Related papers: Discrete logarithm and Diffie-Hellman problems in …
Finding low-weight multiples of a binary polynomial is a difficult problem arising in the context of stream ciphers cryptanalysis. The classical algorithm to solve this problem is based on a time memory trade-off. We will present an…
Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime p and positive integer m=o(sqrt(p)/(log p)^4), outputs an elliptic curve E over the finite field F_p for which the cardinality of E(F_p) is…
We study the Identity Problem, the problem of determining if a finitely generated semigroup of matrices contains the identity matrix; see Problem 3 (Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control Theory'' by…
We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…
We analyse the complexity of solving the discrete logarithm problem and of testing the principality of ideals in a certain class of number fields. We achieve the subexponential complexity in $O(L(1/3,O(1)))$ when both the discriminant and…
Given a set D of tuples defined on a domain Omega, we study differentially private algorithms for constructing a histogram over Omega to approximate the tuple distribution in D. Existing solutions for the problem mostly adopt a hierarchical…
This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately,…
Given a unitary representation of a finite group on a finite-dimensional Hilbert space, we show how to find a state whose translates under the group are distinguishable with the highest probability. We apply this to several quantum oracle…
Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups.…
A succinct histogram captures frequent items and their frequencies across clients and has become increasingly important for large-scale, privacy-sensitive machine learning applications. To develop a rigorous framework to guarantee privacy…
Currently there is an active Post-Quantum Cryptography (PQC) solutions search, which attempts to find cryptographic protocols resistant to attacks by means of for instance Shor polynomial time algorithm for numerical field problems like…
This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its…
Weanalyzethecomputationalcomplexityofanalgorithmtosolve the conjugacy search problem in a certain family of metabelian groups. We prove that in general the time complexity of the conjugacy search problem for these groups is at most…
We extend classical methods of computational complexity to the realm of distributed computing, where they sometimes prove more effective than in their original context. Our focus is on decision problems in the LOCAL model, a setting in…
We introduce a refined differentially private (DP) data structure for kernel density estimation (KDE), offering not only improved privacy-utility tradeoff but also better efficiency over prior results. Specifically, we study the…
Solving the discrete logarithm problem in a finite prime field is an extremely important computing problem in modern cryptography. The hardness of solving the discrete logarithm problem in a finite prime field is the security foundation of…
Following the example of Shor's algorithm for period-finding in the integers, we explore the hidden subgroup problem (HSP) for discrete infinite groups. On the hardness side, we show that HSP is NP-hard for the additive group of rational…
We continue the study of counting complexity begun in [Buergisser, Cucker 04] and [Buergisser, Cucker, Lotz 05] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the…
We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially…
We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class $\operatorname{PTIME}$ of languages computable in polynomial time in terms of differential…