Related papers: The Discrete Logarithm Problem over Prime Fields c…
We show how to reduce the problem of solving members of a certain family of nonlinear differential equations to that of solving some corresponding linear differential equations.
We investigate the space complexity of solving linear systems of equations. While all known deterministic or randomized algorithms solving a square system of $n$ linear equations in $n$ variables require $\Omega(\log^2 n)$ space, Ta-Shma…
In connection with machine arithmetic, we are interested in systems of constraints of the form x + k \leq y + k'. Over integers, the satisfiability problem for such systems is polynomial time. The problem becomes NP complete if we restrict…
Let $\mathbb{Z}^{ab}$ be the ring of integers of $\mathbb{Q}^{ab}$, the maximal abelian extension of $\mathbb{Q}$. We show that there exists an algorithm to decide whether a system of equations and inequations, with integer coefficients,…
This article provides an exposition to the topic of formal moduli problems, emphasizing its connections with differential graded Lie algebras, and mainly following from Jacob Lurie's DAG X: Formal Moduli Problems. As such, this paper should…
We derive the discrete version of the classical Helmholtz condition. Precisely, we state a theorem characterizing second order finite differences equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide…
We analyze the canonical treatment of classical constrained mechanical systems formulated with a discrete time. We prove that under very general conditions, it is possible to introduce nonsingular canonical transformations that preserve the…
We present an algorithm for solving the discrete logarithm problem in Jacobians of families of plane curves whose degrees in $X$ and $Y$ are low with respect to their genera. The finite base fields $\FF_q$ are arbitrary, but their sizes…
We study the Steinberg lattice of the general linear group when reduced modulo a prime different from the defining characteristic.
Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an…
We investigate the discrete Painleve II equation over finite fields. We treat it over local fields and observe that it has a property that is similar to the good reduction over finite fields. We can use this property, which seems to be an…
We consider the problem of computing the monic gcd of two polynomials over a number field L = Q(alpha_1,...,alpha_n). Langemyr and McCallum have already shown how Brown's modular GCD algorithm for polynomials over Q can be modified to work…
We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the…
This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations of a generic autonomous system along a particular solution. The main result of this paper is a compact yet explicit and…
We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified…
A powerful method for solving non-linear first-order ordinary differential equations, which is based on geometrical understanding of the corresponding dynamics of the so called Lie systems, is developed. This method allows us not only to…
The finite Laguerre transform is applied to solve Differential Equations Problems of order higher than two and a one-dimensional steady-state Schr\"{o}dinger equation, by using elementary Linear Algebra methods.
In this paper, we study the discrete logarithm problem in the finite fields $\F_{q^n}$ where $n|q-1$. The field is called a Kummer field or a Kummer extension of $\F_q$. It plays an important role in improving the AKS primality proving…
The bivariate difference field provides an algebraic framework for a sequence satisfying a recurrence of order two. Based on this, we focus on sequences satisfying a recurrence of higher order, and consider the multivariate difference…
We present a deterministic algorithm that, given a prime $p$ and a solution $x \in \mathbb Z$ to the discrete logarithm problem $a^x \equiv b \pmod p$ with $p\nmid a$, efficiently lifts it to a solution modulo $p^k$, i.e., $a^x \equiv b…