110 research outputs found
LLL-reduction for Integer Knapsacks
Given an integer mxn matrix A satisfying certain regularity assumptions, a
well-known integer programming problem asks to find an integer point in the
associated knapsack polytope P(A, b)={x: A x= b, x>=0} or determine that no
such point exists. We obtain a LLL-based polynomial time algorithm that solves
the problem subject to a constraint on the location of the vector b.Comment: improved versio
Lattice based extended formulations for integer linear equality systems
We study different extended formulations for the set in order to tackle the feasibility problem for the set . Here the goal is not to find an improved polyhedral
relaxation of conv, but rather to reformulate in such a way that the new
variables introduced provide good branching directions, and in certain
circumstances permit one to deduce rapidly that the instance is infeasible. For
the case that has one row we analyze the reformulations in more detail.
In particular, we determine the integer width of the extended formulations in
the direction of the last coordinate, and derive a lower bound on the Frobenius
number of . We also suggest how a decomposition of the vector can be
obtained that will provide a useful extended formulation. Our theoretical
results are accompanied by a small computational study.Comment: uses packages amsmath and amssym
Quadratic compact knapsack public-key cryptosystem
AbstractKnapsack-type cryptosystems were among the first public-key cryptographic schemes to be invented. Their NP-completeness nature and the high speed in encryption/decryption made them very attractive. However, these cryptosystems were shown to be vulnerable to the low-density subset-sum attacks or some key-recovery attacks. In this paper, additive knapsack-type public-key cryptography is reconsidered. We propose a knapsack-type public-key cryptosystem by introducing an easy quadratic compact knapsack problem. The system uses the Chinese remainder theorem to disguise the easy knapsack sequence. The encryption function of the system is nonlinear about the message vector. Under the relinearization attack model, the system enjoys a high density. We show that the knapsack cryptosystem is secure against the low-density subset-sum attacks by observing that the underlying compact knapsack problem has exponentially many solutions. It is shown that the proposed cryptosystem is also secure against some brute-force attacks and some known key-recovery attacks including the simultaneous Diophantine approximation attack and the orthogonal lattice attack
Lattice based extended formulations for integer linear equality systems
We study different extended formulations for the set
X^+ = X\cap Z^n_+(X^+)Aaaa$ can be obtained that will provide a useful extended formulation.
Our theoretical results are accompanied by a small computational study
- …