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

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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 ...
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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 ...
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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 ...
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Size upper bound on LLL-reduced basis

Suppose $\Lambda$ is a lattice in $\mathbb{Z}^m$ and let $\{b_1, ..., b_n\}$ be an LLL-reduced basis, rename these vectors as $v_1, ... , v_n$ such that $1 \leq ||v_1|| \leq ... \leq ||v_n||$. How can ...
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0answers
71 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 ...
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1answer
625 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 ...
3
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1answer
159 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 ...
3
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1answer
273 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 ...
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0answers
67 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 ...
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1answer
217 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 ...
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1answer
691 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 ...
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1answer
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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 \...
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1answer
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_{...