199 research outputs found
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
Complexity of the Havas, Majewski, Matthews LLL Hermite Normal Form algorithm
We show that the integers in the HMM LLL HNF algorithm have bit length
O(m.log(m.B)), where m is the number of rows and B is the maximum square length
of a row of the input matrix. This is only a little worse than the estimate
O(m.log(B)) in the LLL algorithm.Comment: 10 page
A Coefficient-Embedding Ideal Lattice can be Embedded into Infinitely Many Polynomial Rings
Many lattice-based crypstosystems employ ideal lattices for high efficiency.
However, the additional algebraic structure of ideal lattices usually makes us
worry about the security, and it is widely believed that the algebraic
structure will help us solve the hard problems in ideal lattices more
efficiently. In this paper, we study the additional algebraic structure of
ideal lattices further and find that a given ideal lattice in some fixed
polynomial ring can be embedded as an ideal in infinitely many different
polynomial rings. We explicitly present all these polynomial rings for any
given ideal lattice. The interesting phenomenon tells us that a single ideal
lattice may have more abundant algebraic structures than we imagine, which will
impact the security of corresponding crypstosystems. For example, it increases
the difficulties to evaluate the security of crypstosystems based on ideal
lattices, since it seems that we need consider all the polynomial rings that
the given ideal lattices can be embedded into if we believe that the algebraic
structure will contribute to solve the corresponding hard problem. It also
inspires us a new method to solve the ideal lattice problems by embedding the
given ideal lattice into another well-studied polynomial ring. As a by-product,
we also introduce an efficient algorithm to identify if a given lattice is an
ideal lattice or not
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
Finding Short Vectors in Structured Lattices with Reduced Quantum Resources
Leading protocols of post-quantum cryptosystems are based on the mathematical
problem of finding short vectors in structured lattices. It is assumed that the
structure of these lattices does not give an advantage for quantum and
classical algorithms attempting to find short vectors. In this work we focus on
cyclic and nega-cyclic lattices and give a quantum algorithmic framework of how
to exploit the symmetries underlying these lattices. This framework leads to a
significant saving in the quantum resources (e.g. qubits count and circuit
depth) required for implementing a quantum algorithm attempting to find short
vectors. We benchmark the proposed framework with the variational quantum
eigensolver, and show that it leads to better results while reducing the qubits
count and the circuit depth. The framework is also applicable to classical
algorithms aimed at finding short vectors in structured lattices, and in this
regard it could be seen as a quantum-inspired approach
Certified lattice reduction
Quadratic form reduction and lattice reduction are fundamental tools in
computational number theory and in computer science, especially in
cryptography. The celebrated Lenstra-Lenstra-Lov\'asz reduction algorithm
(so-called LLL) has been improved in many ways through the past decades and
remains one of the central methods used for reducing integral lattice basis. In
particular, its floating-point variants-where the rational arithmetic required
by Gram-Schmidt orthogonalization is replaced by floating-point arithmetic-are
now the fastest known. However, the systematic study of the reduction theory of
real quadratic forms or, more generally, of real lattices is not widely
represented in the literature. When the problem arises, the lattice is usually
replaced by an integral approximation of (a multiple of) the original lattice,
which is then reduced. While practically useful and proven in some special
cases, this method doesn't offer any guarantee of success in general. In this
work, we present an adaptive-precision version of a generalized LLL algorithm
that covers this case in all generality. In particular, we replace
floating-point arithmetic by Interval Arithmetic to certify the behavior of the
algorithm. We conclude by giving a typical application of the result in
algebraic number theory for the reduction of ideal lattices in number fields.Comment: 23 page
Genetic Algorithms for the Extended GCD Problem
We present several genetic algorithms for solving the extended greatest common divisor problem. After defining the problem and discussing previous work, we will state our results
Computing a Basis for an Integer Lattice
The extended gcd problem takes as input two integers, and asks as output an integer linear combination of the integers that are equal to their gcd. The classical extended Euclidean algorithm and fast variants such as the half-gcd algorithm give efficient algorithmic solutions. In this thesis, we give a fast algorithm to solve the simplest — but not trivial — extension of the scalar extended gcd problem on two integers to the case of integer input matrices.
Given a full column rank (n + 1) × n integer matrix A, we present an algorithm that produces a square nonsingular integer matrix B such that the lattice generated by the rows of B — the set of all integer linear combinations of the rows of B — is equal to the lattice generated by the rows of A. The magnitude of entries in the basis B are guaranteed to be not much larger than those of the input matrix A. The cost of our algorithm to produce B is about the same as that required to multiply together two square integer matrices of dimension n and with the size of entries about that of the input matrix. This running time bound improves by about a factor of n on the fastest previously known algorithm
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