The author uses irrationality and linear independence measures for certain algebraic numbers to derive explicit upper bounds for the solutions of related norm form equations. The Lenstra-Lenstra-Lovász lattice basis reduction algorithm is then utilized to show that the integer solutions to NK/Q(x 4 √ N4 − 1 + y 4 √ N4 + 1 + z) = ±1 (where K = Q( 4 √ N4 − 1, 4 √ N4 + 1)) are given by (x, y, z) =...

let $l$ be a lattice in $zz^n$ of dimension $m$. we prove that there exist integer constants $d$ and $m$ which are basis-independent such that the total degree of any graver element of $l$ is not greater than $m(n-m+1)md$. the case $m=1$ occurs precisely when $l$ is saturated, and in this case the bound is a reformulation of a well-known bound given by several authors. as a corollary, we show t...

In this paper we investigate how the complexity of the shortest vector problem in a lattice Λ depends on the cycle structure of the additive group Z/Λ. We give a proof that the shortest vector problem is NP-complete in the max-norm for n-dimensional lattices Λ where Z/Λ has n−1 cycles. We also give experimental data that show that the LLL algorithm does not perform significantly better on latti...

Lattice reduction is a hard problem of interest to both publickey cryptography and cryptanalysis. Despite its importance, extremely few algorithms are known. The best algorithm known in high dimension is due to Schnorr, proposed in 1987 as a block generalization of the famous LLL algorithm. This paper deals with Schnorr’s algorithm and potential improvements. We prove that Schnorr’s algorithm o...

This paper introduces a number of modifications that allow for significant improvements of parallel LLL reduction. Experiments show that these modifications result in an increase of the speed-up by a factor of more than 1.35 for SVP challenge type lattice bases in comparing the new algorithm with the state-of-the-art parallel LLL algorithm.

Lattices over number elds arise from a variety of sources in algorithmic algebra and more recently cryptography. Similar to the classical case of Z-lattices, the choice of a nice, short (pseudo)-basis is important in many applications. In this article, we provide the rst algorithm that computes such a short (pseudo)-basis. We utilize the LLL algorithm for Z-lattices together with the Bosma-Pohs...

We propose and analyze a Stein variational reduced basis method (SVRB) to solve large-scale PDE-constrained Bayesian inverse problems. To address the computational challenge of drawing numerous samples requiring expensive PDE solves from posterior distribution, we integrate an adaptive goal-oriented model reduction technique with optimization-based gradient descent method. present detailed anal...