22 research outputs found
Satisfying More Than Half of a System of Linear Equations Over GF(2): A Multivariate Approach
In the parameterized problem MaxLin2-AA[k ], we are given a system with variables x1,…,xnx1,…,xn consisting of equations of the form ∏i∈Ixi=b∏i∈Ixi=b, where xi,b∈{−1,1}xi,b∈{−1,1} and I⊆[n]I⊆[n], each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least W/2+kW/2+k, where W is the total weight of all equations and k is the parameter (it is always possible for k=0k=0). We show that MaxLin2-AA[k ] has a kernel with at most View the MathML sourceO(k2logk) variables and can be solved in time 2O(klogk)(nm)O(1)2O(klogk)(nm)O(1). This solves an open problem of Mahajan et al. (2006). The problem Max-r-Lin2-AA[k,rk,r] is the same as MaxLin2-AA[k] with two differences: each equation has at most r variables and r is the second parameter. We prove that Max-r-Lin2-AA[k,rk,r] has a kernel with at most (2k−1)r(2k−1)r variables
Hard Instances of the Constrained Discrete Logarithm Problem
The discrete logarithm problem (DLP) generalizes to the constrained DLP,
where the secret exponent belongs to a set known to the attacker. The
complexity of generic algorithms for solving the constrained DLP depends on the
choice of the set. Motivated by cryptographic applications, we study sets with
succinct representation for which the constrained DLP is hard. We draw on
earlier results due to Erd\"os et al. and Schnorr, develop geometric tools such
as generalized Menelaus' theorem for proving lower bounds on the complexity of
the constrained DLP, and construct sets with succinct representation with
provable non-trivial lower bounds
A quantitative version of the non-abelian idempotent theorem
Suppose that G is a finite group and A is a subset of G such that 1_A has
algebra norm at most M. Then 1_A is a plus/minus sum of at most L cosets of
subgroups of G, and L can be taken to be triply tower in O(M). This is a
quantitative version of the non-abelian idempotent theorem.Comment: 82 pp. Changed the title from `Indicator functions in the
Fourier-Eymard algebra'. Corrected the proof of Lemma 19.1. Expanded the
introduction. Corrected typo
A note on the pyjama problem
This note concerns the so-called pyjama problem, whether it is possible to cover the plane by finitely many rotations of vertical strips of half-width ε. We first prove that there exist no periodic coverings for ε1/5. The question whether ε can be arbitrarily small remains open. © 2013 Elsevier Ltd