. 1
( 29)



>>

Applied Cryptography: Second Edition - Bruce Schneier




Applied Cryptography, Second Edition: Protocols, Algorthms, and Source Code in C
by Bruce Schneier
Wiley Computer Publishing, John Wiley & Sons, Inc.
ISBN: 0471128457 Pub Date: 01/01/96


Foreword By Whitfield Diffie
Preface
About the Author

Chapter 1”Foundations
1.1 Terminology
1.2 Steganography
1.3 Substitution Ciphers and Transposition Ciphers
1.4 Simple XOR
1.5 One-Time Pads
1.6 Computer Algorithms
1.7 Large Numbers

Part I”Cryptographic Protocols
Chapter 2”Protocol Building Blocks
2.1 Introduction to Protocols
2.2 Communications Using Symmetric Cryptography
2.3 One-Way Functions
2.4 One-Way Hash Functions
2.5 Communications Using Public-Key Cryptography
2.6 Digital Signatures
2.7 Digital Signatures with Encryption
2.8 Random and Pseudo-Random-Sequence Generation
Chapter 3”Basic Protocols
3.1 Key Exchange
3.2 Authentication
3.3 Authentication and Key Exchange
3.4 Formal Analysis of Authentication and Key-Exchange Protocols
3.5 Multiple-Key Public-Key Cryptography
3.6 Secret Splitting
3.7 Secret Sharing
3.8 Cryptographic Protection of Databases
Chapter 4”Intermediate Protocols
4.1 Timestamping Services
4.2 Subliminal Channel
4.3 Undeniable Digital Signatures
4.4 Designated Confirmer Signatures
4.5 Proxy Signatures
4.6 Group Signatures
4.7 Fail-Stop Digital Signatures
4.8 Computing with Encrypted Data
4.9 Bit Commitment
4.10 Fair Coin Flips
4.11 Mental Poker
4.12 One-Way Accumulators
4.13 All-or-Nothing Disclosure of Secrets



Page 1 of 666
Applied Cryptography: Second Edition - Bruce Schneier



4.14 Key Escrow
Chapter 5”Advanced Protocols
5.1 Zero-Knowledge Proofs
5.2 Zero-Knowledge Proofs of Identity
5.3 Blind Signatures
5.4 Identity-Based Public-Key Cryptography
5.5 Oblivious Transfer
5.6 Oblivious Signatures
5.7 Simultaneous Contract Signing
5.8 Digital Certified Mail
5.9 Simultaneous Exchange of Secrets
Chapter 6”Esoteric Protocols
6.1 Secure Elections
6.2 Secure Multiparty Computation
6.3 Anonymous Message Broadcast
6.4 Digital Cash

Part II”Cryptographic Techniques
Chapter 7”Key Length
7.1 Symmetric Key Length
7.2 Public-Key Key Length
7.3 Comparing Symmetric and Public-Key Key Length
7.4 Birthday Attacks against One-Way Hash Functions
7.5 How Long Should a Key Be?
7.6 Caveat Emptor
Chapter 8”Key Management
8.1 Generating Keys
8.2 Nonlinear Keyspaces
8.3 Transferring Keys
8.4 Verifying Keys
8.5 Using Keys
8.6 Updating Keys
8.7 Storing Keys
8.8 Backup Keys
8.9 Compromised Keys
8.10 Lifetime of Keys
8.11 Destroying Keys
8.12 Public-Key Key Management
Chapter 9”Algorithm Types and Modes
9.1 Electronic Codebook Mode
9.2 Block Replay
9.3 Cipher Block Chaining Mode
9.4 Stream Ciphers
9.5 Self-Synchronizing Stream Ciphers
9.6 Cipher-Feedback Mode
9.7 Synchronous Stream Ciphers
9.8 Output-Feedback Mode
9.9 Counter Mode
9.10 Other Block-Cipher Modes
9.11 Choosing a Cipher Mode
9.12 Interleaving
9.13 Block Ciphers versus Stream Ciphers




Page 2 of 666
Applied Cryptography: Second Edition - Bruce Schneier



Chapter 10”Using Algorithms
10.1 Choosing an Algorithm
10.2 Public-Key Cryptography versus Symmetric Cryptography
10.3 Encrypting Communications Channels
10.4 Encrypting Data for Storage
10.5 Hardware Encryption versus Software Encryption
10.6 Compression, Encoding, and Encryption
10.7 Detecting Encryption
10.8 Hiding Ciphertext in Ciphertext
10.9 Destroying Information

Part III”Cryptographic Algorithms
Chapter 11”Mathematical Background
11.1 Information Theory
11.2 Complexity Theory
11.3 Number Theory
11.4 Factoring
11.5 Prime Number Generation
11.6 Discrete Logarithms in a Finite Field
Chapter 12”Data Encryption Standard (DES)
12.1 Background
12.2 Description of DES
12.3 Security of DES
12.4 Differential and Linear Cryptanalysis
12.5 The Real Design Criteria
12.6 DES Variants
12.7 How Secure Is DES Today?
Chapter 13”Other Block Ciphers
13.1 Lucifer
13.2 Madryga
13.3 NewDES
13.4 FEAL
13.5 REDOC
13.6 LOKI
13.7 Khufu and Khafre
13.8 RC2
13.9 IDEA
13.10 MMB
13.11 CA-1.1
13.12 Skipjack
Chapter 14”Still Other Block Ciphers
14.1 GOST
14.2 CAST
14.3 Blowfish
14.4 SAFER
14.5 3-Way
14.6 Crab
14.7 SXAL8/MBAL
14.8 RC5
14.9 Other Block Algorithms
14.10 Theory of Block Cipher Design
14.11 Using one-Way Hash Functions




Page 3 of 666
Applied Cryptography: Second Edition - Bruce Schneier



14.12 Choosing a Block Algorithm
Chapter 15”Combining Block Ciphers
15.1 Double Encryption
15.2 Triple Encryption
15.3 Doubling the Block Length
15.4 Other Multiple Encryption Schemes
15.5 CDMF Key Shortening
15.6 Whitening
15.7 Cascading Multiple Block Algorithms
15.8 Combining Multiple Block Algorithms
Chapter 16”Pseudo-Random-Sequence Generators and Stream Ciphers
16.1 Linear Congruential Generators
16.2 Linear Feedback Shift Registers
16.3 Design and Analysis of Stream Ciphers
16.4 Stream Ciphers Using LFSRs
16.5 A5
16.6 Hughes XPD/KPD
16.7 Nanoteq
16.8 Rambutan
16.9 Additive Generators
16.10 Gifford
16.11 Algorithm M
16.12 PKZIP
Chapter 17”Other Stream Ciphers and Real Random-Sequence Generators
17.1 RC4
17.2 SEAL
17.3 WAKE
17.4 Feedback with Carry Shift Registers
17.5 Stream Ciphers Using FCSRs
17.6 Nonlinear-Feedback Shift Registers
17.7 Other Stream Ciphers
17.8 System-Theoretic Approach to Stream-Cipher Design
17.9 Complexity-Theoretic Approach to Stream-Cipher Design
17.10 Other Approaches to Stream-Cipher Design
17.11 Cascading Multiple Stream Ciphers
17.12 Choosing a Stream Cipher
17.13 Generating Multiple Streams from a Single Pseudo-Random-Sequence
Generator
17.14 Real Random-Sequence Generators
Chapter 18”One-Way Hash Functions
18.1 Background
18.2 Snefru
18.3 N- Hash
18.4 MD4
18.5 MD5
18.6 MD2
18.7 Secure Hash Algorithm (SHA)
18.8 RIPE-MD
18.9 HAVAL
18.10 Other One-Way Hash Functions
18.11 One-Way Hash Functions Using Symmetric Block Algorithms
18.12 Using Public-Key Algorithms
18.13 Choosing a One-Way Hash Function



Page 4 of 666
Applied Cryptography: Second Edition - Bruce Schneier



18.14 Message Authentication Codes
Chapter 19”Public-Key Algorithms
19.1 Background
19.2 Knapsack Algorithms
19.3 RSA
19.4 Pohlig-Hellman
19.5 Rabin
19.6 ElGamal
19.7 McEliece
19.8 Elliptic Curve Cryptosystems
19.9 LUC
19.10 Finite Automaton Public-Key Cryptosystems
Chapter 20”Public-Key Digital Signature Algorithms
20.1 Digital Signature Algorithm (DSA)
20.2 DSA Variants
20.3 Gost Digital Signature Algorithm
20.4 Discrete Logarithm Signature Schemes
20.5 Ong-Schnorr-Shamir
20.6 ESIGN
20.7 Cellular Automata
20.8 Other Public-Key Algorithms
Chapter 21”Identification Schemes
21.1 Feige-Fiat-Shamir
21.2 Guillou-Quisquater
21.3 Schnorr
21.4 Converting Identification Schemes to Signature Schemes
Chapter 22”Key-Exchange Algorithms
22.1 Diffie-Hellman
22.2 Station-to-Station Protocol
22.3 Shamir™s Three-Pass Protocol
22.4 COMSET
22.5 Encrypted Key Exchange
22.6 Fortified Key Negotiation
22.7 Conference Key Distribution and Secret Broadcasting
Chapter 23”Special Algorithms for Protocols
23.1 Multiple-Key Public-Key Cryptography
23.2 Secret-Sharing Algorithms
23.3 Subliminal Channel
23.4 Undeniable Digital Signatures
23.5 Designated Confirmer Signatures
23.6 Computing with Encrypted Data
23.7 Fair Coin Flips
23.8 One-Way Accumulators
23.9 All-or-Nothing Disclosure of Secrets
23.10 Fair and Failsafe Cryptosystems
23.11 Zero-Knowledge Proofs of Knowledge
23.12 Blind Signatures
23.13 Oblivious Transfer
23.14 Secure Multiparty Computation
23.15 Probabilistic Encryption
23.16 Quantum Cryptography




Page 5 of 666
Applied Cryptography: Second Edition - Bruce Schneier




Part IV”The Real World
Chapter 24”Example Implementations
24.1 IBM Secret-Key Management Protocol
24.2 MITRENET
24.3 ISDN
24.4 STU-III
24.5 Kerberos
24.6 KryptoKnight
24.7 SESAME
24.8 IBM Common Cryptographic Architecture
24.9 ISO Authentication Framework
24.10 Privacy-Enhanced Mail (PEM)
24.11 Message Security Protocol (MSP)
24.12 Pretty Good Privacy (PGP)
24.13 Smart Cards
24.14 Public-Key Cryptography Standards (PKCS)
24.15 Universal Electronic Payment System (UEPS)
24.16 Clipper
24.17 Capstone
24.18 AT&T Model 3600 Telephone Security Device (TSD)
Chapter 25”Politics
25.1 National Security Agency (NSA)
25.2 National Computer Security Center (NCSC)
25.3 National Institute of Standards and Technology (NIST)
25.4 RSA Data Security, Inc.
25.5 Public Key Partners
25.6 International Association for Cryptologic Research (IACR)
25.7 RACE Integrity Primitives Evaluation (RIPE)
25.8 Conditional Access for Europe (CAFE)
25.9 ISO/IEC 9979
25.10 Professional, Civil Liberties, and Industry Groups
25.11 Sci.crypt
25.12 Cypherpunks
25.13 Patents
25.14 U.S. Export Rules
25.15 Foreign Import and Export of Cryptography
25.16 Legal Issues

Afterword by Matt Blaze
Part V”Source Code
References
Index


Foreword By Whitfield Diffie
The literature of cryptography has a curious history. Secrecy, of course, has always played a central
role, but until the First World War, important developments appeared in print in a more or less
timely fashion and the field moved forward in much the same way as other specialized disciplines. As
late as 1918, one of the most influential cryptanalytic papers of the twentieth century, William F.
Friedman™s monograph The Index of Coincidence and Its Applications in Cryptography, appeared as a



Page 6 of 666
Applied Cryptography: Second Edition - Bruce Schneier



research report of the private Riverbank Laboratories [577]. And this, despite the fact that the work
had been done as part of the war effort. In the same year Edward H. Hebern of Oakland, California
filed the first patent for a rotor machine [710], the device destined to be a mainstay of military
cryptography for nearly 50 years.

After the First World War, however, things began to change. U.S. Army and Navy organizations,
working entirely in secret, began to make fundamental advances in cryptography. During the thirties
and forties a few basic papers did appear in the open literature and several treatises on the subject
were published, but the latter were farther and farther behind the state of the art. By the end of the
war the transition was complete. With one notable exception, the public literature had died. That
exception was Claude Shannon™s paper “The Communication Theory of Secrecy Systems,” which
appeared in the Bell System Technical Journal in 1949 [1432]. It was similar to Friedman™s 1918
paper, in that it grew out of wartime work of Shannon™s. After the Second World War ended it was
declassified, possibly by mistake.

From 1949 until 1967 the cryptographic literature was barren. In that year a different sort of
contribution appeared: David Kahn™s history, The Codebreakers [794]. It didn™t contain any new
technical ideas, but it did contain a remarkably complete history of what had gone before, including
mention of some things that the government still considered secret. The significance of The
Codebreakers lay not just in its remarkable scope, but also in the fact that it enjoyed good sales and
made tens of thousands of people, who had never given the matter a moment™s thought, aware of
cryptography. A trickle of new cryptographic papers began to be written.

At about the same time, Horst Feistel, who had earlier worked on identification friend or foe devices
for the Air Force, took his lifelong passion for cryptography to the IBM Watson Laboratory in
Yorktown Heights, New York. There, he began development of what was to become the U.S. Data
Encryption Standard; by the early 1970s several technical reports on this subject by Feistel and his
colleagues had been made public by IBM [1482,1484,552].

This was the situation when I entered the field in late 1972. The cryptographic literature wasn™t
abundant, but what there was included some very shiny nuggets.

Cryptology presents a difficulty not found in normal academic disciplines: the need for the proper
interaction of cryptography and cryptanalysis. This arises out of the fact that in the absence of real
communications requirements, it is easy to propose a system that appears unbreakable. Many
academic designs are so complex that the would“be cryptanalyst doesn™t know where to start;
exposing flaws in these designs is far harder than designing them in the first place. The result is that
the competitive process, which is one strong motivation in academic research, cannot take hold.

When Martin Hellman and I proposed public“key cryptography in 1975 [496], one of the indirect
aspects of our contribution was to introduce a problem that does not even appear easy to solve. Now
an aspiring cryptosystem designer could produce something that would be recognized as clever”
something that did more than just turn meaningful text into nonsense. The result has been a
spectacular increase in the number of people working in cryptography, the number of meetings held,
and the number of books and papers published.

In my acceptance speech for the Donald E. Fink award”given for the best expository paper to appear
in an IEEE journal”which I received jointly with Hellman in 1980, I told the audience that in writing
“Privacy and Authentication,” I had an experience that I suspected was rare even among the
prominent scholars who populate the IEEE awards ceremony: I had written the paper I had wanted
to study, but could not find, when I first became seriously interested in cryptography. Had I been able
to go to the Stanford bookstore and pick up a modern cryptography text, I would probably have
learned about the field years earlier. But the only things available in the fall of 1972 were a few classic



Page 7 of 666
Applied Cryptography: Second Edition - Bruce Schneier



papers and some obscure technical reports.

The contemporary researcher has no such problem. The problem now is choosing where to start
among the thousands of papers and dozens of books. The contemporary researcher, yes, but what
about the contemporary programmer or engineer who merely wants to use cryptography? Where
does that person turn? Until now, it has been necessary to spend long hours hunting out and then
studying the research literature before being able to design the sort of cryptographic utilities glibly
described in popular articles.

This is the gap that Bruce Schneier™s Applied Cryptography has come to fill. Beginning with the
objectives of communication security and elementary examples of programs used to achieve these
objectives, Schneier gives us a panoramic view of the fruits of 20 years of public research. The title
says it all; from the mundane objective of having a secure conversation the very first time you call
someone to the possibilities of digital money and cryptographically secure elections, this is where
you™ll find it.

Not satisfied that the book was about the real world merely because it went all the way down to the
code, Schneier has included an account of the world in which cryptography is developed and applied,
and discusses entities ranging from the International Association for Cryptologic Research to the
NSA.

When public interest in cryptography was just emerging in the late seventies and early eighties, the
National Security Agency (NSA), America™s official cryptographic organ, made several attempts to
quash it. The first was a letter from a long“time NSA employee allegedly, avowedly, and apparently
acting on his own. The letter was sent to the IEEE and warned that the publication of cryptographic
material was a violation of the International Traffic in Arms Regulations (ITAR). This viewpoint
turned out not even to be supported by the regulations themselves”which contained an explicit
exemption for published material”but gave both the public practice of cryptography and the 1977
Information Theory Workshop lots of unexpected publicity.

A more serious attempt occurred in 1980, when the NSA funded the American Council on Education
to examine the issue with a view to persuading Congress to give it legal control of publications in the
field of cryptography. The results fell far short of NSA™s ambitions and resulted in a program of
voluntary review of cryptographic papers; researchers were requested to ask the NSA™s opinion on
whether disclosure of results would adversely affect the national interest before publication.

As the eighties progressed, pressure focused more on the practice than the study of cryptography.
Existing laws gave the NSA the power, through the Department of State, to regulate the export of
cryptographic equipment. As business became more and more international and the American
fraction of the world market declined, the pressure to have a single product in both domestic and
offshore markets increased. Such single products were subject to export control and thus the NSA
acquired substantial influence not only over what was exported, but also over what was sold in the
United States.

As this is written, a new challenge confronts the public practice of cryptography. The government has
augmented the widely published and available Data Encryption Standard, with a secret algorithm
implemented in tamper“resistant chips. These chips will incorporate a codified mechanism of
government monitoring. The negative aspects of this “key“escrow” program range from a potentially
disastrous impact on personal privacy to the high cost of having to add hardware to products that had
previously encrypted in software. So far key escrow products are enjoying less than stellar sales and
the scheme has attracted widespread negative comment, especially from the independent
cryptographers. Some people, however, see more future in programming than politicking and have
redoubled their efforts to provide the world with strong cryptography that is accessible to public



Page 8 of 666
Applied Cryptography: Second Edition - Bruce Schneier



scrutiny.

A sharp step back from the notion that export control law could supersede the First Amendment
seemed to have been taken in 1980 when the Federal Register announcement of a revision to ITAR
included the statement: “...provision has been added to make it clear that the regulation of the export
of technical data does not purport to interfere with the First Amendment rights of individuals.” But
the fact that tension between the First Amendment and the export control laws has not gone away
should be evident from statements at a conference held by RSA Data Security. NSA™s representative
from the export control office expressed the opinion that people who published cryptographic
programs were “in a grey area” with respect to the law. If that is so, it is a grey area on which the first
edition of this book has shed some light. Export applications for the book itself have been granted,
with acknowledgement that published material lay beyond the authority of the Munitions Control
Board. Applications to export the enclosed programs on disk, however, have been denied.

The shift in the NSA™s strategy, from attempting to control cryptographic research to tightening its
grip on the development and deployment of cryptographic products, is presumably due to its
realization that all the great cryptographic papers in the world do not protect a single bit of traffic.
Sitting on the shelf, this volume may be able to do no better than the books and papers that preceded
it, but sitting next to a workstation, where a programmer is writing cryptographic code, it just may.

Whitfield Diffie
Mountain View, CA




Preface
There are two kinds of cryptography in this world: cryptography that will stop your kid sister from
reading your files, and cryptography that will stop major governments from reading your files. This
book is about the latter.

If I take a letter, lock it in a safe, hide the safe somewhere in New York, then tell you to read the
letter, that™s not security. That™s obscurity. On the other hand, if I take a letter and lock it in a safe,
and then give you the safe along with the design specifications of the safe and a hundred identical
safes with their combinations so that you and the world™s best safecrackers can study the locking
mechanism”and you still can™t open the safe and read the letter”that™s security.

For many years, this sort of cryptography was the exclusive domain of the military. The United
States™ National Security Agency (NSA), and its counterparts in the former Soviet Union, England,
France, Israel, and elsewhere, have spent billions of dollars in the very serious game of securing their
own communications while trying to break everyone else™s. Private individuals, with far less expertise
and budget, have been powerless to protect their own privacy against these governments.

During the last 20 years, public academic research in cryptography has exploded. While classical
cryptography has been long used by ordinary citizens, computer cryptography was the exclusive
domain of the world™s militaries since World War II. Today, state“of“the“art computer cryptography
is practiced outside the secured walls of the military agencies. The layperson can now employ security
practices that can protect against the most powerful of adversaries”security that may protect against
military agencies for years to come.

Do average people really need this kind of security? Yes. They may be planning a political campaign,
discussing taxes, or having an illicit affair. They may be designing a new product, discussing a



Page 9 of 666
Applied Cryptography: Second Edition - Bruce Schneier



marketing strategy, or planning a hostile business takeover. Or they may be living in a country that
does not respect the rights of privacy of its citizens. They may be doing something that they feel
shouldn™t be illegal, but is. For whatever reason, the data and communications are personal, private,
and no one else™s business.

This book is being published in a tumultuous time. In 1994, the Clinton administration approved the
Escrowed Encryption Standard (including the Clipper chip and Fortezza card) and signed the Digital
Telephony bill into law. Both of these initiatives try to ensure the government™s ability to conduct
electronic surveillance.

Some dangerously Orwellian assumptions are at work here: that the government has the right to
listen to private communications, and that there is something wrong with a private citizen trying to
keep a secret from the government. Law enforcement has always been able to conduct court“
authorized surveillance if possible, but this is the first time that the people have been forced to take
active measures to make themselves available for surveillance. These initiatives are not simply
government proposals in some obscure area; they are preemptive and unilateral attempts to usurp
powers that previously belonged to the people.

Clipper and Digital Telephony do not protect privacy; they force individuals to unconditionally trust
that the government will respect their privacy. The same law enforcement authorities who illegally
tapped Martin Luther King Jr.™s phones can easily tap a phone protected with Clipper. In the recent
past, local police authorities have either been charged criminally or sued civilly in numerous
jurisdictions”Maryland, Connecticut, Vermont, Georgia, Missouri, and Nevada”for conducting
illegal wiretaps. It™s a poor idea to deploy a technology that could some day facilitate a police state.

The lesson here is that it is insufficient to protect ourselves with laws; we need to protect ourselves
with mathematics. Encryption is too important to be left solely to governments.

This book gives you the tools you need to protect your own privacy; cryptography products may be
declared illegal, but the information will never be.

How to Read This Book

I wrote Applied Cryptography to be both a lively introduction to the field of cryptography and a
comprehensive reference. I have tried to keep the text readable without sacrificing accuracy. This
book is not intended to be a mathematical text. Although I have not deliberately given any false
information, I do play fast and loose with theory. For those interested in formalism, there are copious
references to the academic literature.

Chapter 1 introduces cryptography, defines many terms, and briefly discusses precomputer
cryptography.

Chapters 2 through 6 (Part I) describe cryptographic protocols: what people can do with
cryptography. The protocols range from the simple (sending encrypted messages from one person to
another) to the complex (flipping a coin over the telephone) to the esoteric (secure and anonymous
digital money exchange). Some of these protocols are obvious; others are almost amazing.
Cryptography can solve a lot of problems that most people never realized it could.

Chapters 7 through 10 (Part II) discuss cryptographic techniques. All four chapters in this section are
important for even the most basic uses of cryptography. Chapters 7 and 8 are about keys: how long a
key should be in order to be secure, how to generate keys, how to store keys, how to dispose of keys,
and so on. Key management is the hardest part of cryptography and often the Achilles™ heel of an
otherwise secure system. Chapter 9 discusses different ways of using cryptographic algorithms, and



Page 10 of 666
Applied Cryptography: Second Edition - Bruce Schneier



Chapter 10 gives the odds and ends of algorithms: how to choose, implement, and use algorithms.

Chapters 11 through 23 (Part III) list algorithms. Chapter 11 provides the mathematical background.
This chapter is only required if you are interested in public“key algorithms. If you just want to
implement DES (or something similar), you can skip ahead. Chapter 12 discusses DES: the algorithm,
its history, its security, and some variants. Chapters 13, 14, and 15 discuss other block algorithms; if
you want something more secure than DES, skip to the section on IDEA and triple“DES. If you want
to read about a bunch of algorithms, some of which may be more secure than DES, read the whole
chapter. Chapters 16 and 17 discuss stream algorithms. Chapter 18 focuses on one“way hash
functions; MD5 and SHA are the most common, although I discuss many more. Chapter 19 discusses
public“key encryption algorithms, Chapter 20 discusses public“key digital signature algorithms,
Chapter 21 discusses public“key identification algorithms, and Chapter 22 discusses public“key key
exchange algorithms. The important algorithms are RSA, DSA, Fiat“Shamir, and Diffie“Hellman,
respectively. Chapter 23 has more esoteric public“key algorithms and protocols; the math in this
chapter is quite complicated, so wear your seat belt.

Chapters 24 and 25 (Part IV) turn to the real world of cryptography. Chapter 24 discusses some of
the current implementations of these algorithms and protocols, while Chapter 25 touches on some of
the political issues surrounding cryptography. These chapters are by no means intended to be
comprehensive.

Also included are source code listings for 10 algorithms discussed in Part III. I was unable to include
all the code I wanted to due to space limitations, and cryptographic source code cannot otherwise be
exported. (Amazingly enough, the State Department allowed export of the first edition of this book
with source code, but denied export for a computer disk with the exact same source code on it. Go
figure.) An associated source code disk set includes much more source code than I could fit in this
book; it is probably the largest collection of cryptographic source code outside a military institution. I
can only send source code disks to U.S. and Canadian citizens living in the U.S. and Canada, but
hopefully that will change someday. If you are interested in implementing or playing with the
cryptographic algorithms in this book, get the disk. See the last page of the book for details.

One criticism of this book is that its encyclopedic nature takes away from its readability. This is true,
but I wanted to provide a single reference for those who might come across an algorithm in the
academic literature or in a product. For those who are more interested in a tutorial, I apologize. A lot
is being done in the field; this is the first time so much of it has been gathered between two covers.
Even so, space considerations forced me to leave many things out. I covered topics that I felt were
important, practical, or interesting. If I couldn™t cover a topic in depth, I gave references to articles
and papers that did.

I have done my best to hunt down and eradicate all errors in this book, but many have assured me
that it is an impossible task. Certainly, the second edition has far fewer errors than the first. An
errata listing is available from me and will be periodically posted to the Usenet newsgroup sci.crypt. If
any reader finds an error, please let me know. I™ll send the first person to find each error in the book
a free copy of the source code disk.




About the Author
BRUCE SCHNEIER is president of Counterpane Systems, an Oak Park, Illinois consulting firm
specializing in cryptography and computer security. Bruce is also the author of E“Mail Security (John
Wiley & Sons, 1995) and Protect Your Macintosh (Peachpit Press, 1994); and has written dozens of




Page 11 of 666
Applied Cryptography: Second Edition - Bruce Schneier



articles on cryptography for major magazines. He is a contributing editor to Dr. Dobb™s Journal,
where he edits the “Algorithms Alley” column, and a contributing editor to Computer and
Communications Security Reviews. Bruce serves on the board of directors of the International
Association for Cryptologic Research, is a member of the Advisory Board for the Electronic Privacy
Information Center, and is on the program committee for the New Security Paradigms Workshop. In
addition, he finds time to give frequent lectures on cryptography, computer security, and privacy.

Acknowledgments

The list of people who had a hand in this book may seem unending, but all are worthy of mention. I
would like to thank Don Alvarez, Ross Anderson, Dave Balenson, Karl Barrus, Steve Bellovin, Dan
Bernstein, Eli Biham, Joan Boyar, Karen Cooper, Whit Diffie, Joan Feigenbaum, Phil Karn, Neal
Koblitz, Xuejia Lai, Tom Leranth, Mike Markowitz, Ralph Merkle, Bill Patton, Peter Pearson,
Charles Pfleeger, Ken Pizzini, Bart Preneel, Mark Riordan, Joachim Schurman, and Marc Schwartz
for reading and editing all or parts of the first edition; Marc Vauclair for translating the first edition
into French; Abe Abraham, Ross Anderson, Dave Banisar, Steve Bellovin, Eli Biham, Matt Bishop,
Matt Blaze, Gary Carter, Jan Camenisch, Claude CrŽpeau, Joan Daemen, Jorge Davila, Ed Dawson,
Whit Diffie, Carl Ellison, Joan Feigenbaum, Niels Ferguson, Matt Franklin, Rosario Gennaro, Dieter
Gollmann, Mark Goresky, Richard Graveman, Stuart Haber, Jingman He, Bob Hogue, Kenneth
Iversen, Markus Jakobsson, Burt Kaliski, Phil Karn, John Kelsey, John Kennedy, Lars Knudsen,
Paul Kocher, John Ladwig, Xuejia Lai, Arjen Lenstra, Paul Leyland, Mike Markowitz, Jim Massey,
Bruce McNair, William Hugh Murray, Roger Needham, Clif Neuman, Kaisa Nyberg, Luke
O™Connor, Peter Pearson, RenŽ Peralta, Bart Preneel, Yisrael Radai, Matt Robshaw, Michael Roe,
Phil Rogaway, Avi Rubin, Paul Rubin, Selwyn Russell, Kazue Sako, Mahmoud Salmasizadeh,
Markus Stadler, Dmitry Titov, Jimmy Upton, Marc Vauclair, Serge Vaudenay, Gideon Yuval, Glen
Zorn, and several anonymous government employees for reading and editing all or parts of the second
edition; Lawrie Brown, Leisa Condie, Joan Daemen, Peter Gutmann, Alan Insley, Chris Johnston,
John Kelsey, Xuejia Lai, Bill Leininger, Mike Markowitz, Richard Outerbridge, Peter Pearson, Ken
Pizzini, Colin Plumb, RSA Data Security, Inc., Michael Roe, Michael Wood, and Phil Zimmermann
for providing source code; Paul MacNerland for creating the figures for the first edition; Karen
Cooper for copyedit ing the second edition; Beth Friedman for proofreading the second edition; Carol
Kennedy for indexing the second edition; the readers of sci.crypt and the Cypherpunks mailing list
for commenting on ideas, answering questions, and finding errors in the first edition; Randy Seuss for
providing Internet access; Jeff Duntemann and Jon Erickson for helping me get started; assorted
random Insleys for the impetus, encouragement, support, conversations, friendship, and dinners; and
AT&T Bell Labs for firing me and making this all possible. All these people helped to create a far
better book than I could have created alone.

Bruce Schneier
Oak Park, Ill.
schneier@counterpane.com



Chapter 1
Foundations
1.1 Terminology

Sender and Receiver

Suppose a sender wants to send a message to a receiver. Moreover, this sender wants to send the




Page 12 of 666
Applied Cryptography: Second Edition - Bruce Schneier



message securely: She wants to make sure an eavesdropper cannot read the message.

Messages and Encryption

A message is plaintext (sometimes called cleartext). The process of disguising a message in such a way
as to hide its substance is encryption. An encrypted message is ciphertext. The process of turning
ciphertext back into plaintext is decryption. This is all shown in Figure 1.1.

(If you want to follow the ISO 7498-2 standard, use the terms “encipher” and “decipher.” It seems
that some cultures find the terms “encrypt” and “decrypt” offensive, as they refer to dead bodies.)

The art and science of keeping messages secure is cryptography, and it is practiced by
cryptographers. Cryptanalysts are practitioners of cryptanalysis, the art and science of breaking
ciphertext; that is, seeing through the disguise. The branch of mathematics encompassing both
cryptography and cryptanalysis is cryptology and its practitioners are cryptologists. Modern
cryptologists are generally trained in theoretical mathematics”they have to be.




Figure 1.1 Encryption and Decryption.

Plaintext is denoted by M, for message, or P, for plaintext. It can be a stream of bits, a text file, a
bitmap, a stream of digitized voice, a digital video image...whatever. As far as a computer is
concerned, M is simply binary data. (After this chapter, this book concerns itself with binary data and
computer cryptography.) The plaintext can be intended for either transmission or storage. In any
case, M is the message to be encrypted.

Ciphertext is denoted by C. It is also binary data: sometimes the same size as M, sometimes larger. (By
combining encryption with compression, C may be smaller than M. However, encryption does not
accomplish this.) The encryption function E, operates on M to produce C. Or, in mathematical
notation:

E(M) = C

In the reverse process, the decryption function D operates on C to produce M:

D(C) = M

Since the whole point of encrypting and then decrypting a message is to recover the original plaintext,
the following identity must hold true:

D(E(M)) = M

Authentication, Integrity, and Nonrepudiation

In addition to providing confidentiality, cryptography is often asked to do other jobs:

” Authentication. It should be possible for the receiver of a message to ascertain its
origin; an intruder should not be able to masquerade as someone else.
” Integrity. It should be possible for the receiver of a message to verify that it has not
been modified in transit; an intruder should not be able to substitute a false message for a
legitimate one.



Page 13 of 666
Applied Cryptography: Second Edition - Bruce Schneier



” Nonrepudiation. A sender should not be able to falsely deny later that he sent a
message.

These are vital requirements for social interaction on computers, and are analogous to face-to-face
interactions. That someone is who he says he is...that someone™s credentials”whether a driver™s
license, a medical degree, or a passport”are valid...that a document purporting to come from a
person actually came from that person.... These are the things that authentication, integrity, and
nonrepudiation provide.

Algorithms and Keys

A cryptographic algorithm, also called a cipher, is the mathematical function used for encryption and
decryption. (Generally, there are two related functions: one for encryption and the other for
decryption.)

If the security of an algorithm is based on keeping the way that algorithm works a secret, it is a
restricted algorithm. Restricted algorithms have historical interest, but are woefully inadequate by
today™s standards. A large or changing group of users cannot use them, because every time a user
leaves the group everyone else must switch to a different algorithm. If someone accidentally reveals
the secret, everyone must change their algorithm.

Even more damning, restricted algorithms allow no quality control or standardization. Every group
of users must have their own unique algorithm. Such a group can™t use off-the-shelf hardware or
software products; an eavesdropper can buy the same product and learn the algorithm. They have to
write their own algorithms and implementations. If no one in the group is a good cryptographer, then
they won™t know if they have a secure algorithm.

Despite these major drawbacks, restricted algorithms are enormously popular for low-security
applications. Users either don™t realize or don™t care about the security problems inherent in their
system.

Modern cryptography solves this problem with a key, denoted by K. This key might be any one of a
large number of values. The range of possible values of the key is called the keyspace. Both the
encryption and decryption operations use this key (i.e., they are dependent on the key and this fact is
denoted by the k subscript), so the functions now become:

EK(M) = C
DK(C) = M

Those functions have the property that (see Figure 1.2):

DK(EK(M)) = M

Some algorithms use a different encryption key and decryption key (see Figure 1.3). That is, the
encryption key, K1, is different from the corresponding decryption key, K2. In this case:

EK1(M) = C
DK2(C) = M
DK2(EK1 (M)) = M




Page 14 of 666
Applied Cryptography: Second Edition - Bruce Schneier




All of the security in these algorithms is based in the key (or keys); none is based in the details of the
algorithm. This means that the algorithm can be published and analyzed. Products using the
algorithm can be mass-produced. It doesn™t matter if an eavesdropper knows your algorithm; if she
doesn™t know your particular key, she can™t read your messages.




Figure 1.2 Encryption and decryption with a key.




Figure 1.3 Encryption and decryption with two different keys.

A cryptosystem is an algorithm, plus all possible plaintexts, ciphertexts, and keys.

Symmetric Algorithms

There are two general types of key-based algorithms: symmetric and public-key. Symmetric
algorithms, sometimes called conventional algorithms, are algorithms where the encryption key can
be calculated from the decryption key and vice versa. In most symmetric algorithms, the encryption
key and the decryption key are the same. These algorithms, also called secret-key algorithms, single-
key algorithms, or one-key algorithms, require that the sender and receiver agree on a key before they
can communicate securely. The security of a symmetric algorithm rests in the key; divulging the key
means that anyone could encrypt and decrypt messages. As long as the communication needs to
remain secret, the key must remain secret.

Encryption and decryption with a symmetric algorithm are denoted by:

EK(M) = C
DK(C) = M


Symmetric algorithms can be divided into two categories. Some operate on the plaintext a single bit
(or sometimes byte) at a time; these are called stream algorithms or stream ciphers. Others operate on
the plaintext in groups of bits. The groups of bits are called blocks, and the algorithms are called
block algorithms or block ciphers. For modern computer algorithms, a typical block size is 64 bits”
large enough to preclude analysis and small enough to be workable. (Before computers, algorithms
generally operated on plaintext one character at a time. You can think of this as a stream algorithm
operating on a stream of characters.)

Public-Key Algorithms

Public-key algorithms (also called asymmetric algorithms) are designed so that the key used for
encryption is different from the key used for decryption. Furthermore, the decryption key cannot (at
least in any reasonable amount of time) be calculated from the encryption key. The algorithms are
called “public-key” because the encryption key can be made public: A complete stranger can use the
encryption key to encrypt a message, but only a specific person with the corresponding decryption
key can decrypt the message. In these systems, the encryption key is often called the public key, and




Page 15 of 666
Applied Cryptography: Second Edition - Bruce Schneier



the decryption key is often called the private key. The private key is sometimes also called the secret
key, but to avoid confusion with symmetric algorithms, that tag won™t be used here.

Encryption using public key K is denoted by:

EK(M) = C

Even though the public key and private key are different, decryption with the corresponding private
key is denoted by:

DK(C) = M

Sometimes, messages will be encrypted with the private key and decrypted with the public key; this is
used in digital signatures (see Section 2.6). Despite the possible confusion, these operations are
denoted by, respectively:

EK(M) = C
DK(C) = M

Cryptanalysis

The whole point of cryptography is to keep the plaintext (or the key, or both) secret from
eavesdroppers (also called adversaries, attackers, interceptors, interlopers, intruders, opponents, or
simply the enemy). Eavesdroppers are assumed to have complete access to the communications
between the sender and receiver.

Cryptanalysis is the science of recovering the plaintext of a message without access to the key.
Successful cryptanalysis may recover the plaintext or the key. It also may find weaknesses in a
cryptosystem that eventually lead to the previous results. (The loss of a key through noncryptanalytic
means is called a compromise.)

An attempted cryptanalysis is called an attack. A fundamental assumption in cryptanalysis, first
enunciated by the Dutchman A. Kerckhoffs in the nineteenth century, is that the secrecy must reside
entirely in the key [794]. Kerckhoffs assumes that the cryptanalyst has complete details of the
cryptographic algorithm and implementation. (Of course, one would assume that the CIA does not
make a habit of telling Mossad about its cryptographic algorithms, but Mossad probably finds out
anyway.) While real-world cryptanalysts don™t always have such detailed information, it™s a good
assumption to make. If others can™t break an algorithm, even with knowledge of how it works, then
they certainly won™t be able to break it without that knowledge.

There are four general types of cryptanalytic attacks. Of course, each of them assumes that the
cryptanalyst has complete knowledge of the encryption algorithm used:

1. Ciphertext-only attack. The cryptanalyst has the ciphertext of several messages, all of
which have been encrypted using the same encryption algorithm. The cryptanalyst™s job is to
recover the plaintext of as many messages as possible, or better yet to deduce the key (or keys)
used to encrypt the messages, in order to decrypt other messages encrypted with the same keys.
Given: C1 = Ek(P1), C2 = Ek(P2),...Ci = Ek(Pi)
Deduce: Either P1, P2,...Pi; k; or an algorithm to infer Pi+1 from Ci+1 = Ek(Pi+1)
2. Known-plaintext attack. The cryptanalyst has access not only to the ciphertext of several
messages, but also to the plaintext of those messages. His job is to deduce the key (or keys) used to



Page 16 of 666
Applied Cryptography: Second Edition - Bruce Schneier



encrypt the messages or an algorithm to decrypt any new messages encrypted with the same key (or
keys).
Given: P1, C1 = Ek(P1), P2, C2 = Ek(P2),...Pi, Ci = Ek(Pi)
Deduce: Either k, or an algorithm to infer Pi+1 from Ci+1 = Ek(Pi+1)
3. Chosen-plaintext attack. The cryptanalyst not only has access to the ciphertext and
associated plaintext for several messages, but he also chooses the plaintext that gets encrypted. This
is more powerful than a known-plaintext attack, because the cryptanalyst can choose specific
plaintext blocks to encrypt, ones that might yield more information about the key. His job is to
deduce the key (or keys) used to encrypt the messages or an algorithm to decrypt any new messages
encrypted with the same key (or keys).
Given: P1, C1 = Ek(P1), P2, C2 = Ek(P2),...Pi, Ci = Ek(Pi), where the cryptanalyst gets to
choose P1, P2,...Pi
Deduce: Either k, or an algorithm to infer Pi+1 from Ci+1 = Ek(Pi+1)
4. Adaptive-chosen-plaintext attack. This is a special case of a chosen-plaintext attack.
Not only can the cryptanalyst choose the plaintext that is encrypted, but he can also modify his
choice based on the results of previous encryption. In a chosen-plaintext attack, a cryptanalyst
might just be able to choose one large block of plaintext to be encrypted; in an adaptive-chosen-
plaintext attack he can choose a smaller block of plaintext and then choose another based on the
results of the first, and so forth.


There are at least three other types of cryptanalytic attack.

5. Chosen-ciphertext attack. The cryptanalyst can choose different ciphertexts to be
decrypted and has access to the decrypted plaintext. For example, the cryptanalyst has access to
a tamperproof box that does automatic decryption. His job is to deduce the key.
Given: C1, P1 = Dk(C1), C2, P2 = Dk(C2),...Ci, Pi = Dk(Ci)
Deduce: k

This attack is primarily applicable to public-key algorithms and will be discussed in Section
19.3. A chosen-ciphertext attack is sometimes effective against a symmetric algorithm as well.
(Sometimes a chosen-plaintext attack and a chosen-ciphertext attack are together known as a
chosen-text attack.)
6. Chosen-key attack. This attack doesn™t mean that the cryptanalyst can choose the key;
it means that he has some knowledge about the relationship between different keys. It™s strange
and obscure, not very practical, and discussed in Section 12.4.
7. Rubber-hose cryptanalysis. The cryptanalyst threatens, blackmails, or tortures
someone until they give him the key. Bribery is sometimes referred to as a purchase-key attack.
These are all very powerful attacks and often the best way to break an algorithm.

Known-plaintext attacks and chosen-plaintext attacks are more common than you might think. It is
not unheard-of for a cryptanalyst to get a plaintext message that has been encrypted or to bribe
someone to encrypt a chosen message. You may not even have to bribe someone; if you give a message
to an ambassador, you will probably find that it gets encrypted and sent back to his country for
consideration. Many messages have standard beginnings and endings that might be known to the
cryptanalyst. Encrypted source code is especially vulnerable because of the regular appearance of
keywords: #define, struct, else, return. Encrypted executable code has the same kinds of problems:
functions, loop structures, and so on. Known-plaintext attacks (and even chosen-plaintext attacks)
were successfully used against both the Germans and the Japanese during World War II. David
Kahn™s books [794,795,796] have historical examples of these kinds of attacks.




Page 17 of 666
Applied Cryptography: Second Edition - Bruce Schneier




And don™t forget Kerckhoffs™s assumption: If the strength of your new cryptosystem relies on the fact
that the attacker does not know the algorithm™s inner workings, you™re sunk. If you believe that
keeping the algorithm™s insides secret improves the security of your cryptosystem more than letting
the academic community analyze it, you™re wrong. And if you think that someone won™t disassemble
your code and reverse-engineer your algorithm, you™re naïve. (In 1994 this happened with the RC4
algorithm”see Section 17.1.) The best algorithms we have are the ones that have been made public,
have been attacked by the world™s best cryptographers for years, and are still unbreakable. (The
National Security Agency keeps their algorithms secret from outsiders, but they have the best
cryptographers in the world working within their walls”you don™t. Additionally, they discuss their
algorithms with one another, relying on peer review to uncover any weaknesses in their work.)

Cryptanalysts don™t always have access to the algorithms, as when the United States broke the
Japanese diplomatic code PURPLE during World War II [794]”but they often do. If the algorithm is
being used in a commercial security program, it is simply a matter of time and money to disassemble
the program and recover the algorithm. If the algorithm is being used in a military communications
system, it is simply a matter of time and money to buy (or steal) the equipment and reverse-engineer
the algorithm.

Those who claim to have an unbreakable cipher simply because they can™t break it are either geniuses
or fools. Unfortunately, there are more of the latter in the world. Beware of people who extol the
virtues of their algorithms, but refuse to make them public; trusting their algorithms is like trusting
snake oil.

Good cryptographers rely on peer review to separate the good algorithms from the bad.

Security of Algorithms

Different algorithms offer different degrees of security; it depends on how hard they are to break. If
the cost required to break an algorithm is greater than the value of the encrypted data, then you™re
probably safe. If the time required to break an algorithm is longer than the time the encrypted data
must remain secret, then you™re probably safe. If the amount of data encrypted with a single key is
less than the amount of data necessary to break the algorithm, then you™re probably safe.

I say “probably” because there is always a chance of new breakthroughs in cryptanalysis. On the
other hand, the value of most data decreases over time. It is important that the value of the data
always remain less than the cost to break the security protecting it.

Lars Knudsen classified these different categories of breaking an algorithm. In decreasing order of
severity [858]:

1. Total break. A cryptanalyst finds the key, K, such that DK(C) = P.
2. Global deduction. A cryptanalyst finds an alternate algorithm, A, equivalent to DK(C),
without knowing K.
3. Instance (or local) deduction. A cryptanalyst finds the plaintext of an intercepted
ciphertext.
4. Information deduction. A cryptanalyst gains some information about the key or
plaintext. This information could be a few bits of the key, some information about the form of
the plaintext, and so forth.

An algorithm is unconditionally secure if, no matter how much ciphertext a cryptanalyst has, there is
not enough information to recover the plaintext. In point of fact, only a one-time pad (see Section 1.5)



Page 18 of 666
Applied Cryptography: Second Edition - Bruce Schneier



is unbreakable given infinite resources. All other cryptosystems are breakable in a ciphertext-only
attack, simply by trying every possible key one by one and checking whether the resulting plaintext is
meaningful. This is called a brute-force attack (see Section 7.1).

Cryptography is more concerned with cryptosystems that are computationally infeasible to break. An
algorithm is considered computationally secure (sometimes called strong) if it cannot be broken with
available resources, either current or future. Exactly what constitutes “available resources” is open to
interpretation.

You can measure the complexity (see Section 11.1) of an attack in different ways:

1. Data complexity. The amount of data needed as input to the attack.
2. Processing complexity. The time needed to perform the attack. This is often called the
work factor.
3. Storage requirements. The amount of memory needed to do the attack.

As a rule of thumb, the complexity of an attack is taken to be the minimum of these three factors.
Some attacks involve trading off the three complexities: A faster attack might be possible at the
expense of a greater storage requirement.

Complexities are expressed as orders of magnitude. If an algorithm has a processing complexity of
2128, then 2128 operations are required to break the algorithm. (These operations may be complex and
time-consuming.) Still, if you assume that you have enough computing speed to perform a million
operations every second and you set a million parallel processors against the task, it will still take over
1019 years to recover the key. That™s a billion times the age of the universe.


While the complexity of an attack is constant (until some cryptanalyst finds a better attack, of course),
computing power is anything but. There have been phenomenal advances in computing power during
the last half-century and there is no reason to think this trend won™t continue. Many cryptanalytic
attacks are perfect for parallel machines: The task can be broken down into billions of tiny pieces and
none of the processors need to interact with each other. Pronouncing an algorithm secure simply
because it is infeasible to break, given current technology, is dicey at best. Good cryptosystems are
designed to be infeasible to break with the computing power that is expected to evolve many years in
the future.

Historical Terms

Historically, a code refers to a cryptosystem that deals with linguistic units: words, phrases, sentences,
and so forth. For example, the word “OCELOT” might be the ciphertext for the entire phrase
“TURN LEFT 90 DEGREES,” the word “LOLLIPOP” might be the ciphertext for “TURN RIGHT
90 DEGREES,” and the words “BENT EAR” might be the ciphertext for “HOWITZER.” Codes of
this type are not discussed in this book; see [794,795]. Codes are only useful for specialized
circumstances. Ciphers are useful for any circumstance. If your code has no entry for
“ANTEATERS,” then you can™t say it. You can say anything with a cipher.

1.2 Steganography

Steganography serves to hide secret messages in other messages, such that the secret™s very existence
is concealed. Generally the sender writes an innocuous message and then conceals a secret message on
the same piece of paper. Historical tricks include invisible inks, tiny pin punctures on selected
characters, minute differences between handwritten characters, pencil marks on typewritten



Page 19 of 666
Applied Cryptography: Second Edition - Bruce Schneier



characters, grilles which cover most of the message except for a few characters, and so on.

More recently, people are hiding secret messages in graphic images. Replace the least significant bit of
each byte of the image with the bits of the message. The graphical image won™t change appreciably”
most graphics standards specify more gradations of color than the human eye can notice”and the
message can be stripped out at the receiving end. You can store a 64-kilobyte message in a 1024 —
1024 grey-scale picture this way. Several public-domain programs do this sort of thing.

Peter Wayner™s mimic functions obfuscate messages. These functions modify a message so that its
statistical profile resembles that of something else: the classifieds section of The New York Times, a
play by Shakespeare, or a newsgroup on the Internet [1584,1585]. This type of steganography won™t
fool a person, but it might fool some big computers scanning the Internet for interesting messages.

1.3 Substitution Ciphers and Transposition Ciphers

Before computers, cryptography consisted of character-based algorithms. Different cryptographic
algorithms either substituted characters for one another or transposed characters with one another.
The better algorithms did both, many times each.

Things are more complex these days, but the philosophy remains the same. The primary change is
that algorithms work on bits instead of characters. This is actually just a change in the alphabet size:
from 26 elements to two elements. Most good cryptographic algorithms still combine elements of
substitution and transposition.

Substitution Ciphers

A substitution cipher is one in which each character in the plaintext is substituted for another
character in the ciphertext. The receiver inverts the substitution on the ciphertext to recover the
plaintext.

In classical cryptography, there are four types of substitution ciphers:

” A simple substitution cipher, or monoalphabetic cipher, is one in which each character
of the plaintext is replaced with a corresponding character of ciphertext. The cryptograms in
newspapers are simple substitution ciphers.
” A homophonic substitution cipher is like a simple substitution cryptosystem, except a
single character of plaintext can map to one of several characters of ciphertext. For example,
“A” could correspond to either 5, 13, 25, or 56, “B” could correspond to either 7, 19, 31, or 42,
and so on.
” A polygram substitution cipher is one in which blocks of characters are encrypted in
groups. For example, “ABA” could correspond to “RTQ,” “ABB” could correspond to “SLL,”
and so on.
” A polyalphabetic substitution cipher is made up of multiple simple substitution ciphers.
For example, there might be five different simple substitution ciphers used; the particular one
used changes with the position of each character of the plaintext.

The famous Caesar Cipher, in which each plaintext character is replaced by the character three to the
right modulo 26 (“A” is replaced by “D,” “B” is replaced by “E,”..., “W” is replaced by “Z,” “X” is
replaced by “A,” “Y” is replaced by “B,” and “Z” is replaced by “C”) is a simple substitution cipher.
It™s actually even simpler, because the ciphertext alphabet is a rotation of the plaintext alphabet and
not an arbitrary permutation.




Page 20 of 666
Applied Cryptography: Second Edition - Bruce Schneier




ROT13 is a simple encryption program commonly found on UNIX systems; it is also a simple
substitution cipher. In this cipher, “A” is replaced by “N,” “B” is replaced by “O,” and so on. Every
letter is rotated 13 places.

Encrypting a file twice with ROT13 restores the original file.

P = ROT13 (ROT13 (P))

ROT13 is not intended for security; it is often used in Usenet posts to hide potentially offensive text, to
avoid giving away the solution to a puzzle, and so forth.

Simple substitution ciphers can be easily broken because the cipher does not hide the underlying
frequencies of the different letters of the plaintext. All it takes is about 25 English characters before a
good cryptanalyst can reconstruct the plaintext [1434]. An algorithm for solving these sorts of ciphers
can be found in [578,587,1600,78,1475,1236,880]. A good computer algorithm is [703].

Homophonic substitution ciphers were used as early as 1401 by the Duchy of Mantua [794]. They are
much more complicated to break than simple substitution ciphers, but still do not obscure all of the
statistical properties of the plaintext language. With a known-plaintext attack, the ciphers are trivial
to break. A ciphertext-only attack is harder, but only takes a few seconds on a computer. Details are
in [1261].

Polygram substitution ciphers are ciphers in which groups of letters are encrypted together. The
Playfair cipher, invented in 1854, was used by the British during World War I [794]. It encrypts pairs
of letters together. Its cryptanalysis is discussed in [587,1475,880]. The Hill cipher is another example
of a polygram substitution cipher [732]. Sometimes you see Huffman coding used as a cipher; this is
an insecure polygram substitution cipher.

Polyalphabetic substitution ciphers were invented by Leon Battista in 1568 [794]. They were used by
the Union army during the American Civil War. Despite the fact that they can be broken easily
[819,577,587,794] (especially with the help of computers), many commercial computer security
products use ciphers of this form [1387,1390,1502]. (Details on how to break this encryption scheme,
as used in WordPerfect, can be found in [135,139].) The Vigenère cipher, first published in 1586, and
the Beaufort cipher are also examples of polyalphabetic substitution ciphers.

Polyalphabetic substitution ciphers have multiple one-letter keys, each of which is used to encrypt one
letter of the plaintext. The first key encrypts the first letter of the plaintext, the second key encrypts
the second letter of the plaintext, and so on. After all the keys are used, the keys are recycled. If there
were 20 one-letter keys, then every twentieth letter would be encrypted with the same key. This is
called the period of the cipher. In classical cryptography, ciphers with longer periods were
significantly harder to break than ciphers with short periods. There are computer techniques that can
easily break substitution ciphers with very long periods.

A running-key cipher”sometimes called a book cipher”in which one text is used to encrypt another
text, is another example of this sort of cipher. Even though this cipher has a period the length of the
text, it can also be broken easily [576,794].


Transposition Ciphers

In a transposition cipher the plaintext remains the same, but the order of characters is shuffled
around. In a simple columnar transposition cipher, the plaintext is written horizontally onto a piece of



Page 21 of 666
Applied Cryptography: Second Edition - Bruce Schneier



graph paper of fixed width and the ciphertext is read off vertically (see Figure 1.4). Decryption is a
matter of writing the ciphertext vertically onto a piece of graph paper of identical width and then
reading the plaintext off horizontally.

Cryptanalysis of these ciphers is discussed in [587,1475]. Since the letters of the ciphertext are the
same as those of the plaintext, a frequency analysis on the ciphertext would reveal that each letter has
approximately the same likelihood as in English. This gives a very good clue to a cryptanalyst, who
can then use a variety of techniques to determine the right ordering of the letters to obtain the
plaintext. Putting the ciphertext through a second transposition cipher greatly enhances security.
There are even more complicated transposition ciphers, but computers can break almost all of them.

The German ADFGVX cipher, used during World War I, is a transposition cipher combined with a
simple substitution. It was a very complex algorithm for its day but was broken by Georges Painvin, a
French cryptanalyst [794].

Although many modern algorithms use transposition, it is troublesome because it requires a lot of
memory and sometimes requires messages to be only certain lengths. Substitution is far more
common.

Rotor Machines

In the 1920s, various mechanical encryption devices were invented to automate the process of
encryption. Most were based on the concept of a rotor, a mechanical wheel wired to perform a general
substitution.

A rotor machine has a keyboard and a series of rotors, and implements a version of the Vigenère
cipher. Each rotor is an arbitrary permutation of the alphabet, has 26 positions, and performs a
simple substitution. For example, a rotor might be wired to substitute “F” for “A,” “U” for “B,” “L”
for “C,” and so on. And the output pins of one rotor are connected to the input pins of the next.




Figure 1.4 Columnar transposition cipher.

For example, in a 4-rotor machine the first rotor might substitute “F” for “A,” the second might
substitute “Y” for “F,” the third might substitute “E” for “Y,” and the fourth might substitute “C”
for “E”; “C” would be the output ciphertext. Then some of the rotors shift, so next time the
substitutions will be different.

It is the combination of several rotors and the gears moving them that makes the machine secure.
Because the rotors all move at different rates, the period for an n-rotor machine is 26n. Some rotor
machines can also have a different number of positions on each rotor, further frustrating
cryptanalysis.

The best-known rotor device is the Enigma. The Enigma was used by the Germans during World
War II. The idea was invented by Arthur Scherbius and Arvid Gerhard Damm in Europe. It was
patented in the United States by Arthur Scherbius [1383]. The Germans beefed up the basic design
considerably for wartime use.




Page 22 of 666
Applied Cryptography: Second Edition - Bruce Schneier




The German Enigma had three rotors, chosen from a set of five, a plugboard that slightly permuted
the plaintext, and a reflecting rotor that caused each rotor to operate on each plaintext letter twice. As
complicated as the Enigma was, it was broken during World War II. First, a team of Polish
cryptographers broke the German Enigma and explained their attack to the British. The Germans
modified their Enigma as the war progressed, and the British continued to cryptanalyze the new
versions. For explanations of how rotor ciphers work and how they were broken, see
[794,86,448,498,446,880,1315,1587,690]. Two fascinating accounts of how the Enigma was broken are
[735,796].

Further Reading

This is not a book about classical cryptography, so I will not dwell further on these subjects. Two
excellent precomputer cryptology books are [587,1475]; [448] presents some modern cryptanalysis of
cipher machines. Dorothy Denning discusses many of these ciphers in [456] and [880] has some fairly
complex mathematical analysis of the same ciphers. Another older cryptography text, which discusses
analog cryptography, is [99]. An article that presents a good overview of the subject is [579]. David
Kahn™s historical cryptography books are also excellent [794,795,796].

1.4 Simple XOR

XOR is exclusive-or operation: ˜^™ in C or in mathematical notation. It™s a standard operation on
bits:

0 0=0
0 1=1
1 0=1
1 1=0

Also note that:

a a=0
a b b=a

The simple-XOR algorithm is really an embarrassment; it™s nothing more than a Vigenère
polyalphabetic cipher. It™s here only because of its prevalence in commercial software packages, at
least those in the MS-DOS and Macintosh worlds [1502,1387]. Unfortunately, if a software security
program proclaims that it has a “proprietary” encryption algorithm”significantly faster than DES”
the odds are that it is some variant of this.

/* Usage: crypto key input_file output_file */

void main (int argc, char *argv[])

{
FILE *fi, *fo;
char *cp;
int c;

if ((cp = argv[1]) && *cp!='\0') {
if ((fi = fopen(argv[2], “rb”)) != NULL) {
if ((fo = fopen(argv[3], “wb”)) != NULL) {
while ((c = getc(fi)) != EOF) {
if (!*cp) cp = argv[1];
c ^= *(cp++);
putc(c,fo);




Page 23 of 666
Applied Cryptography: Second Edition - Bruce Schneier



}
fclose(fo);
}
fclose(fi);
}
}

. 1
( 29)



>>