Lattice Basis Reduction techniques based on the LLL …tonchev/Khadka.pdf · Introduction Goal Basis Reduction Lattice Diﬀusion and Sublattice Fusion Algorithm Hill Climbing Algorithm

IntroductionGoal

Basis ReductionLattice Diffusion and Sublattice Fusion Algorithm

Hill Climbing AlgorithmExperiment & Results

ReferencesThe End.

Lattice Basis Reduction techniques based on theLLL algorithm

Bal K. Khadka

Michigan Technological University, Houghton, Michigan 49931, USA

August 26 - August 30, 2015

Bal K. Khadka (August 26 - August 30, 2015) Algebraic Combinatorics and Applications

IntroductionGoal



ReferencesThe End.

Contents

1 Introduction

2 Goal

3 Basis Reduction

4 Lattice Diffusion and Sublattice Fusion Algorithm

5 Hill Climbing Algorithm

6 Experiment & Results

7 References

8 The End.


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Introduction

Lattice Basis:Given m linearly independent vectors B = (b1, b2, ..., bm)

T inEuclidean n−space Rn where m ≤ n, the lattice L generated bythem is defined as L(B) = {

∑xibi |xi ∈ Z}. That is

L(B) = {xTB | x ∈ Zm}.

Let B = (b1, b2, ..., bm)T be a basis of L ⊂ Rn and U be an

integral unimodular matrix (an m ×m integer matrix havingdeterminant ±1), then UB is another basis of L.


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Introduction

Lattice Basis:Given m linearly independent vectors B = (b1, b2, ..., bm)

T inEuclidean n−space Rn where m ≤ n, the lattice L generated bythem is defined as L(B) = {

∑xibi |xi ∈ Z}. That is

L(B) = {xTB | x ∈ Zm}.

Let B = (b1, b2, ..., bm)T be a basis of L ⊂ Rn and U be an

integral unimodular matrix (an m ×m integer matrix havingdeterminant ±1), then UB is another basis of L.


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Introduction

Minkowski Convex Body Theorem:A convex set S ⊂ Rn which is symmetric about origin and withvolume greater than m2ndet(L) contains at least m non-zerodistinct lattice pairs ±x1, ±x2, . . . .± xm.

Corollary:If L ⊂ Rn is an n dimensional lattice wih determinant det(L) thenthere is a nonzero b ∈ L such that | b |≤ 2√

π[Γ(n2 + 1)]

1n (det(L))

1n .


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Introduction

Minkowski Convex Body Theorem:A convex set S ⊂ Rn which is symmetric about origin and withvolume greater than m2ndet(L) contains at least m non-zerodistinct lattice pairs ±x1, ±x2, . . . .± xm.

Corollary:If L ⊂ Rn is an n dimensional lattice wih determinant det(L) thenthere is a nonzero b ∈ L such that | b |≤ 2√

π[Γ(n2 + 1)]

1n (det(L))

1n .


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Goal

Given an integral lattice basis of a lattice L ⊂ Rn as input, to finda vector in the lattice L with a minimal Euclidean norm.


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Gram-Schimdt Orthogonalization (GSO)

Given a basis B = (b1, b2, ..., bm)T for a vector space Rn, we

can use GSO process to construct an orthogonal basisB∗ = (b∗1, b

∗2, ..., b

∗m)

T such that b∗1 = b1 and

b∗i = bi −i−1∑j=1

µijb∗j (2 ≤ i ≤ m),

µij =〈bi , b∗j 〉〈b∗j , b∗j 〉

(1 ≤ j < i ≤ n).

B = MB∗, where M = (µij) is a lower triangular matrix.

For any non zero lattice vector b ∈ L ⊂ Rn we have| b |≥ min {| b∗1 |, · · · , | b∗n |}.


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∗2, ..., b

∗m)


b∗i = bi −i−1∑j=1

µijb∗j (2 ≤ i ≤ m),

µij =〈bi , b∗j 〉〈b∗j , b∗j 〉

(1 ≤ j < i ≤ n).




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∗2, ..., b

∗m)


b∗i = bi −i−1∑j=1

µijb∗j (2 ≤ i ≤ m),

µij =〈bi , b∗j 〉〈b∗j , b∗j 〉

(1 ≤ j < i ≤ n).




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LLL Reduced Bases

Let L(B) with B = (b1, b2, ..., bm) be a lattice in Rn withGSO vector b∗1, b

∗2, ..., b

∗m. The basis B is called α reduced (or

LLL-reduced with the reduction parameter α ∈ (14 , 1)) if the

following conditions hold:

a) |µi ,j | ≤ 12 for 1 ≤ j < i ≤ m,

b) |b∗i + µi ,i−1b∗i−1|2 ≥ α|b∗i−1|2 for 2 ≤ i ≤ m.


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LLL Reduced Bases


∗2, ..., b




a) |µi ,j | ≤ 12 for 1 ≤ j < i ≤ m,



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LLL Reduced Bases


∗2, ..., b




a) |µi ,j | ≤ 12 for 1 ≤ j < i ≤ m,



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Lattice Diffusion and Sublattice Fusion Algorithm

Input: Basis B = (b)m×n of L ⊂ Rn, β < m→ Block Sizeand N, M → parameters

Take N permutation matrices Pj , (1 ≤ j ≤ N) with radiusclose to m.

M ←M∪i=1

βi ↑ Sort{LLL(PjB)|length of (LLL(PjB)) is

minimum for 1 ≤ j ≤ N}.

B ′ ← LLL(M)

Output: a vector of minimal length


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M ←M∪i=1



B ′ ← LLL(M)



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M ←M∪i=1



B ′ ← LLL(M)



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M ←M∪i=1



B ′ ← LLL(M)



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M ←M∪i=1



B ′ ← LLL(M)



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Hill Climbing Algorithm

Begin: Basis B = (b)m×n of L ⊂ Rn.

Take k permutation matrices Pj , (2 ≤ j ≤ k) such thatd(Pj , Im) = r (r ≤ m ) . where d is a hamming distance.

B ← {LLL(PjB)|length of (LLL(PjB)) is minimum for1 ≤ j ≤ k}.

End if the desired bound is achieved, or no furtherimprovement is observed.

else,

go to the step 2.


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else,

go to the step 2.


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else,

go to the step 2.


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else,

go to the step 2.


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else,

go to the step 2.


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else,

go to the step 2.


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Our experiment

Case 1: we constructed hadamard matrices and inflated usingintegral unimodular matrices.

Case 2: we picked Ideal lattices from online resources.

We used Hill climbing/lattice diffusion and sublattice fusionalgorithm to get the desired approximated shortest vector.

We successfully reduced B to find the competitive shortestvectors.


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Our experiment






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Our experiment






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Our experiment






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Results

Inflated Hadamard Matrix

Ideal Lattice

ASVP Hall of Fame


http://math.fau.edu/bkhadka/Research%20Interests_files/Inflated%20hadamard%20matrix.txt

http://math.fau.edu/bkhadka/Research%20Interests_files/id52i53.txt

http://www.latticechallenge.org/ideallattice-challenge/alphasvphalloffame.php

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Results


Ideal Lattice

ASVP Hall of Fame





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Results


Ideal Lattice

ASVP Hall of Fame





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References

Lattice Basis Reduction by Murray R. BermnerLLL reduction using NTL library by Victor Shouphttp://www.latticechallenge.org/ideallattice-challenge/in-dex.php


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The End

Any Question?

Thank You!


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The End

Any Question?

Thank You!


Lattice Basis Reduction techniques based on the LLL …tonchev/Khadka.pdf · Introduction Goal Basis Reduction Lattice Diﬀusion and Sublattice Fusion Algorithm Hill Climbing Algorithm

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