Questions tagged [lenstra-lenstra-lovasz]
The Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm efficiently finds a short, nearly orthogonal lattice basis form an arbitrary one
13
questions
1
vote
2
answers
82
views
Example of bad basis for lattices (worst-case for LLL)
Summary. Given some dimension $n$ (say $n=50$), is it possible to describe explicitly a lattice $L$ and a basis $B$ of $L$ such that
$$ \frac{ \| LLL(B)_1 \| }{ \lambda_1(L) } > 1.02^n $$
where $...
- 141
0
votes
1
answer
85
views
NTL: Solve the closest vector problem for non-square matrix using LLL/Nearest Plane Algorithm
Assume I have a matrix $A \in \mathbb{Z}^{m \times n}$, $m > n$, which forms a basis of a lattice. Given a vector target vector $t = Ax + e$, $t,e \in \mathbb{Z}^m$,$x \in \mathbb{Z}^n$, I want to ...
- 337
2
votes
0
answers
89
views
Howgrave-Graham lattice attack on NTRU
I am lookin for a good example to illustrate this attack on NTRU using low parameters but I failed to do that, The attack consist to use LLL reduction on A basis of NTRU Lattice, let us use the column ...
- 139
0
votes
0
answers
48
views
Gram-Schmidt upper triangular basis
I'm trying to understand the Gram-Schmidt Orthogonalization process. Below, there is an explanation that a lattice basis can be described by an upper triangular vector.
It is often convenient to ...
- 33
3
votes
0
answers
73
views
Gram-Schmidt coefficients in LLL algorithm
To my understanding the LLL lattice reduction algorithm starts with a set of integer vectors $\{b_1, \dots, b_2\}$, which span a lattice, and tries to generate a new basis of shorter vectors of the ...
- 31
8
votes
1
answer
791
views
Introduction to LLL algorithm applied to linear modular inequalities
What is the Lenstra–Lenstra–Lovász lattice basis reduction algorithm about?
How is it applied to solve for $x\pmod m$ a system of modular inequalities $(u_i\,x+v_i\bmod m)<w$ for $0\le i<n$?
I'm ...
- 122k
3
votes
1
answer
201
views
Lattice reduction question regarding the capability of LLL and BKZ
I've been reading How to estimate the hardness of SIS instances? and following some of its sources, and I want to confirm a few things.
LLL algorithm runs in polynomial time, but isn't capable of ...
- 6,551
3
votes
1
answer
320
views
LLL - Lattice Reduced Basis Algorithm question?
I have two related questions:
Version 1: Let $B=\{b_1,b_2,\dots,b_n\}$ be an orthogonal basis for $R^n$. What is the associated reduced basis obtained by applying LLL algorithm to $B$?
I know how ...
- 859
1
vote
0
answers
72
views
What is the minimal angle between two LLL reduced vectors?
What is the minimal angle between two LLL reduced vectors?
It seems it should be 60 degree as $|\mu_{i,j}| \leq \frac{1}{2}$.
If we make the upper bound of $\mu_{i,j}$ by 1/3, can we get better ...
- 219
5
votes
1
answer
238
views
$L^3$ Grover search of NTRU variants
I was reading a text on cryptology by Wayne Patterson and came across the $L^3$ algorithm which reduces integer lattices with respect to their base. I've also read on the NIST CFP A8 that attacks ...
- 214
4
votes
1
answer
769
views
Significance of Gram-Schmidt coefficients in LLL algorithm
Let $\{ {\bf v}_1,{\bf v}_2 \}$ be two linearly independent vectors. An orthogonal base $\{{\bf u}_1,{\bf u}_2 \}$ of the vector space $\mathrm{span}\{ {\bf v}_1,{\bf v}_2 \}$ can be computed using ...
- 849
8
votes
2
answers
2k
views
Why is the Lovász condition used in the LLL algorithm?
The LLL algorithm is used to approximate the Shortest Vector Problem, i.e., it outputs a reduced basis. Such a basis will satisfy two conditions:
$$ \forall i \gt j. \quad \lvert\mu_{ij}\rvert \le \...
- 849
9
votes
1
answer
2k
views
Problem with LLL reduction on truncated LCG schemes
I am struggling to apply Freize et al. paper to break a truncated LCG.
A truncated LCG is a pseudo random generator that outputs the $n$ leading bits $y_i$ of $x_i$, where $(x_i)$ is such that $x_{...
- 285