Table Of ContentCOMPLEXITY OF
LATTICE PROBLEMS
A Cryptographic
Perspective
Daniele Micciancio
Shafi Goldwasser
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COMPLEXITY OF LATTICEP ROBLEMS
A CryptograpPheircs pective
THE KLUWER INTERNATIONAL SERIES
IN ENGINEERING AND COMPUTER SCIENCE
COMPLEXITY OF LATTICE PROBLEMS
A Cryptographic Perspective
Daniele Micciancio
UniverosfCi atyl iforSnainaD ,i ego
Sbafi Goldwasser
TheM assachusettso fTI encshtniotluotgye,
Cambridge
SPRINGERS CIENCE+BUSINESMSE DIA,L LC
ISBN9 78-1-4613-5293-I8S BN9 78-1-4615-08(9e7B-o7o k)
DOl 10.1007/978-1-4615-0897-7
Library of Congress Cataloging-in-Publication Data
A C.I.CPa.t alorgeuceo frodtr h ibso oki sa vailable
fromt heL ibraorfCy o ngress.
Copyright © 2002S pringSecri encBeu+s ineMsesd iaN ew York
Originaplulbyl ishbeydK luweArc ademiPcu blishienr s2 002
Softcoverre prionft thhaer dcov1esrte diti2o0n0 2
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retriseyvsalt eomrt ransmiitnta endyf ormo rb ya nym eansm,e chanicpahlo,t o
copyinrge,c ordionrgo ,t herwiwsiet,h otuhte p riowrr ittpeenr missoifot nh e
publisShperri,n gSecri ence+BusiMneedsisaL ,L C.
Printed on acid-free paper.
Contents
Preface ix
1. BASICS 1
1 Lattices 1
1.1 Determinant 6
1.2 Successive minima 7
1.3 Minkowski's theorems 11
2 Computational problems 14
2.1 Complexity Theory 15
2.2 Some lattice problems 17
2.3 Hardness of approximation 19
3 Notes 21
2. APPROXIMATION ALGORITHMS 23
1 Solving SVP in dimension 2 24
1.1 Reduced basis 24
1.2 Gauss' algorithm 27
1.3 Running time analysis 30
2 Approximating SVP in dimension 32
n
2.1 Reduced basis 32
2.2 The LLL basis reduction algorithm 34
2.3 Running time analysis 36
3 Approximating CVP in dimension 40
n
4 Notes 42
3. CLOSEST VECTOR PROBLEM 45
1 Decision versus Search 46
2 NP -completeness 48
COMPLEXITY OF LATTICE PROBLEMS
VI
3 SVP is not harder than CVP 52
3.1 Deterministic reduction 53
3.2 Randomized Reduction 56
4 Inapproximability of CVP 58
4.1 Polylogarithmic factor 58
4.2 Larger factors 61
5 CVP with preprocessing 64
6 Notes 67
4. SHORTEST VECTOR PROBLEM 69
1 Kannan's homogenization technique 70
2 The Ajtai-Micciancio embedding 77
3 NP-hardness of SVP 83
3.1 Hardness under randomized reductions 83
3.2 Hardness under nonuniform reductions 85
3.3 Hardness under deterministic reductions 86
4 Notes 87
5. SPHERE PACKINGS 91
1 Packing Points in Small Spheres 94
2 The Exponential Sphere Packing 96
2.1 The Schnorr-Adleman prime number lattice 97
2.2 Finding clusters 99
2.3 Some additional properties 104
3 Integer Lattices 105
4 Deterministic construction 108
5 Notes 110
6. LOW-DEGREE HYPERGRAPHS 111
1 Sauer's Lemma 112
2 Weak probabilistic construction 114
2.1 The exponential bound 115
2.2 Well spread hypergraphs 118
2.3 Proof of the weak theorem 121
3 Strong probabilistic construction 122
4 Notes 124
7. BASIS REDUCTION PROBLEMS 125
1 Successive minima and Minkowski's reduction 125
Contents
Vll
2 Orthogonality defect and KZ reduction 131
3 Small rectangles and the covering radius 136
4 Notes 141
8. CRYPTOGRAPHIC FUNCTIONS 143
1 General techniques 146
1.1 Lattices, sublattices and groups 147
1.2 Discrepancy 153
1.3 Statistical distance 157
2 Collision resistant hash functions 161
2.1 The construction 162
2.2 Collision resistance 164
2.3 The iterative step 168
2.4 Almost perfect lattices 182
3 Encryption Functions 184
3.1 The GGH scheme 185
3.2 The HNF technique 187
3.3 The Ajtai-Dwork cryptosystem 189
3.4 NTRU 191
4 Notes 194
9. INTERACTIVE PROOF SYSTEMS 195
1 Closest vector problem 198
1.1 Proof of the soundness claim 201
1.2 Conclusion 204
2 Shortest vector problem 204
3 Treating other norms 206
4 What does it mean? 208
5 Notes 210
211
References
Index
219
Preface
Lattices are geometric objects that can be pictorially described as the
set of intersection points of an infinite, regular n-dimensional grid. De
spite their apparent simplicity, lattices hide a rich combinatorial struc
ture, which has attracted the attention of great mathematicians over the
last two centuries. Not surprisingly, lattices have found numerous ap
plications in mathematics and computer science, ranging from number
theory and Diophantine approximation, to combinatorial optimization
and cryptography.
The study of lattices, specifically from a computational point of view,
was marked by two major breakthroughs: the development of the LLL
lattice reduction algorithm by Lenstra, Lenstra and Lov:isz in the early
80's, and Ajtai's discovery of a connection between the worst-case and
average-case hardness of certain lattice problems in the late 90's.
The LLL algorithm, despite the relatively poor quality of the solution
it gives in the worst case, allowed to devise polynomial time solutions
to many classical problems in computer science. These include, solving
integer programs in a fixed number of variables, factoring polynomials
over the rationals, breaking knapsack based cryptosystems, and finding
solutions to many other Diophantine and cryptanalysis problems.
Ajtai's discovery suggested a completely different way to use lattices in
cryptography. Instead of using algorithmic solutions to computationally
tractable lattice approximation problems to break cryptosystems, Ajtai's
work shows how to use the existence of computationally intractable-to
approximate lattice problems to build cryptosystems which are
impossi
ble break. Namely, design cryptographic functions that are provably
to
as hard to break as it is to solve a computationally hard lattice problem.
Whereas in complexity theory we say that a problem is hard if it is
hard for the worst case instance, in cryptography a problem is deemed
hard only if it is hard in the average case (i.e., for all but a negligible
IX
COMPLEXITY OF LATTICE PROBLEMS
X
fraction of the instances). The novelty in Ajtai's result, is that he shows
how to build a cryptographic function which is as hard to break on
the average (e.g., over the random choices of the function instance) as
it is to solve the worst case instance of a certain lattice problem. This
achievement is unique to lattice theory at this time, and points to lattices
as an ideal source of hardness for cryptographic purposes.
These new constructive applications of lattices, are deeply rooted in
complexity theory, and were followed by a sharp increase in the study
of lattices from a computational complexity point of view. This led to
the resolution of several long standing open problems in the area. Most
notably, the NP-hardness of the shortest vector problem in its exact and
approximate versions. We present a self contained exposition of this
latter result as well as other results on the computational complexity of
lattice problems.
We did not attempt to cover everything known about lattices, as this
would have filled several volumes. Rather, we selected a few represen
tative topics, based on our personal taste and research experience. Re
grettably, a topic which we neglect is duality and transference theorems.
With this notable exception, we believe that most of the current ideas
relevant to lattice based cryptography appear within in some form or
another.
Many research questions regarding lattices and their cryptographic
usage remain open. We hope that this book will help make lattice based
cryptography more accessible to a wider audience, and ultimately yield
further progress in this exciting research area.
Acknowledgments. Part of the material presented in this book is
based on joint work of the authors with Shai Halevi, Oded Goldreich,
Muli Safra and Jean-Pierre Seifert. Many other people have indirectly
contributed to this book, either through their work, or through many
conversations with the authors. Among them, we would like to men
tion Miklos Ajtai, Ravi Kannan, Amit Sahai, Claus Schnorr, Madhu
Sudan and Salil Vadhan. We would like to thank all our coauthors and
colleagues that have made this book possible.
The first author would like to thank also the National Science Foun
dation and Chris and Warren Hellman for partially supporting this work
under NSF Career Award CCR-0093029 and a 2001-02 Hellman Fellow
ship.
DANIELE MICCIANCIO
