Cryptanalytic Attacks on RSA

269 Pages·2008·4.325 MB·English

by  Song Y. Yan PhD (auth.)

Cryptanalytic Attacks
on RSA
Cryptanalytic Attacks
on RSA
by
Song Y. Yan
University of Bedfordshire, UK
and
Massachusetts Institute of Technology, USA
Song Y. Yan, PhD
Professor of Computer Science and Mathematics
Director, Institute for Research in Applicable
Computing
University of Bedfordshire
Bedfordshire LU1 3JU
UK
[email protected]
and
Visiting Professor
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139-4307
USA
[email protected]

ISBN-13: 978-0-387-48741-0 e-ISBN-13: 978-0-387-48742-7

Library of Congress Control Number: 2007934650

© 2008 Springer Science+Business Media, LLC.
All rights reserved. This work may not be translated or copied in whole or in part without the written
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The use in this publication of trade names, trademarks, service marks and similar terms, even if they
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Printed on acid-free paper.

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springer.com
Dedicated to Professor Glyn James
on the occasion of his 70th birthday
with gratitude for his encouragement and support
Table of Contents
Preface ....................................................... xi
Notations..................................................... xv
1. Computational/Mathematical Preliminaries.............. 1
1.1 Introduction ............................................... 1
1.2 Computability, Complexity and Intractability .................. 4
1.3 Efficient Number-Theoretic Algorithms........................ 15
1.4 Intractable Number-Theoretic Problems ....................... 41
1.5 Chapter Notes and Further Reading .......................... 54
2. RSA Public-Key Cryptography .......................... 55
2.1 Introduction ............................................... 55
2.2 Public-Key Cryptography ................................... 60
2.3 RSA Public-Key Cryptography............................... 66
2.4 RSA Problem and RSA Assumption .......................... 71
2.5 RSA-Type Crytposystems ................................... 73
2.6 Chapter Notes and Further Readings.......................... 88
3. Integer Factorization Attacks ............................ 91
3.1 Introduction ............................................... 91
3.2 Fermat Factoring Attack .................................... 93
3.3 The “p±1” and ECM Attacks ............................... 94
3.4 Quadratic Sieve Attack...................................... 98
3.5 Successful QS Attack ....................................... 103
3.6 Number Field Sieve Attack .................................. 105
3.7 Chapter Notes and Further Reading .......................... 110
4. Discrete Logarithm Attacks .............................. 111
4.1 Introduction ............................................... 111
4.2 Baby-Step Giant-Step Attack ................................ 115
4.3 Silver–Pohlig–Hellman Attack................................ 118
4.4 Index Calculus Attacks...................................... 122
4.5 Xedni Calculus Attack ...................................... 127
viii Table of Contents
4.6 Chapter Notes and Further Reading .......................... 132
5. Quantum Computing Attacks ............................ 135
5.1 Introduction ............................................... 135
5.2 Order Finding Problem ..................................... 137
5.3 Quantum Order Finding Attack .............................. 139
5.4 Quantum Integer Factorization Attack ........................ 142
5.5 Quantum Discrete Logarithm Attack.......................... 146
5.6 Chapter Notes and Further Reading .......................... 148
6. Simple Elementary Attacks .............................. 149
6.1 Introduction ............................................... 149
6.2 Guessing Plaintext Attacks .................................. 150
6.3 Blinding Attack on RSA Signatures........................... 151
6.4 Guessing φ(N) Attack ...................................... 152
6.5 Guessing d Attack .......................................... 155
6.6 eth Root Attack ............................................ 159
6.7 Common Modulus Attack ................................... 161
6.8 Fixed-Point Attack ......................................... 164
6.9 Chapter Notes and Further Readings.......................... 166
7. Public Exponent Attacks................................. 169
7.1 Introduction ............................................... 169
7.2 A Theorem of Coppersmith.................................. 170
7.3 Short e Attacks for Same Messages ........................... 173
7.4 Short e Attacks for Related Messages ......................... 177
7.5 Lattice Attack for Stereotyped Messages....................... 183
7.6 Chapter Notes and Further Reading .......................... 187
8. Private Exponent Attacks................................ 189
8.1 Introduction ............................................... 189
8.2 Diophantine Attack......................................... 190
8.3 Extended Diophantine Attacks ............................... 195
8.4 Small Private CRT-Exponent Attacks......................... 198
8.5 Partial Private Key Exposure Attacks......................... 201
8.6 Chapter Notes and Further Reading .......................... 205
9. Side-Channel Attacks .................................... 207
9.1 Introduction ............................................... 207
9.2 Modular Exponentiation Revisited............................ 208
9.3 Timing Attacks ............................................ 209
9.4 Time Attacks on RSA in OpenSSL ........................... 212
9.5 Power (Analysis) Attacks.................................... 215
9.6 Random Fault Attacks ...................................... 216
9.7 Chapter Notes and Further Reading .......................... 222
Table of Contents ix
10. The Road Ahead......................................... 223
10.1 Introduction ............................................... 223
10.2 Elliptic Curve-Based Cryptography ........................... 224
10.3 Coding-Based Cryptography ................................. 225
10.4 Lattice-Based Cryptography ................................. 227
10.5 Quantum Cryptography..................................... 229
10.6 Conclusions................................................ 230
10.7 Chapter Notes and Further Reading .......................... 232
Bibliography.................................................. 233
Index......................................................... 251
About the Author ............................................ 255
Preface
The art of war teaches us to rely not on the likelihood of the enemy’s
not coming, but on our own readiness to receive him; not on the
chance of his not attacking, but rather on the fact that we have made
our position unassailable.
Sun Tzu
The Art of War (500 BC)
The book is about the cryptanalytic attacks on RSA. RSA is the first work-
able and practical public-key cryptographic system, invented in 1977 and
published in 1978, by Rivest, Shamir and Adleman, then all at the Mas-
sachusetts Institute of Technology (MIT), and is still the most widely used
cryptographic systems in e.g., online transactions, emails, smartcards, and
more generally electronic and mobile commerce over the Internet, for which
its three inventors received the year 2002 Turing Award, a prize considered
to be the equivalent Nobel Prize for Computer Science. The security of RSA
relies on the computational intractability of the Integer Factorization Prob-
lem (IFP), for which, no efficient (i.e., polynomial-time) algorithm is known.
To get an idea how difficult the integer factorization is, let us consider the
following 2048 bits (617 digits) composite number, known as RSA-2048:
251959084756578934940271832400483985714292821262040320277771378360
436620207075955562640185258807844069182906412495150821892985591491
761845028084891200728449926873928072877767359714183472702618963750
149718246911650776133798590957000973304597488084284017974291006424
586918171951187461215151726546322822168699875491824224336372590851
418654620435767984233871847744479207399342365848238242811981638150
106748104516603773060562016196762561338441436038339044149526344321
901146575444541784240209246165157233507787077498171257724679629263
863563732899121548314381678998850404453640235273819513786365643912
12010397122822120720357.
It is a product of two prime numbers. The RSA Data Security Incorpora-
tion currently offers a $200,000 prize for the first person or group finding
xii Preface
its two prime factors. The basic idea of RSA encryption and decryption is,
surprisingly, rather simple:
C ≡Me (mod N), M ≡Cd (mod N),
whereN =pqwithpandqprime,M,C,eanddaretheplaintext,ciphertext,
encryptionexponentanddecryptionexponent,respectively.Notethateandd
mustbesatisfiedwiththeconditionthated≡1(mod φ(N)),whereφ(N)=
(p−1)(q−1) is Euler’s φ-function. Let, for example, e = 65537, N be the
above mentioned number RSA-2048, and C the following number:
218598056144555493024019389629177159753811144728543422921500499254
181211032562087679022259831067991286101190897695119357754765408522
697956638242922870637083231694404873947694078432775781998614979942
064361669462614088852741600217233052059574880668463536030287944235
822627708134997061064700771693064600712629809165416998449992925313
374281387325903328781863209595468701560742767599157207314869432305
892651836189508103764678721683360183118994273706398707795480800698
501878875875150532123738006235671958527639461339868604410378449818
383913059864587128396200112815989134558427750667427151537609736712
04647757116059031684587.
To recover M from C one requires to find d; to find d one needs to calculate
φ(N);tocalculateφ(N)oneneedstofactorN.Butunfortunately,factorizing
N isintractablewhenN islarge(inthepresentcase,N isa2048-bitnumber,
which is far beyond the computing power of any factoring algorithm on any
computer at present); no polynomial-time factoring algorithm is known so
far. Thus, RSA is secure and C is safe since it is difficult to recover M from
C without factoring N. This is essentially the whole idea of RSA! One can
trytodecrypttheabovegivenRSAciphertextC ortrytofactorthenumber
RSA-2048inordertogetanideahowdifficultitistobreakRSAortofactor
a large number.
The book consists of ten chapters. Chapter 1 presents some computational
and mathematical preliminaries, particularly the theory and practice of
tractable and intractable computations in number theory. Chapter 2 intro-
ducesthebasicconceptsandtheoryoftheRSAcryptographicsystemandits
variants in a broad sense. As the security of RSA is based on the intractabil-
ity of the Integer Factorization Problem (IFP), which is also closely related
to the Discrete Logarithm Problem (DLP), the attacks based on solutions
to IFP problem are discussed in Chapter 3, whereas the attacks based on
solutionstoDLPproblemarediscussedinChapter4.Asquantumalgorithm
is applicable to both the IFP problem and the DLP problem, Chapter 5 will
discuss some quantum attacks on RSA via quantum order finding, quantum
factoringandquantumdiscretelogarithmsolving.Chapter6concentrateson
somesimpleelementarynumber-theoreticattacksonRSA,includinge.g.,for-
wardattack,shortplaintextattack,commonmodulusattackandfixed-point
Preface xiii
attack.Itiscommonthattospeed-upthecomputationofRSAencryption,a
short public exponent e is often used. It is also true for the RSA decryption
if a short private exponent d is used. However, the use of short exponent e
or d can be dangerous. So, in Chapter 7 we shall discuss some cryptanalytic
attacks on the short RSA public exponent e, whereas in Chapter 8 we shall
discuss some attacks on the short RSA private exponent d. In Chapter 9, a
completelydifferenttypeofattacks,namely,theside-channelattacksonRSA,
are discussed. Unlike the mathematical/algorithmic attacks in the previous
chapters, side-channel attacks do not exploit the mathematical properties or
weakness of the RSA algorithm/system itself, but exploit the hardware im-
plementation issues of the system. In other words, these attacks are nothing
to do with the RSA algorithm/system itself but have something to do with
thehardwareimplementationoftheRSAalgorithm/system.Chapter10,the
final chapter, presents some quantum resistant, non-factoring based crypto-
graphic systems as an alternative/replacement to RSA, such as lattice based
and code-based cryptosystems, so that once RSA is proved to be insecure,
there is an immediate replacement to the insecure RSA.
The book is self-contained and the materials presented in the book have
been extensively classroom tested for various courses in Cryptography and
CryptanalysisatAstonandCoventryUniversitiesinEngland,andtheSouth
ChinaUniversityofTechnologyandNankaiUniversityinChina.Manyparts
of the materials in the book have also been presented in seminars in various
universitiesaroundtheworld.Hence,thebookissuitableeitherasaresearch
reference for public-key cryptology in general and for RSA cryptology in
particular, or as a graduate text in the field.
Acknowledgments
TheauthorwouldliketothankProfSushilJajodiaofGeorgeMasonUniver-
sity,USA,ProfGlynJamesandDrAnneJamesofCoventryUniversity,UK,
Prof Stephen Cook of the University of Toronto, Canada, and Prof Richard
Brent of Oxford University and Australian National University for their en-
couragement,supportandhelpduringthewritingofthebook.Specialthanks
mustalsobegiventoSusanLagerstrom-FifeandSharonPalleschi,theeditors
at Springer in Boston, USA, for their encouragement, support and help.
Parts of book were written while the author visited the following three
places: the Department of Computer Science at the University of Toronto
(UT) in March-April 2005, hosted by Prof Stephen Cook and supported by
UTandtheRoyalSocietyLondon,theMathematicalScienceInstituteatthe
AustralianNationalUniversity(ANU)inOctober-November2006,hostedby
Prof Richard Brent and supported by ANU and the Royal Society London,