by designing the control objective in its Lie algebra. Control on Lie groups is challenging due to its nonlinear nature and difficulties in system parameterization. Existing methods to…
In [5], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively in [1] to study the algebraic curves for…
We present a randomized quantum algorithm for polynomial factorization over finite fields. For polynomials of degree $n$ over a finite field $\F_q$, the average-case complexity of our algorithm is an expected $O(n^{1 + o(1)} \log^{2 +…
We find equations for the higher dimensional analogue of the modular curve X_0(3) using Mumford's algebraic formalism of algebraic theta functions. As a consequence, we derive a method for the construction of genus 2 hyperelliptic curves…
Let f be an arbitrary positive integer valued function. The goal of this note is to show that one can construct a finitely generated group in which the discrete log problem is polynomially equivalent to computing the function f. In…
Unit group computations are a cryptographic primitive for which one has a fast quantum algorithm, but the required number of qubits is $\tilde O(m^5)$. In this work we propose a modification of the algorithm for which the number of qubits…
We obtain better algorithms for computing more balanced orientations and degree splits in LOCAL. Important to our result is a connection to the hypergraph sinkless orientation problem [BMNSU, SODA'25] We design an algorithm of complexity…
Amongst the most remarkable successes of quantum computation are Shor's efficient quantum algorithms for the computational tasks of integer factorisation and the evaluation of discrete logarithms. In this article we review the essential…
We consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain $\{0,1\}^n$ over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of…
We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of…
This paper is the first version of a project of classifying all superelliptic curves of genus $g \leq 48$ according to their automorphism group. We determine the parametric equations in each family, the corresponding signature of the group,…
We describe an efficient algorithm to compute a pseudotriangulation of a finite planar family of pairwise disjoint convex bodies presented by its chirotope. The design of the algorithm relies on a deepening of the theory of visibility…
We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let $f$ be an $m$-bit Boolean function and consider an $n$-bit function $F$ obtained by applying $f$ to conjunctions of…
Genus 2 curves are useful in cryptography for both discrete-log based and pairing-based systems, but a method is required to compute genus 2 curves such that the Jacobian has a given number of points. Currently, all known methods involve…
Let p be a prime and let C be a genus one curve over a number field k representing an element of order dividing p in the Shafarevich-Tate group of its Jacobian. We describe an algorithm which computes the set of D in the Shafarevich-Tate…