This is a Chapter from the Handbook of Applied Cryptography, by A. Menezes, P. van
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c 1997 by CRC Press, Inc.
Chapter 1Overview of Cryptography
Contents in Brief
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Information security and cryptography . . . . . . . . . . . . . . 2
1.3 Background on functions . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Basic terminology and concepts . . . . . . . . . . . . . . . . . . . 11
1.5 Symmetric-key encryption . . . . . . . . . . . . . . . . . . . . . 15
1.6 Digital signatures . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Authentication and identiﬁcation . . . . . . . . . . . . . . . . . . 24
1.8 Public-key cryptography . . . . . . . . . . . . . . . . . . . . . . 25
1.9 Hash functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.10 Protocols and mechanisms . . . . . . . . . . . . . . . . . . . . . 33
1.11 Key establishment, management, and certiﬁcation . . . . . . . . . 35
1.12 Pseudorandom numbers and sequences . . . . . . . . . . . . . . 39
1.13 Classes of attacks and security models . . . . . . . . . . . . . . . 41
1.14 Notes and further references . . . . . . . . . . . . . . . . . . . . 45
1.1 Introduction
Cryptography has a long and fascinating history. The most complete non-technical account
of the subject is Kahn’s The Codebreakers. This book traces cryptography from its initial
and limited use by the Egyptians some 4000 years ago, to the twentieth century where it
played a crucial role in the outcome of both world wars. Completed in 1963, Kahn’s book
covers those aspects of the history which were most signiﬁcant (up to that time) to the development
of the subject. The predominant practitioners of the art were those associated with
the military, the diplomatic service and government in general. Cryptography was used as
a tool to protect national secrets and strategies.
The proliferation of computers and communications systems in the 1960s brought with
it a demand from the private sector for means to protect information in digital form and to
provide security services. Beginning with the work of Feistel at IBM in the early 1970s and
culminating in 1977 with the adoption as a U.S. Federal Information Processing Standard
for encrypting unclassiﬁed information, DES, the Data Encryption Standard, is the most
well-known cryptographic mechanism in history. It remains the standard means for securing
electronic commerce for many ﬁnancial institutions around the world.
The most strikingdevelopmentin the historyof cryptographycame in 1976when Difﬁe
and Hellman published New Directions in Cryptography. This paper introduced the revolutionary
concept of public-key cryptography and also provided a new and ingenious method
1
2 Ch. 1 Overview of Cryptography
for key exchange, the security of which is based on the intractability of the discrete logarithm
problem. Although the authors had no practical realization of a public-key encryption
scheme at the time, the idea was clear and it generated extensive interest and activity
in the cryptographic community. In 1978 Rivest, Shamir, and Adleman discovered the ﬁrst
practical public-key encryption and signature scheme, now referred to as RSA. The RSA
scheme is based on another hard mathematical problem, the intractability of factoring large
integers. This application of a hard mathematical problem to cryptography revitalized efforts
to ﬁnd more efﬁcient methods to factor. The 1980s saw major advances in this area
but none which rendered the RSA system insecure. Another class of powerful and practical
public-key schemes was found by ElGamal in 1985. These are also based on the discrete
logarithm problem.
One of the most signiﬁcant contributions provided by public-key cryptography is the
digital signature. In 1991 the ﬁrst international standard for digital signatures (ISO/IEC
9796) was adopted. It is based on the RSA public-key scheme. In 1994 the U.S. Government
adopted the Digital Signature Standard, a mechanism based on the ElGamal publickey
scheme.
The search for new public-key schemes, improvements to existing cryptographic mechanisms,
and proofs of security continues at a rapid pace. Various standards and infrastructures
involving cryptography are being put in place. Security products are being developed
to address the security needs of an information intensive society.
The purpose of this book is to give an up-to-date treatise of the principles, techniques,
and algorithms of interest in cryptographic practice. Emphasis has been placed on those
aspects which are most practical and applied. The reader will be made aware of the basic
issues and pointed to speciﬁc related research in the literature where more indepth discussions
can be found. Due to the volume of material which is covered, most results will be
stated without proofs. This also serves the purpose of not obscuring the very applied nature
of the subject. This book is intended for both implementers and researchers. It describes
algorithms, systems, and their interactions.
Chapter 1 is a tutorial on the many and various aspects of cryptography. It does not
attempt to convey all of the details and subtleties inherent to the subject. Its purpose is to
introducethe basic issues and principlesand to pointthe reader to appropriatechapters in the
book for more comprehensive treatments. Speciﬁc techniques are avoided in this chapter.
1.2 Information security and cryptography
The concept of information will be taken to be an understood quantity. To introduce cryptography,
an understanding of issues related to information security in general is necessary.
Information security manifests itself in many ways according to the situation and requirement.
Regardless of who is involved, to one degree or another, all parties to a transaction
must have conﬁdence that certain objectives associated with informationsecurity have been
met. Some of these objectives are listed in Table 1.1.
Over the centuries, an elaborate set of protocols and mechanisms has been created to
deal with information security issues when the information is conveyed by physical documents.
Often the objectives of information security cannot solely be achieved through
mathematical algorithms and protocols alone, but require procedural techniques and abidance
of laws to achieve the desired result. For example, privacy of letters is provided by
sealed envelopes delivered by an accepted mail service. The physical security of the envelope
is, for practical necessity, limited and so laws are enacted which make it a criminal
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.2 Information security and cryptography 3
privacy
or conﬁdentiality
keeping information secret from all but those who are authorized
to see it.
data integrity ensuring information has not been altered by unauthorized or
unknown means.
entity authentication
or identiﬁcation
corroboration of the identity of an entity (e.g., a person, a
computer terminal, a credit card, etc.).
message
authentication
corroborating the source of information; also known as data
origin authentication.
signature a means to bind information to an entity.
authorization conveyance, to another entity, of ofﬁcial sanction to do or be
something.
validation a means to provide timeliness of authorization to use or manipulate
information or resources.
access control restricting access to resources to privileged entities.
certiﬁcation endorsement of information by a trusted entity.
timestamping recording the time of creation or existence of information.
witnessing verifying the creation or existence of information by an entity
other than the creator.
receipt acknowledgement that information has been received.
conﬁrmation acknowledgement that services have been provided.
ownership a means to provide an entity with the legal right to use or
transfer a resource to others.
anonymity concealing the identity of an entity involved in some process.
non-repudiation preventing the denial of previous commitments or actions.
revocation retraction of certiﬁcation or authorization.
Table 1.1: Some information security objectives.
offense to open mail for which one is not authorized. It is sometimes the case that security
is achieved not through the information itself but through the physical document recording
it. For example, paper currency requires special inks and material to prevent counterfeiting.
Conceptually, the way information is recorded has not changed dramatically over time.
Whereas information was typically stored and transmitted on paper, much of it now resides
on magnetic media and is transmitted via telecommunications systems, some wireless.
What has changed dramatically is the ability to copy and alter information. One can
make thousands of identical copies of a piece of information stored electronically and each
is indistinguishable from the original. With information on paper, this is much more difﬁcult.
What is needed then for a society where information is mostly stored and transmitted
in electronic form is a means to ensure information security which is independent of the
physical medium recording or conveying it and such that the objectives of information security
rely solely on digital information itself.
One of the fundamental tools used in information security is the signature. It is a building
block for many other services such as non-repudiation, data origin authentication, identiﬁcation,
and witnessing, to mention a few. Having learned the basics in writing, an individual
is taught how to produce a handwritten signature for the purpose of identiﬁcation.
At contract age the signature evolves to take on a very integral part of the person’s identity.
This signature is intended to be unique to the individual and serve as a means to identify,
authorize, and validate. With electronic information the concept of a signature needs to be
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
4 Ch. 1 Overview of Cryptography
redressed; it cannot simply be something unique to the signer and independent of the information
signed. Electronic replication of it is so simple that appending a signature to a
document not signed by the originator of the signature is almost a triviality.
Analogues of the “paper protocols” currently in use are required. Hopefully these new
electronic based protocols are at least as good as those they replace. There is a unique opportunity
for society to introduce new and more efﬁcient ways of ensuring information security.
Much can be learned from the evolution of the paper based system, mimicking those
aspects which have served us well and removing the inefﬁciencies.
Achieving information security in an electronic society requires a vast array of technical
and legal skills. There is, however, no guarantee that all of the information security objectives
deemed necessary can be adequately met. The technical means is provided through
cryptography.
1.1 Deﬁnition Cryptography is the study of mathematical techniques related to aspects of information
security such as conﬁdentiality, data integrity, entity authentication, and data origin
authentication.
Cryptography is not the only means of providing information security, but rather one set of
techniques.
Cryptographic goals
Of all the information security objectives listed in Table 1.1, the following four form a
frameworkuponwhich the others will be derived: (1) privacyor conﬁdentiality(§1.5, §1.8);
(2) data integrity (§1.9); (3) authentication (§1.7); and (4) non-repudiation (§1.6).
1. Conﬁdentiality is a service used to keep the content of information from all but those
authorized to have it. Secrecy is a term synonymous with conﬁdentiality and privacy.
There are numerous approaches to providing conﬁdentiality, ranging from physical
protection to mathematical algorithms which render data unintelligible.
2. Data integrity is a service which addresses the unauthorized alteration of data. To
assure data integrity, one must have the ability to detect data manipulation by unauthorized
parties. Data manipulation includes such things as insertion, deletion, and
substitution.
3. Authenticationis a service related to identiﬁcation. This function applies to both entities
andinformationitself. Two parties enteringinto a communicationshouldidentify
each other. Informationdelivered over a channel should be authenticated as to origin,
date of origin, data content, time sent, etc. For these reasons this aspect of cryptography
is usually subdivided into two major classes: entity authentication and data
origin authentication. Data origin authentication implicitly provides data integrity
(for if a message is modiﬁed, the source has changed).
4. Non-repudiationis a service which preventsan entity fromdenyingpreviouscommitments
or actions. When disputes arise due to an entity denying that certain actions
were taken, a means to resolve the situation is necessary. For example, one entity
may authorize the purchase of property by another entity and later deny such authorization
was granted. A procedure involving a trusted third party is needed to resolve
the dispute.
A fundamental goal of cryptography is to adequately address these four areas in both
theory and practice. Cryptography is about the prevention and detection of cheating and
other malicious activities.
This book describes a number of basic cryptographic tools (primitives) used to provide
information security. Examples of primitives include encryption schemes (§1.5 and §1.8),
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.2 Information security and cryptography 5
hash functions (§1.9), and digital signature schemes (§1.6). Figure 1.1 provides a schematic
listing of the primitives considered and how they relate. Many of these will be brieﬂy introduced
in this chapter, with detailed discussion left to later chapters. These primitives should
Symmetric-key
ciphers
Primitives
Unkeyed
Arbitrary length
hash functions
hash functions (MACs)
Arbitrary length
ciphers
Block
Stream
ciphers
Pseudorandom
sequences
Random sequences
Public-key
Primitives
Public-key
ciphers
Identiﬁcation primitives
Signatures
Identiﬁcation primitives
Primitives
Security Symmetric-key
Primitives
One-way permutations
Signatures
Figure 1.1: A taxonomy of cryptographic primitives.
be evaluated with respect to various criteria such as:
1. level of security. This is usually difﬁcult to quantify. Often it is given in terms of the
number of operations required (using the best methods currently known) to defeat the
intended objective. Typically the level of security is deﬁned by an upper bound on
the amount of work necessary to defeat the objective. This is sometimes called the
work factor (see §1.13.4).
2. functionality. Primitives will need to be combined to meet various information security
objectives. Which primitives are most effective for a given objective will be
determined by the basic properties of the primitives.
3. methods of operation. Primitives, when applied in various ways and with various inputs,
will typically exhibit different characteristics; thus, one primitive could provide
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
6 Ch. 1 Overview of Cryptography
very different functionality depending on its mode of operation or usage.
4. performance. This refers to the efﬁciency of a primitive in a particular mode of operation.
(For example, an encryption algorithm may be rated by the number of bits
per second which it can encrypt.)
5. ease of implementation. This refers to the difﬁculty of realizing the primitive in a
practical instantiation. This might include the complexity of implementing the primitive
in either a software or hardware environment.
The relative importance of various criteria is very much dependent on the application
and resources available. For example, in an environmentwhere computing power is limited
one may have to trade off a very high level of security for better performance of the system
as a whole.
Cryptography, over the ages, has been an art practised by many who have devised ad
hoc techniques to meet some of the information security requirements. The last twenty
years have been a period of transition as the discipline moved from an art to a science. There
are now several international scientiﬁc conferences devoted exclusively to cryptography
and also an international scientiﬁc organization, the International Association for Cryptologic
Research (IACR), aimed at fostering research in the area.
This book is about cryptography: the theory, the practice, and the standards.
1.3 Background on functions
While this book is not a treatise on abstract mathematics, a familiarity with basic mathematical
concepts will prove to be useful. One concept which is absolutely fundamental to
cryptography is that of a function in the mathematical sense. A function is alternately referred
to as a mapping or a transformation.
1.3.1 Functions (1-1, one-way, trapdoor one-way)
A set consists of distinct objects which are called elements of the set. For example, a set X
might consist of the elements a, b, c, and this is denoted X = {a, b, c}.
1.2 Deﬁnition A function is deﬁned by two sets X and Y and a rule f which assigns to each
element in X precisely one element in Y . The set X is called the domain of the function
and Y the codomain. If x is an element of X (usually written x ∈ X) the image of x is the
element in Y which the rule f associates with x; the image y of x is denoted by y = f(x).
Standard notation for a function f from set X to set Y is f : X −→ Y . If y ∈ Y , then a
preimage of y is an element x ∈ X for which f(x) = y. The set of all elements in Y which
have at least one preimage is called the image of f, denoted Im(f).
1.3 Example (function) Consider the sets X = {a, b, c}, Y = {1, 2, 3, 4}, and the rule f
from X to Y deﬁned as f(a) = 2, f(b) = 4, f(c) = 1. Figure 1.2 shows a schematic of
the sets X, Y and the function f. The preimage of the element 2 is a. The image of f is
{1, 2, 4}.
Thinking of a function in terms of the schematic (sometimes called a functional diagram)
given in Figure 1.2, each element in the domain X has precisely one arrowed line
originating from it. Each element in the codomain Y can have any number of arrowed lines
incident to it (including zero lines).
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.3 Background on functions 7
1
3
4
c
b
a
2
f
YX
Figure 1.2: A function f from a set X of three elements to a set Y of four elements.
Often only the domain X and the rule f are given and the codomain is assumed to be
the image of f. This point is illustrated with two examples.
1.4 Example (function) Take X = {1, 2, 3, . . . , 10} and let f be the rule that for each x ∈ X,
f(x) = rx, where rx is the remainder when x2
is divided by 11. Explicitly then
f(1) = 1 f(2) = 4 f(3) = 9 f(4) = 5 f(5) = 3
f(6) = 3 f(7) = 5 f(8) = 9 f(9) = 4 f(10) = 1.
The image of f is the set Y = {1, 3, 4, 5, 9}.
1.5 Example (function) Take X = {1, 2, 3, . . . , 1050
} and let f be the rule f(x) = rx, where
rx is the remainder when x2
is divided by 1050
+ 1 for all x ∈ X. Here it is not feasible
to write down f explicitly as in Example 1.4, but nonetheless the function is completely
speciﬁed by the domain and the mathematical description of the rule f.
(i) 1-1 functions
1.6 Deﬁnition A function (or transformation) is 1 − 1 (one-to-one) if each element in the
codomain Y is the image of at most one element in the domain X.
1.7 Deﬁnition A function (or transformation) is onto if each element in the codomain Y is
the image of at least one element in the domain. Equivalently, a function f : X −→ Y is
onto if Im(f) = Y .
1.8 Deﬁnition If a function f : X −→ Y is 1−1 and Im(f) = Y , then f is called a bijection.
1.9 Fact If f : X −→ Y is 1 − 1 then f : X −→ Im(f) is a bijection. In particular, if
f : X −→ Y is 1 − 1, and X and Y are ﬁnite sets of the same size, then f is a bijection.
In terms of the schematic representation, if f is a bijection, then each element in Y
has exactly one arrowed line incident with it. The functions described in Examples 1.3 and
1.4 are not bijections. In Example 1.3 the element 3 is not the image of any element in the
domain. In Example 1.4 each element in the codomain has two preimages.
1.10 Deﬁnition If f is a bijection from X to Y then it is a simple matter to deﬁne a bijection g
from Y to X as follows: for each y ∈ Y deﬁne g(y) = x where x ∈ X and f(x) = y. This
function g obtained from f is called the inverse function of f and is denoted by g = f−1
.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
8 Ch. 1 Overview of Cryptography
b
c
d
e
2
3
4
5
1
2
3
4
5
b
c
d
e
1a a
f
X Y
g
XY
Figure 1.3: A bijection f and its inverse g = f−1
.
1.11 Example (inverse function) Let X = {a, b, c, d, e}, and Y = {1, 2, 3, 4, 5}, and consider
the rule f given by the arrowed edges in Figure 1.3. f is a bijection and its inverse g is
formedsimply byreversingthe arrows on the edges. The domainof g is Y and the codomain
is X.
Note that if f is a bijection, then so is f−1
. In cryptography bijections are used as
the tool for encrypting messages and the inverse transformations are used to decrypt. This
will be made clearer in §1.4 when some basic terminology is introduced. Notice that if the
transformations were not bijections then it would not be possible to always decrypt to a
unique message.
(ii) One-way functions
There are certain types of functions which play signiﬁcant roles in cryptography. At the
expense of rigor, an intuitive deﬁnition of a one-way function is given.
1.12 Deﬁnition A function f from a set X to a set Y is called a one-way function if f(x) is
“easy” to compute for all x ∈ X but for “essentially all” elements y ∈ Im(f) it is “computationally
infeasible” to ﬁnd any x ∈ X such that f(x) = y.
1.13 Note (clariﬁcation of terms in Deﬁnition 1.12)
(i) A rigorous deﬁnition of the terms “easy” and “computationally infeasible” is necessary
but would detract from the simple idea that is being conveyed. For the purpose
of this chapter, the intuitive meaning will sufﬁce.
(ii) The phrase “for essentially all elements in Y ” refers to the fact that there are a few
values y ∈ Y for which it is easy to ﬁnd an x ∈ X such that y = f(x). For example,
one may compute y = f(x) for a small number of x values and then for these, the
inverse is known by table look-up. An alternate way to describe this property of a
one-way function is the following: for a random y ∈ Im(f) it is computationally
infeasible to ﬁnd any x ∈ X such that f(x) = y.
The concept of a one-way function is illustrated through the following examples.
1.14 Example (one-way function) Take X = {1, 2, 3, . . . , 16} and deﬁne f(x) = rx for all
x ∈ X where rx is the remainder when 3x
is divided by 17. Explicitly,
x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
f(x) 3 9 10 13 5 15 11 16 14 8 7 4 12 2 6 1
Given a number between 1 and 16, it is relatively easy to ﬁnd the image of it under f. However,
given a number such as 7, without having the table in front of you, it is harder to ﬁnd
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.3 Background on functions 9
x given that f(x) = 7. Of course, if the number you are given is 3 then it is clear that x = 1
is what you need; but for most of the elements in the codomain it is not that easy.
One must keep in mind that this is an example which uses very small numbers; the
important point here is that there is a difference in the amount of work to compute f(x)
and the amount of work to ﬁnd x given f(x). Even for very large numbers, f(x) can be
computed efﬁciently using the repeated square-and-multiply algorithm (Algorithm 2.143),
whereas the process of ﬁnding x from f(x) is much harder.
1.15 Example (one-way function) A prime number is a positive integer greater than 1 whose
only positive integer divisors are 1 and itself. Select primes p = 48611, q = 53993, form
n = pq = 2624653723, and let X = {1, 2, 3, . . . , n − 1}. Deﬁne a function f on X
by f(x) = rx for each x ∈ X, where rx is the remainder when x3
is divided by n. For
instance, f(2489991) = 1981394214 since 24899913
= 5881949859 · n + 1981394214.
Computing f(x) is a relatively simple thing to do, but to reverse the procedureis much more
difﬁcult; that is, given a remainder to ﬁnd the value x which was originally cubed (raised
to the third power). This procedure is referred to as the computation of a modular cube root
with modulus n. If the factors of n are unknown and large, this is a difﬁcult problem; however,
if the factors p and q of n are known then there is an efﬁcient algorithm for computing
modular cube roots. (See §8.2.2(i) for details.)
Example 1.15 leads one to consider another type of function which will prove to be
fundamental in later developments.
(iii) Trapdoor one-way functions
1.16 Deﬁnition A trapdoor one-way function is a one-way function f : X −→ Y with the
additional property that given some extra information (called the trapdoor information) it
becomes feasible to ﬁnd for any given y ∈ Im(f), an x ∈ X such that f(x) = y.
Example 1.15 illustrates the concept of a trapdoor one-way function. With the additional
information of the factors of n = 2624653723 (namely, p = 48611 and q = 53993,
each of which is ﬁve decimal digits long) it becomes much easier to invert the function.
The factors of 2624653723 are large enough that ﬁnding them by hand computation would
be difﬁcult. Of course, any reasonable computer program could ﬁnd the factors relatively
quickly. If, on the other hand, one selects p and q to be very large distinct prime numbers
(each having about 100 decimal digits) then, by today’s standards, it is a difﬁcult problem,
even with the most powerful computers, to deduce p and q simply from n. This is the wellknown
integer factorization problem (see §3.2) and a source of many trapdoor one-way
functions.
It remains to be rigorously established whether there actually are any (true) one-way
functions. That is to say, no one has yet deﬁnitively proved the existence of such functions
under reasonable (and rigorous) deﬁnitions of “easy” and “computationally infeasible”.
Since the existence of one-way functions is still unknown, the existence of trapdoor
one-way functions is also unknown. However, there are a number of good candidates for
one-way and trapdoor one-way functions. Many of these are discussed in this book, with
emphasis given to those which are practical.
One-way and trapdoor one-way functions are the basis for public-key cryptography
(discussed in §1.8). The importance of these concepts will become clearer when their application
to cryptographic techniques is considered. It will be worthwhile to keep the abstract
concepts of this section in mind as concrete methods are presented.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
10 Ch. 1 Overview of Cryptography
1.3.2 Permutations
Permutations are functions which are often used in various cryptographic constructs.
1.17 Deﬁnition Let S be a ﬁnite set of elements. A permutation p on S is a bijection (Deﬁnition
1.8) from S to itself (i.e., p: S −→ S).
1.18 Example (permutation) Let S = {1, 2, 3, 4, 5}. A permutation p: S −→ S is deﬁned as
follows:
p(1) = 3, p(2) = 5, p(3) = 4, p(4) = 2, p(5) = 1.
A permutation can be described in various ways. It can be displayed as above or as an array:
p =
1 2 3 4 5
3 5 4 2 1
, (1.1)
where the top row in the array is the domain and the bottom row is the image under the
mapping p. Of course, other representations are possible.
Since permutations are bijections, they have inverses. If a permutation is written as an
array (see 1.1), its inverse is easily found by interchangingthe rows in the array and reordering
the elements in the new top row if desired (the bottom row would have to be reordered
correspondingly). The inverse of p in Example 1.18 is p−1
=
1 2 3 4 5
5 4 1 3 2
.
1.19 Example (permutation) Let X be the set of integers {0, 1, 2, . . . , pq − 1} where p and q
are distinct large primes (for example, p and q are each about 100 decimal digits long), and
suppose that neither p−1 nor q−1 is divisible by 3. Then the function p(x) = rx, where rx
is the remainder when x3
is divided by pq, can be shown to be a permutation. Determining
the inverse permutation is computationally infeasible by today’s standards unless p and q
are known (cf. Example 1.15).
1.3.3 Involutions
Another type of function which will be referred to in §1.5.3 is an involution. Involutions
have the property that they are their own inverses.
1.20 Deﬁnition Let S be a ﬁnite set and let f be a bijection from S to S (i.e., f : S −→ S).
The function f is called an involution if f = f−1
. An equivalent way of stating this is
f(f(x)) = x for all x ∈ S.
1.21 Example (involution) Figure 1.4 is an example of an involution. In the diagram of an
involution, note that if j is the image of i then i is the image of j.
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.4 Basic terminology and concepts 11
1
2
3
4
5
2
3
4
5
1
S S
Figure 1.4: An involution on a set S of 5 elements.
1.4 Basic terminology and concepts
The scientiﬁc study of any discipline must be built upon rigorous deﬁnitions arising from
fundamental concepts. What follows is a list of terms and basic concepts used throughout
this book. Where appropriate, rigor has been sacriﬁced (here in Chapter 1) for the sake of
clarity.
Encryption domains and codomains
• A denotes a ﬁnite set called the alphabet of deﬁnition. For example, A = {0, 1}, the
binary alphabet, is a frequently used alphabet of deﬁnition. Note that any alphabet
can be encodedin terms of the binary alphabet. For example, since there are 32 binary
strings of length ﬁve, each letter of the English alphabet can be assigned a unique
binary string of length ﬁve.
• M denotes a set called the message space. M consists of strings of symbols from
an alphabet of deﬁnition. An element of M is called a plaintext message or simply
a plaintext. For example, M may consist of binary strings, English text, computer
code, etc.
• C denotes a set called the ciphertext space. C consists of strings of symbols from an
alphabet of deﬁnition, which may differ from the alphabet of deﬁnition for M. An
element of C is called a ciphertext.
Encryption and decryption transformations
• K denotes a set called the key space. An element of K is called a key.
• Each element e ∈ K uniquely determines a bijection from M to C, denoted by Ee.
Ee is called an encryption function or an encryption transformation. Note that Ee
must be a bijection if the process is to be reversed and a unique plaintext message
recovered for each distinct ciphertext.1
• For each d ∈ K, Dd denotes a bijection from C to M (i.e., Dd : C −→ M). Dd is
called a decryption function or decryption transformation.
• The process of applying the transformation Ee to a message m ∈ M is usually referred
to as encrypting m or the encryption of m.
• The process of applying the transformation Dd to a ciphertext c is usually referred to
as decrypting c or the decryption of c.
1More generality is obtained if Ee is simply deﬁned as a 1 − 1 transformation from M to C. That is to say,
Ee is a bijection from M to Im(Ee) where Im(Ee) is a subset of C.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
12 Ch. 1 Overview of Cryptography
• An encryption scheme consists of a set {Ee : e ∈ K} of encryption transformations
and a corresponding set {Dd : d ∈ K} of decryption transformations with the property
that for each e ∈ K there is a unique key d ∈ K such that Dd = E−1
e ; that is,
Dd(Ee(m)) = m for all m ∈ M. An encryption scheme is sometimes referred to
as a cipher.
• The keys e and d in the preceding deﬁnition are referred to as a key pair and sometimes
denoted by (e, d). Note that e and d could be the same.
• To construct an encryption scheme requires one to select a message space M, a ciphertext
space C, a key space K, a set of encryption transformations {Ee : e ∈ K},
and a corresponding set of decryption transformations {Dd : d ∈ K}.
Achieving conﬁdentiality
An encryption scheme may be used as follows for the purpose of achieving conﬁdentiality.
Two parties Alice and Bob ﬁrst secretly choose or secretly exchange a key pair (e, d). At a
subsequent point in time, if Alice wishes to send a message m ∈ M to Bob, she computes
c = Ee(m) and transmits this to Bob. Upon receiving c, Bob computes Dd(c) = m and
hence recovers the original message m.
The question arises as to why keys are necessary. (Why not just choose one encryption
function and its corresponding decryption function?) Having transformations which are
very similar but characterized by keys means that if some particular encryption/decryption
transformation is revealed then one does not have to redesign the entire scheme but simply
change the key. It is sound cryptographic practice to change the key (encryption/decryption
transformation) frequently. As a physical analogue, consider an ordinary resettable combination
lock. The structure of the lock is available to anyone who wishes to purchase one but
the combination is chosen and set by the owner. If the owner suspects that the combination
has been revealed he can easily reset it without replacing the physical mechanism.
1.22 Example (encryption scheme) Let M = {m1, m2, m3} and C = {c1, c2, c3}. There
are precisely 3! = 6 bijections from M to C. The key space K = {1, 2, 3, 4, 5, 6} has
six elements in it, each specifying one of the transformations. Figure 1.5 illustrates the six
encryption functions which are denoted by Ei, 1 ≤ i ≤ 6. Alice and Bob agree on a trans-
E1
m1
m2
m3
c1
c2
E2
m1
m2
m3
m1
m2
m3
E3
E4
m1
m2
m3
m1
m2
m3
E5
m1
m2
m3
E6
c1
c2
c1
c2c2
c1
c1
c2
c1
c2
c3 c3 c3
c3 c3 c3
Figure 1.5: Schematic of a simple encryption scheme.
formation, say E1. To encrypt the message m1, Alice computes E1(m1) = c3 and sends
c3 to Bob. Bob decrypts c3 by reversing the arrows on the diagram for E1 and observing
that c3 points to m1.
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.4 Basic terminology and concepts 13
When M is a small set, the functional diagram is a simple visual means to describe the
mapping. In cryptography, the set M is typically of astronomical proportions and, as such,
the visual description is infeasible. What is required, in these cases, is some other simple
means to describe the encryption and decryption transformations, such as mathematical al-
gorithms.
Figure 1.6 provides a simple model of a two-party communication using encryption.
m
c
m
Dd(c) = mEe(m) = c
plaintext
source
Alice Bob
UNSECURED CHANNEL
Adversary
decryptionencryption
destination
Figure 1.6: Schematic of a two-party communication using encryption.
Communication participants
Referring to Figure 1.6, the following terminology is deﬁned.
• An entity or party is someone or something which sends, receives, or manipulates
information. Alice and Bob are entities in Example 1.22. An entity may be a person,
a computer terminal, etc.
• A sender is an entity in a two-partycommunicationwhich is the legitimate transmitter
of information. In Figure 1.6, the sender is Alice.
• A receiver is an entity in a two-party communication which is the intended recipient
of information. In Figure 1.6, the receiver is Bob.
• An adversary is an entity in a two-party communication which is neither the sender
nor receiver, and which tries to defeat the informationsecurity service being provided
between the sender and receiver. Various other names are synonymous with adversary
such as enemy, attacker, opponent,tapper, eavesdropper,intruder, and interloper.
An adversary will often attempt to play the role of either the legitimate sender or the
legitimate receiver.
Channels
• A channel is a means of conveying information from one entity to another.
• A physically secure channel or secure channel is one which is not physically accessible
to the adversary.
• An unsecured channel is one from which parties other than those for which the information
is intended can reorder, delete, insert, or read.
• A secured channelis one from which an adversarydoes not have the ability to reorder,
delete, insert, or read.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
14 Ch. 1 Overview of Cryptography
One should note the subtle difference between a physically secure channel and a secured
channel – a secured channel may be secured by physical or cryptographic techniques,
the latter being the topic of this book. Certain channels are assumed to be physically secure.
These include trusted couriers, personal contact between communicatingparties, and a dedicated
communication link, to name a few.
Security
A fundamental premise in cryptography is that the sets M, C, K, {Ee : e ∈ K}, {Dd : d ∈
K} are public knowledge. When two parties wish to communicate securely using an encryption
scheme, the only thing that they keep secret is the particular key pair (e, d) which
they are using, and which they must select. One can gain additional security by keeping the
class of encryption and decryption transformations secret but one should not base the security
of the entire scheme on this approach. History has shown that maintaining the secrecy
of the transformations is very difﬁcult indeed.
1.23 Deﬁnition An encryption scheme is said to be breakable if a third party, without prior
knowledge of the key pair (e, d), can systematically recover plaintext from corresponding
ciphertext within some appropriate time frame.
An appropriate time frame will be a function of the useful lifespan of the data being
protected. For example, an instruction to buy a certain stock may only need to be kept secret
for a few minutes whereas state secrets may need to remain conﬁdential indeﬁnitely.
An encryption scheme can be broken by trying all possible keys to see which one the
communicating parties are using (assuming that the class of encryption functions is public
knowledge). This is called an exhaustive search of the key space. It follows then that the
numberof keys (i.e., the size of the key space) should be large enoughto make this approach
computationallyinfeasible. It is the objectiveof a designerof an encryptionscheme that this
be the best approach to break the system.
Frequently cited in the literature are Kerckhoffs’ desiderata, a set of requirements for
cipher systems. They are given here essentially as Kerckhoffs originally stated them:
1. the system should be, if not theoretically unbreakable, unbreakable in practice;
2. compromise of the system details should not inconvenience the correspondents;
3. the key should be rememberable without notes and easily changed;
4. the cryptogram should be transmissible by telegraph;
5. the encryption apparatus should be portable and operable by a single person; and
6. the system should be easy, requiring neither the knowledge of a long list of rules nor
mental strain.
This list of requirementswas articulatedin 1883and, forthe most part, remains useful today.
Point 2 allows that the class of encryption transformations being used be publicly known
and that the security of the system should reside only in the key chosen.
Information security in general
So far the terminology has been restricted to encryption and decryption with the goal of privacy
in mind. Information security is much broader, encompassing such things as authentication
and data integrity. A few more general deﬁnitions, pertinent to discussions later in
the book, are given next.
• An information security service is a method to provide some speciﬁc aspect of security.
For example, integrity of transmitted data is a security objective, and a method
to ensure this aspect is an information security service.
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.5 Symmetric-key encryption 15
• Breaking an information security service (which often involves more than simply encryption)
implies defeating the objective of the intended service.
• A passive adversary is an adversary who is capable only of reading information from
an unsecured channel.
• An active adversary is an adversary who may also transmit, alter, or delete information
on an unsecured channel.
Cryptology
• Cryptanalysis is the study of mathematical techniques for attempting to defeat cryptographic
techniques, and, more generally, information security services.
• A cryptanalyst is someone who engages in cryptanalysis.
• Cryptology is the study of cryptography (Deﬁnition 1.1) and cryptanalysis.
• A cryptosystem is a general term referring to a set of cryptographic primitives used
to provide information security services. Most often the term is used in conjunction
with primitives providing conﬁdentiality, i.e., encryption.
Cryptographic techniques are typically divided into two generic types: symmetric-key
and public-key. Encryption methods of these types will be discussed separately in §1.5 and
§1.8. Other deﬁnitions and terminology will be introduced as required.
1.5 Symmetric-key encryption
§1.5 considers symmetric-key encryption. Public-key encryption is the topic of §1.8.
1.5.1 Overview of block ciphers and stream ciphers
1.24 Deﬁnition Consider an encryption scheme consisting of the sets of encryption and decryption
transformations {Ee : e ∈ K} and {Dd : d ∈ K}, respectively, where K is the key
space. The encryption scheme is said to be symmetric-key if for each associated encryption/decryption
key pair (e, d), it is computationally “easy” to determine d knowing only e,
and to determine e from d.
Since e = d in most practical symmetric-key encryption schemes, the term symmetrickey
becomes appropriate. Other terms used in the literature are single-key, one-key, private-
key,2
and conventional encryption. Example 1.25 illustrates the idea of symmetric-key en-
cryption.
1.25 Example (symmetric-key encryption) Let A = {A, B, C, . . . , X, Y, Z} be the English
alphabet. Let M and C be the set of all strings of length ﬁve over A. The key e is chosen
to be a permutation on A. To encrypt, an English message is broken up into groups each
having ﬁve letters (with appropriate padding if the length of the message is not a multiple
of ﬁve) and a permutation e is applied to each letter one at a time. To decrypt, the inverse
permutation d = e−1
is applied to each letter of the ciphertext. For instance, suppose that
the key e is chosen to be the permutation which maps each letter to the one which is three
positions to its right, as shown below
e =
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
D E F G H I J K L M N O P Q R S T U V W X Y Z A B C
2Private key is a term also used in quite a different context (see §1.8). The term will be reserved for the latter
usage in this book.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
16 Ch. 1 Overview of Cryptography
A message
m = THISC IPHER ISCER TAINL YNOTS ECURE
is encrypted to
c = Ee(m) = WKLVF LSKHU LVFHU WDLQO BQRWV HFXUH.
A two-party communication using symmetric-key encryption can be described by the
block diagram of Figure 1.7, which is Figure 1.6 with the addition of the secure (both con-
m
e
c
SECURE CHANNEL
Dd(c) = mEe(m) = c
e
m
UNSECURED CHANNEL
encryption
plaintext
source
Alice
Adversary
source
key
decryption
destination
Bob
Figure 1.7: Two-party communication using encryption, with a secure channel for key exchange.
The decryption key d can be efﬁciently computed from the encryption key e.
ﬁdential and authentic) channel. One of the major issues with symmetric-key systems is to
ﬁnd an efﬁcient method to agree upon and exchange keys securely. This problem is referred
to as the key distribution problem (see Chapters 12 and 13).
It is assumed that all parties knowthe set of encryption/decryptiontransformations(i.e.,
they all know the encryptionscheme). As has been emphasized several times the only information
which should be required to be kept secret is the key d. However, in symmetric-key
encryption, this means that the key e must also be kept secret, as d can be deduced from
e. In Figure 1.7 the encryption key e is transported from one entity to the other with the
understanding that both can construct the decryption key d.
There are two classes of symmetric-key encryption schemes which are commonly distinguished:
block ciphers and stream ciphers.
1.26 Deﬁnition A block cipher is an encryption scheme which breaks up the plaintext messages
to be transmitted into strings (called blocks) of a ﬁxed length t over an alphabet A,
and encrypts one block at a time.
Most well-known symmetric-key encryption techniques are block ciphers. A number
of examples of these are given in Chapter 7. Two important classes of block ciphers are
substitution ciphers and transposition ciphers (§1.5.2). Product ciphers (§1.5.3) combine
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.5 Symmetric-key encryption 17
these. Stream ciphers are considered in §1.5.4, while comments on the key space follow in
§1.5.5.
1.5.2 Substitution ciphers and transposition ciphers
Substitution ciphers are block ciphers which replace symbols (or groups of symbols) by
other symbols or groups of symbols.
Simple substitution ciphers
1.27 Deﬁnition Let A be an alphabet of q symbols and M be the set of all strings of length
t over A. Let K be the set of all permutations on the set A. Deﬁne for each e ∈ K an
encryption transformation Ee as:
Ee(m) = (e(m1)e(m2) · · · e(mt)) = (c1c2 · · · ct) = c,
where m = (m1m2 · · · mt) ∈ M. In other words, for each symbol in a t-tuple, replace
(substitute) it by another symbol from A according to some ﬁxed permutation e. To decrypt
c = (c1c2 · · · ct) compute the inverse permutation d = e−1
and
Dd(c) = (d(c1)d(c2) · · · d(ct)) = (m1m2 · · · mt) = m.
Ee is called a simple substitution cipher or a mono-alphabetic substitution cipher.
The number of distinct substitution ciphers is q! and is independent of the block size in
the cipher. Example 1.25 is an example of a simple substitution cipher of block length ﬁve.
Simple substitution ciphers over small block sizes provide inadequate security even
when the key space is extremely large. If the alphabet is the English alphabet as in Example
1.25, then the size of the key space is 26! ≈ 4 × 1026
, yet the key being used can be
determined quite easily by examining a modest amount of ciphertext. This follows from the
simple observation that the distribution of letter frequencies is preserved in the ciphertext.
For example, the letter E occurs more frequently than the other letters in ordinary English
text. Hence the letter occurring most frequently in a sequence of ciphertext blocks is most
likely to correspond to the letter E in the plaintext. By observing a modest quantity of ciphertext
blocks, a cryptanalyst can determine the key.
Homophonic substitution ciphers
1.28 Deﬁnition To each symbol a ∈ A, associate a set H(a) of strings of t symbols, with
the restriction that the sets H(a), a ∈ A, be pairwise disjoint. A homophonic substitution
cipher replaces each symbol a in a plaintext message block with a randomly chosen string
from H(a). To decrypt a string c of t symbols, one must determine an a ∈ A such that
c ∈ H(a). The key for the cipher consists of the sets H(a).
1.29 Example (homophonic substitution cipher) Consider A = {a, b}, H(a) = {00, 10}, and
H(b) = {01, 11}. The plaintext message block ab encrypts to one of the following: 0001,
0011, 1001, 1011. Observe that the codomain of the encryption function (for messages of
length two) consists of the following pairwise disjoint sets of four-element bitstrings:
aa −→ {0000, 0010, 1000, 1010}
ab −→ {0001, 0011, 1001, 1011}
ba −→ {0100, 0110, 1100, 1110}
bb −→ {0101, 0111, 1101, 1111}
Any 4-bitstring uniquely identiﬁes a codomain element, and hence a plaintext message.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
18 Ch. 1 Overview of Cryptography
Often the symbols do not occur with equal frequency in plaintext messages. With a
simple substitution cipher this non-uniformfrequency property is reﬂected in the ciphertext
as illustrated in Example 1.25. A homophonic cipher can be used to make the frequency of
occurrence of ciphertext symbols more uniform, at the expense of data expansion. Decryption
is not as easily performed as it is for simple substitution ciphers.
Polyalphabetic substitution ciphers
1.30 Deﬁnition A polyalphabetic substitution cipher is a block cipher with block length t over
an alphabet A having the following properties:
(i) the key space K consists of all ordered sets of t permutations (p1, p2, . . . , pt), where
each permutation pi is deﬁned on the set A;
(ii) encryption of the message m = (m1m2 · · · mt) under the key e = (p1, p2, . . . , pt)
is given by Ee(m) = (p1(m1)p2(m2) · · · pt(mt)); and
(iii) the decryption key associated with e = (p1, p2, . . . , pt) is d = (p−1
1 , p−1
2 , . . . , p−1
t ).
1.31 Example (Vigen`ere cipher) Let A = {A, B, C, . . . , X, Y, Z} and t = 3. Choose e =
(p1, p2, p3), where p1 maps each letter to the letter three positions to its right in the alphabet,
p2 to the one seven positions to its right, and p3 ten positions to its right. If
m = THI SCI PHE RIS CER TAI NLY NOT SEC URE
then
c = Ee(m) = WOS VJS SOO UPC FLB WHS QSI QVD VLM XYO.
Polyalphabeticciphers have the advantageover simple substitution ciphers that symbol
frequencies are not preserved. In the example above, the letter E is encrypted to both O and
L. However, polyalphabetic ciphers are not signiﬁcantly more difﬁcult to cryptanalyze, the
approach being similar to the simple substitution cipher. In fact, once the block length t is
determined, the ciphertext letters can be divided into t groups (where group i, 1 ≤ i ≤ t,
consists of those ciphertext letters derived using permutation pi), and a frequency analysis
can be done on each group.
Transposition ciphers
Another class of symmetric-key ciphers is the simple transposition cipher, which simply
permutes the symbols in a block.
1.32 Deﬁnition Consider a symmetric-keyblock encryptionscheme with block length t. Let K
be the set of all permutations on the set {1, 2, . . . , t}. For each e ∈ K deﬁne the encryption
function
Ee(m) = (me(1)me(2) · · · me(t))
where m = (m1m2 · · · mt) ∈ M, the message space. The set of all such transformations
is called a simple transposition cipher. The decryption key corresponding to e is the inverse
permutationd = e−1
. To decryptc = (c1c2 · · · ct), computeDd(c) = (cd(1)cd(2) · · · cd(t)).
A simple transposition cipher preserves the number of symbols of a given type within
a block, and thus is easily cryptanalyzed.
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.5 Symmetric-key encryption 19
1.5.3 Composition of ciphers
In order to describe product ciphers, the concept of composition of functions is introduced.
Compositions are a convenient way of constructing more complicated functions from simpler
ones.
Composition of functions
1.33 Deﬁnition Let S, T , and U be ﬁnite sets and let f : S −→ T and g: T −→ U be functions.
The composition of g with f, denoted g ◦ f (or simply gf), is a function from S to
U as illustrated in Figure 1.8 and deﬁned by (g ◦ f)(x) = g(f(x)) for all x ∈ S.
s
t
u
v
1
2
3
4
s
t
u
v
a
b
c
a
b
c
S
T U S U
g ◦ f
f g
Figure 1.8: The composition g ◦ f of functions g and f.
Composition can be easily extended to more than two functions. For functions f1, f2,
. . . , ft, one can deﬁne ft ◦· · ·◦f2 ◦f1, provided that the domain of ft equals the codomain
of ft−1 and so on.
Compositions and involutions
Involutionswere introducedin §1.3.3 as a simple class of functions with an interesting property:
Ek(Ek(x)) = x for all x in the domain of Ek; that is, Ek ◦Ek is the identity function.
1.34 Remark (composition of involutions) The composition of two involutions is not necessarily
an involution, as illustrated in Figure 1.9. However, involutions may be composed to get
somewhat more complicated functions whose inverses are easy to ﬁnd. This is an important
feature for decryption. For example if Ek1 , Ek2 , . . . , Ekt are involutions then the inverse
of Ek = Ek1 Ek2 · · · Ekt is E−1
k = Ekt Ekt−1 · · · Ek1 , the composition of the involutions
in the reverse order.
1
2
3
4 4
3
2
1
4
3
2
1 1
2
3
4 4
2
1
3
4
3
2
1
f g g ◦ f
Figure 1.9: The composition g ◦ f of involutions g and f is not an involution.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
20 Ch. 1 Overview of Cryptography
Product ciphers
Simple substitution and transposition ciphers individually do not provide a very high level
of security. However, by combining these transformations it is possible to obtain strong ciphers.
As will be seen in Chapter 7 some of the most practical and effective symmetric-key
systems are product ciphers. One example of a product cipher is a composition of t ≥ 2
transformations Ek1 Ek2 · · · Ekt where each Eki , 1 ≤ i ≤ t, is either a substitution or a
transposition cipher. For the purpose of this introduction, let the composition of a substitution
and a transposition be called a round.
1.35 Example (product cipher) Let M = C = K be the set of all binary strings of length six.
The number of elements in M is 26
= 64. Let m = (m1m2 · · · m6) and deﬁne
E
(1)
k (m) = m ⊕ k, where k ∈ K,
E(2)
(m) = (m4m5m6m1m2m3).
Here, ⊕ is the exclusive-OR (XOR) operation deﬁned as follows: 0 ⊕ 0 = 0, 0 ⊕ 1 = 1,
1 ⊕ 0 = 1, 1 ⊕ 1 = 0. E
(1)
k is a polyalphabetic substitution cipher and E(2)
is a transposition
cipher (not involving the key). The product E
(1)
k E(2)
is a round. While here the
transposition cipher is very simple and is not determined by the key, this need not be the
case.
1.36 Remark (confusion and diffusion) A substitution in a round is said to add confusion to the
encryption process whereas a transposition is said to add diffusion. Confusion is intended
to make the relationship between the key and ciphertext as complex as possible. Diffusion
refers to rearranging or spreading out the bits in the message so that any redundancy in the
plaintext is spread out over the ciphertext. A round then can be said to add both confusion
and diffusion to the encryption. Most modern block cipher systems apply a number of
rounds in succession to encrypt plaintext.
1.5.4 Stream ciphers
Stream ciphers form an important class of symmetric-key encryption schemes. They are, in
one sense, very simple block ciphers having block length equal to one. What makes them
useful is the fact that the encryption transformation can change for each symbol of plaintext
being encrypted. In situations where transmission errors are highly probable, stream
ciphers are advantageous because they have no error propagation. They can also be used
when the data must be processed one symbol at a time (e.g., if the equipment has no memory
or buffering of data is limited).
1.37 Deﬁnition Let K be the key space for a set of encryption transformations. A sequence of
symbols e1e2e3 · · · ei ∈ K, is called a keystream.
1.38 Deﬁnition Let A be an alphabet of q symbols and let Ee be a simple substitution cipher
with block length 1 where e ∈ K. Let m1m2m3 · · · be a plaintext string and let e1e2e3 · · ·
be a keystream from K. A stream cipher takes the plaintext string and produces a ciphertext
string c1c2c3 · · · where ci = Eei (mi). If di denotes the inverse of ei, then Ddi (ci) = mi
decrypts the ciphertext string.
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.5 Symmetric-key encryption 21
A stream cipher applies simple encryption transformations according to the keystream
being used. The keystream could be generated at random, or by an algorithm which generates
the keystream from an initial small keystream (called a seed), or from a seed and
previous ciphertext symbols. Such an algorithm is called a keystream generator.
The Vernam cipher
A motivating factor for the Vernam cipher was its simplicity and ease of implementation.
1.39 Deﬁnition The Vernam Cipher is a stream cipher deﬁned on the alphabet A = {0, 1}. A
binary message m1m2 · · · mt is operated on by a binary key string k1k2 · · · kt of the same
length to produce a ciphertext string c1c2 · · · ct where
ci = mi ⊕ ki, 1 ≤ i ≤ t.
If the key string is randomly chosen and never used again, the Vernam cipher is called a
one-time system or a one-time pad.
To see how the Vernam cipher corresponds to Deﬁnition 1.38, observe that there are
precisely two substitution ciphers on the set A. One is simply the identity map E0 which
sends 0 to 0 and 1 to 1; the other E1 sends 0 to 1 and 1 to 0. When the keystream contains
a 0, apply E0 to the corresponding plaintext symbol; otherwise, apply E1.
If the key string is reused there are ways to attack the system. For example, if c1c2 · · · ct
and c1c2 · · · ct are two ciphertext strings produced by the same keystream k1k2 · · · kt then
ci = mi ⊕ ki, ci = mi ⊕ ki
and ci ⊕ ci = mi ⊕ mi. The redundancy in the latter may permit cryptanalysis.
The one-time pad can be shown to be theoretically unbreakable. That is, if a cryptanalyst
has a ciphertext string c1c2 · · · ct encrypted using a random key string which has been
used only once, the cryptanalyst can do no better than guess at the plaintext being any binary
string of length t (i.e., t-bit binary strings are equally likely as plaintext). It has been
proventhat to realize an unbreakablesystem requires a random key of the same length as the
message. This reduces the practicality of the system in all but a few specialized situations.
Reportedly until very recently the communication line between Moscow and Washington
was secured by a one-time pad. Transport of the key was done by trusted courier.
1.5.5 The key space
The size of the key space is the number of encryption/decryptionkey pairs that are available
in the cipher system. A key is typically a compact way to specify the encryption transformation
(from the set of all encryption transformations) to be used. For example, a transposition
cipher of block length t has t! encryption functions from which to select. Each can
be simply described by a permutation which is called the key.
It is a great temptation to relate the security of the encryption scheme to the size of the
key space. The following statement is important to remember.
1.40 Fact A necessary, but usually not sufﬁcient, condition for an encryption scheme to be secure
is that the key space be large enough to preclude exhaustive search.
For instance, the simple substitution cipher in Example 1.25 has a key space of size
26! ≈ 4 × 1026
. The polyalphabetic substitution cipher of Example 1.31 has a key space
of size (26!)3
≈ 7 × 1079
. Exhaustive search of either key space is completely infeasible,
yet both ciphers are relatively weak and provide little security.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
22 Ch. 1 Overview of Cryptography
1.6 Digital signatures
A cryptographic primitive which is fundamental in authentication, authorization, and nonrepudiation
is the digital signature. The purpose of a digital signature is to provide a means
for an entity to bind its identity to a piece of information. The process of signing entails
transforming the message and some secret information held by the entity into a tag called
a signature. A generic description follows.
Nomenclature and set-up
• M is the set of messages which can be signed.
• S is a set of elements called signatures, possibly binary strings of a ﬁxed length.
• SA is a transformation from the message set M to the signature set S, and is called
a signing transformation for entity A.3
The transformation SA is kept secret by A,
and will be used to create signatures for messages from M.
• VA is a transformation from the set M × S to the set {true, false}.4
VA is called
a veriﬁcation transformation for A’s signatures, is publicly known, and is used by
other entities to verify signatures created by A.
1.41 Deﬁnition The transformations SA and VA provide a digital signature scheme for A. Occasionally
the term digital signature mechanism is used.
1.42 Example (digital signature scheme) M = {m1, m2, m3} and S = {s1, s2, s3}. The left
side of Figure 1.10 displays a signing function SA from the set M and, the right side, the
corresponding veriﬁcation function VA.
SA
VA
False
True
m1
m2
m3 s2
s1
s3
(m1, s1)
(m1, s2)
(m1, s3)
(m2, s1)
(m2, s2)
(m2, s3)
(m3, s1)
(m3, s2)
(m3, s3)
Figure 1.10: A signing and veriﬁcation function for a digital signature scheme.
3The names of Alice and Bob are usually abbreviated to A and B, respectively.
4M × S consists of all pairs (m, s) where m ∈ M, s ∈ S, called the Cartesian product of M and S.
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.6 Digital signatures 23
Signing procedure
Entity A (the signer) creates a signature for a message m ∈ M by doing the following:
1. Compute s = SA(m).
2. Transmit the pair (m, s). s is called the signature for message m.
Veriﬁcation procedure
To verify that a signature s on a message m was created by A, an entity B (the veriﬁer)
performs the following steps:
1. Obtain the veriﬁcation function VA of A.
2. Compute u = VA(m, s).
3. Accept the signature as having been created by A if u = true, and reject the signature
if u = false.
1.43 Remark (concise representation) The transformations SA and VA are typically characterized
more compactly by a key; that is, there is a class of signing and veriﬁcation algorithms
publicly known, and each algorithm is identiﬁed by a key. Thus the signing algorithm SA
of A is determined by a key kA and A is only required to keep kA secret. Similarly, the
veriﬁcation algorithm VA of A is determined by a key lA which is made public.
1.44 Remark (handwritten signatures) Handwritten signatures could be interpreted as a special
class of digital signatures. To see this, take the set of signatures S to contain only one
element which is the handwritten signature of A, denoted by sA. The veriﬁcation function
simply checks if the signature on a message purportedly signed by A is sA.
An undesirable feature in Remark 1.44 is that the signature is not message-dependent.
Hence, further constraints are imposed on digital signature mechanisms as next discussed.
Properties required for signing and veriﬁcation functions
There are several properties which the signing and veriﬁcation transformationsmust satisfy.
(a) s is a valid signature of A on message m if and only if VA(m, s) = true.
(b) It is computationally infeasible for any entity other than A to ﬁnd, for any m ∈ M,
an s ∈ S such that VA(m, s) = true.
Figure 1.10 graphically displays property (a). There is an arrowed line in the diagram
for VA from (mi, sj) to true provided there is an arrowed line from mi to sj in the diagram
for SA. Property (b) provides the security for the method – the signature uniquely binds A
to the message which is signed.
No one has yet formally proved that digital signature schemes satisfying (b) exist (although
existence is widely believed to be true); however, there are some very good candidates.
§1.8.3 introduces a particular class of digital signatures which arise from publickey
encryption techniques. Chapter 11 describes a number of digital signature mechanisms
which are believed to satisfy the two properties cited above. Although the description of a
digital signature given in this section is quite general, it can be broadened further, as presented
in §11.2.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
24 Ch. 1 Overview of Cryptography
1.7 Authentication and identiﬁcation
Authentication is a term which is used (and often abused) in a very broad sense. By itself
it has little meaning other than to convey the idea that some means has been provided to
guarantee that entities are who they claim to be, or that information has not been manipulated
by unauthorized parties. Authentication is speciﬁc to the security objective which
one is trying to achieve. Examples of speciﬁc objectives include access control, entity authentication,
message authentication, data integrity, non-repudiation, and key authentication.
These instances of authentication are dealt with at length in Chapters 9 through 13.
For the purposes of this chapter, it sufﬁces to give a brief introduction to authentication by
describing several of the most obvious applications.
Authentication is one of the most important of all information security objectives. Until
the mid 1970s it was generally believed that secrecy and authentication were intrinsically
connected. With the discovery of hash functions (§1.9) and digital signatures (§1.6), it was
realized that secrecy and authentication were truly separate and independent information
security objectives. It may at ﬁrst not seem important to separate the two but there are situations
where it is not only useful but essential. For example, if a two-party communication
between Alice and Bob is to take place where Alice is in one country and Bob in another,
the host countries might not permit secrecy on the channel; one or both countries might
want the ability to monitor all communications. Alice and Bob, however, would like to be
assured of the identity of each other, and of the integrity and origin of the information they
send and receive.
The preceding scenario illustrates several independent aspects of authentication. If Alice
and Bob desire assurance of each other’s identity, there are two possibilities to consider.
1. Alice and Bob could be communicating with no appreciable time delay. That is, they
are both active in the communication in “real time”.
2. Alice or Bob could be exchanging messages with some delay. That is, messages
might be routed through various networks, stored, and forwarded at some later time.
In the ﬁrst instance Alice and Bob would want to verify identities in real time. This
might be accomplished by Alice sending Bob some challenge, to which Bob is the only
entity which can respond correctly. Bob could perform a similar action to identify Alice.
This type of authentication is commonly referred to as entity authentication or more simply
identiﬁcation.
For the second possibility, it is not convenient to challenge and await response, and
moreover the communication path may be only in one direction. Different techniques are
now required to authenticate the originator of the message. This form of authentication is
called data origin authentication.
1.7.1 Identiﬁcation
1.45 Deﬁnition An identiﬁcation or entity authentication technique assures one party (through
acquisition of corroborative evidence) of both the identity of a second party involved, and
that the second was active at the time the evidence was created or acquired.
Typically the only data transmitted is that necessary to identify the communicating parties.
The entities are both active in the communication, giving a timeliness guarantee.
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.8 Public-key cryptography 25
1.46 Example (identiﬁcation) A calls B on the telephone. If A and B know each other then
entity authentication is provided through voice recognition. Although not foolproof, this
works effectively in practice.
1.47 Example (identiﬁcation) Person A provides to a banking machine a personal identiﬁcation
number (PIN) along with a magnetic stripe card containing information about A. The
banking machine uses the information on the card and the PIN to verify the identity of the
card holder. If veriﬁcation succeeds, A is given access to various services offered by the
machine.
Example 1.46 is an instance of mutual authentication whereas Example 1.47 only provides
unilateral authentication. Numerous mechanisms and protocols devised to provide
mutual or unilateral authentication are discussed in Chapter 10.
1.7.2 Data origin authentication
1.48 Deﬁnition Data origin authentication or message authentication techniques provide to
one party which receives a message assurance (through corroborativeevidence) of the identity
of the party which originated the message.
Often a message is provided to B along with additional information so that B can determine
the identity of the entity who originated the message. This form of authentication
typically provides no guarantee of timeliness, but is useful in situations where one of the
parties is not active in the communication.
1.49 Example (need for data origin authentication) A sends to B an electronic mail message
(e-mail). The message may travel throughvarious network communicationssystems and be
stored for B to retrieve at some later time. A and B are usually not in direct communication.
B would like some means to verify that the message received and purportedly created by
A did indeed originate from A.
Data origin authentication implicitly provides data integrity since, if the message was
modiﬁed during transmission, A would no longer be the originator.
1.8 Public-key cryptography
The concept of public-key encryption is simple and elegant, but has far-reaching conse-
quences.
1.8.1 Public-key encryption
Let {Ee : e ∈ K} be a set of encryption transformations, and let {Dd : d ∈ K} be the set of
corresponding decryption transformations, where K is the key space. Consider any pair of
associated encryption/decryption transformations (Ee, Dd) and suppose that each pair has
the property that knowing Ee it is computationally infeasible, given a random ciphertext
c ∈ C, to ﬁnd the message m ∈ M such that Ee(m) = c. This property implies that given
e it is infeasible to determine the corresponding decryption key d. (Of course e and d are
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
26 Ch. 1 Overview of Cryptography
simply means to describe the encryption and decryption functions, respectively.) Ee is being
viewed here as a trapdoor one-way function (Deﬁnition 1.16) with d being the trapdoor
information necessary to compute the inverse function and hence allow decryption. This is
unlike symmetric-key ciphers where e and d are essentially the same.
Under these assumptions, consider the two-party communication between Alice and
Bob illustrated in Figure 1.11. Bob selects the key pair (e, d). Bob sends the encryption key
e (called the public key) to Alice over any channel but keeps the decryptionkey d (called the
private key) secure and secret. Alice may subsequently send a message m to Bob by applying
the encryption transformation determined by Bob’s public key to get c = Ee(m). Bob
decrypts the ciphertext c by applying the inverse transformation Dd uniquely determined
by d.
e
m
c
Dd(c) = m
d
m
UNSECURED CHANNEL
Ee(m) = c UNSECURED CHANNEL
Alice Bob
encryption
destination
plaintext
source
key
source
decryption
Passive
Adversary
Figure 1.11: Encryption using public-key techniques.
Notice how Figure 1.11 differs from Figure 1.7 for a symmetric-key cipher. Here the
encryption key is transmitted to Alice over an unsecured channel. This unsecured channel
may be the same channel on which the ciphertext is being transmitted (but see §1.8.2).
Since the encryption key e need not be kept secret, it may be made public. Any entity
can subsequently send encrypted messages to Bob which only Bob can decrypt. Figure 1.12
illustrates this idea, where A1, A2, and A3 are distinct entities. Note that if A1 destroys
message m1 after encrypting it to c1, then even A1 cannot recover m1 from c1.
As a physical analogue, consider a metal box with the lid secured by a combination
lock. The combination is known only to Bob. If the lock is left open and made publicly
available then anyone can place a message inside and lock the lid. Only Bob can retrieve
the message. Even the entity which placed the message into the box is unable to retrieve it.
Public-key encryption, as described here, assumes that knowledge of the public key e
does not allow computation of the private key d. In other words, this assumes the existence
of trapdoor one-way functions (§1.3.1(iii)).
1.50 Deﬁnition Consider an encryptionscheme consistingof the sets of encryptionand decrypc
1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.8 Public-key cryptography 27
c2
Dd(c1) = m1
e
Dd(c2) = m2
Dd(c3) = m3
Ee(m2) = c2
e
c1
c3
Ee(m1) = c1
Ee(m3) = c3
e
A1
A2
A3
Bob
Figure 1.12: Schematic use of public-key encryption.
tion transformations{Ee : e ∈ K} and {Dd : d ∈ K}, respectively. The encryption method
is said to be a public-key encryption scheme if for each associated encryption/decryption
pair (e, d), one key e (the public key) is made publicly available, while the other d (the private
key) is kept secret. For the scheme to be secure, it must be computationally infeasible
to compute d from e.
1.51 Remark (private key vs. secret key) To avoid ambiguity, a common convention is to use
the term private key in association with public-key cryptosystems, and secret key in association
with symmetric-key cryptosystems. This may be motivated by the following line of
thought: it takes two or more parties to share a secret, but a key is truly private only when
one party alone knows it.
There are many schemes known which are widely believed to be secure public-key
encryption methods, but none have been mathematically proven to be secure independent
of qualifyingassumptions. This is not unlike the symmetric-keycase where the only system
which has been proven secure is the one-time pad (§1.5.4).
1.8.2 The necessity of authentication in public-key systems
It would appearthat public-keycryptographyis an ideal system, not requiringa secure channel
to pass the encryption key. This would imply that two entities could communicate over
an unsecured channel without ever having met to exchange keys. Unfortunately, this is not
the case. Figure 1.13 illustrates how an active adversary can defeat the system (decrypt
messages intended for a second entity) without breaking the encryption system. This is a
type of impersonation and is an example of protocol failure (see §1.10). In this scenario
the adversary impersonates entity B by sending entity A a public key e which A assumes
(incorrectly) to be the public key of B. The adversary intercepts encrypted messages from
A to B, decrypts with its own private key d , re-encrypts the message under B’s public key
e, and sends it on to B. This highlights the necessity to authenticate public keys to achieve
data origin authentication of the public keys themselves. A must be convinced that she is
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
28 Ch. 1 Overview of Cryptography
encrypting under the legitimate public key of B. Fortunately, public-key techniques also
allow an elegant solution to this problem (see §1.11).
e
m
c
e
c
m
A
B
Ee (m) = c
m
Dd(c) = m
d
d
Ee(m) = c
Dd (c ) = m
destination
key
source
plaintext
source
encryption
decryption
Adversary
key
source
encryption
decryption
Figure 1.13: An impersonation attack on a two-party communication.
1.8.3 Digital signatures from reversible public-key encryption
This section considers a class of digital signature schemes which is based on public-key
encryption systems of a particular type.
Suppose Ee is a public-key encryption transformation with message space M and ciphertext
space C. Suppose further that M = C. If Dd is the decryption transformation
corresponding to Ee then since Ee and Dd are both permutations, one has
Dd(Ee(m)) = Ee(Dd(m)) = m, for all m ∈ M.
A public-key encryption scheme of this type is called reversible.5
Note that it is essential
that M = C for this to be a valid equality for all m ∈ M; otherwise, Dd(m) will be
meaningless for m ∈ C.
5There is a broader class of digital signatures which can be informally described as arising from irreversible
cryptographic algorithms. These are described in §11.2.
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.8 Public-key cryptography 29
Construction for a digital signature scheme
1. Let M be the message space for the signature scheme.
2. Let C = M be the signature space S.
3. Let (e, d) be a key pair for the public-key encryption scheme.
4. Deﬁne the signing function SA to be Dd. That is, the signature for a message m ∈ M
is s = Dd(m).
5. Deﬁne the veriﬁcation function VA by
VA(m, s) =
true, if Ee(s) = m,
false, otherwise.
The signature scheme can be simpliﬁed further if A only signs messages having a special
structure, and this structure is publicly known. Let M be a subset of M where elements
of M have a well-deﬁned special structure, such that M contains only a negligible
fraction of messages from the set. For example, suppose that M consists of all binary
strings of length 2t for some positive integer t. Let M be the subset of M consisting of all
strings where the ﬁrst t bits are replicated in the last t positions (e.g., 101101 would be in
M for t = 3). If A only signs messages within the subset M , these are easily recognized
by a veriﬁer.
Redeﬁne the veriﬁcation function VA as
VA(s) =
true, if Ee(s) ∈ M ,
false, otherwise.
Under this new scenario A only needs to transmit the signature s since the message m =
Ee(s) can be recovered by applying the veriﬁcation function. Such a scheme is called a
digital signature scheme with message recovery. Figure 1.14 illustrates how this signature
function is used. The feature of selecting messages of special structure is referred to as
selecting messages with redundancy.
Dd(m) = s
e
Ee(s)
if m ∈ M
key
message
M
m d
m
Accept
source
source
s
Veriﬁer B
Signer A
Figure 1.14: A digital signature scheme with message recovery.
The modiﬁcation presented above is more than a simpliﬁcation; it is absolutely crucial
if one hopes to meet the requirement of property (b) of signing and veriﬁcation functions
(see page 23). To see why this is the case, note that any entity B can select a random element
s ∈ S as a signature and apply Ee to get u = Ee(s), since S = M and Ee is public
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
30 Ch. 1 Overview of Cryptography
knowledge. B may then take the message m = u and the signature on m to be s and transmits
(m, s). It is easy to check that s will verify as a signature created by A for m but in
which A has had no part. In this case B has forged a signature of A. This is an example of
what is called existential forgery. (B has produced A’s signature on some message likely
not of B’s choosing.)
If M contains only a negligible fraction of messages from M, then the probability of
some entity forging a signature of A in this manner is negligibly small.
1.52 Remark (digital signatures vs. conﬁdentiality) Although digital signature schemes based
on reversible public-key encryption are attractive, they require an encryption method as a
primitive. There are situations where a digital signature mechanism is required but encryption
is forbidden. In such cases these digital signature schemes are inappropriate.
Digital signatures in practice
For digital signatures to be useful in practice, concrete realizations of the preceding concepts
should have certain additional properties. A digital signature must
1. be easy to compute by the signer (the signing function should be easy to apply);
2. be easy to verify by anyone (the veriﬁcation function should be easy to apply); and
3. have an appropriate lifespan, i.e., be computationally secure from forgery until the
signature is no longer necessary for its original purpose.
Resolution of disputes
The purpose of a digital signature (or any signature method) is to permit the resolution of
disputes. For example, an entity A could at some point deny having signed a message or
some other entity B could falsely claim that a signature on a message was produced by A.
In order to overcome such problems a trusted third party (TTP) or judge is required. The
TTP must be some entity which all parties involved agree upon in advance.
If A denies that a message m held by B was signed by A, then B should be able to
present the signature sA for m to the TTP along with m. The TTP rules in favor of B if
VA(m, sA) = true and in favor of A otherwise. B will accept the decision if B is conﬁdent
that the TTP has the same verifyingtransformationVA as A does. A will accept the decision
if A is conﬁdent that the TTP used VA and that SA has not been compromised. Therefore,
fair resolution of disputes requires that the following criteria are met.
Requirements for resolution of disputed signatures
1. SA and VA have properties (a) and (b) of page 23.
2. The TTP has an authentic copy of VA.
3. The signing transformation SA has been kept secret and remains secure.
These properties are necessary but in practice it might not be possible to guarantee
them. For example, the assumption that SA and VA have the desired characteristics given
in property 1 might turn out to be false for a particular signature scheme. Another possibility
is that A claims falsely that SA was compromised. To overcome these problems requires
an agreed method to validate the time period for which A will accept responsibility
for the veriﬁcation transformation. An analogue of this situation can be made with credit
card revocation. The holder of a card is responsible until the holder notiﬁes the card issuing
company that the card has been lost or stolen. §13.8.2 gives a more indepth discussion of
these problems and possible solutions.
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.8 Public-key cryptography 31
1.8.4 Symmetric-key vs. public-key cryptography
Symmetric-key and public-key encryption schemes have various advantages and disadvantages,
some of which are common to both. This section highlights a number of these and
summarizes features pointed out in previous sections.
(i) Advantages of symmetric-key cryptography
1. Symmetric-key ciphers can be designed to have high rates of data throughput. Some
hardware implementations achieve encrypt rates of hundreds of megabytes per second,
while software implementations may attain throughput rates in the megabytes
per second range.
2. Keys for symmetric-key ciphers are relatively short.
3. Symmetric-key ciphers can be employed as primitives to construct various cryptographic
mechanisms including pseudorandom number generators (see Chapter 5),
hash functions (see Chapter 9), and computationally efﬁcient digital signature schemes
(see Chapter 11), to name just a few.
4. Symmetric-key ciphers can be composed to produce stronger ciphers. Simple transformations
which are easy to analyze, but on their own weak, can be used to construct
strong product ciphers.
5. Symmetric-key encryption is perceived to have an extensive history, although it must
be acknowledged that, notwithstanding the invention of rotor machines earlier, much
of the knowledge in this area has been acquired subsequent to the invention of the
digital computer, and, in particular, the design of the Data Encryption Standard (see
Chapter 7) in the early 1970s.
(ii) Disadvantages of symmetric-key cryptography
1. In a two-party communication, the key must remain secret at both ends.
2. In a large network, there are many key pairs to be managed. Consequently, effective
key managementrequires the use of an unconditionallytrusted TTP (Deﬁnition 1.65).
3. In a two-party communication between entities A and B, sound cryptographic practice
dictates that the key be changed frequently, and perhaps for each communication
session.
4. Digital signature mechanisms arising from symmetric-key encryption typically require
either large keys for the public veriﬁcation function or the use of a TTP (see
Chapter 11).
(iii) Advantages of public-key cryptography
1. Only the private key must be kept secret (authenticity of public keys must, however,
be guaranteed).
2. The administration of keys on a network requires the presence of only a functionally
trusted TTP (Deﬁnition 1.66) as opposed to an unconditionally trusted TTP. Depending
on the mode of usage, the TTP might only be required in an “off-line” manner,
as opposed to in real time.
3. Depending on the mode of usage, a private key/public key pair may remain unchanged
for considerable periods of time, e.g., many sessions (even several years).
4. Many public-key schemes yield relatively efﬁcient digital signature mechanisms.
The key used to describe the public veriﬁcation function is typically much smaller
than for the symmetric-key counterpart.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
32 Ch. 1 Overview of Cryptography
5. In a large network, the number of keys necessary may be considerably smaller than
in the symmetric-key scenario.
(iv) Disadvantages of public-key encryption
1. Throughput rates for the most popular public-key encryption methods are several orders
of magnitude slower than the best known symmetric-key schemes.
2. Key sizes are typically much larger than those required for symmetric-keyencryption
(see Remark 1.53), and the size of public-key signatures is larger than that of tags
providing data origin authentication from symmetric-key techniques.
3. No public-key scheme has been proven to be secure (the same can be said for block
ciphers). The most effective public-key encryption schemes found to date have their
security based on the presumeddifﬁcultyof a small set of number-theoreticproblems.
4. Public-key cryptography does not have as extensive a history as symmetric-key encryption,
being discovered only in the mid 1970s.6
Summary of comparison
Symmetric-key and public-key encryption have a number of complementary advantages.
Current cryptographic systems exploit the strengths of each. An example will serve to il-
lustrate.
Public-key encryption techniques may be used to establish a key for a symmetric-key
system being used by communicating entities A and B. In this scenario A and B can take
advantage of the long term nature of the public/private keys of the public-key scheme and
the performance efﬁciencies of the symmetric-key scheme. Since data encryption is frequently
the most time consuming part of the encryption process, the public-key scheme for
key establishment is a small fraction of the total encryption process between A and B.
To date, the computational performance of public-key encryption is inferior to that of
symmetric-key encryption. There is, however, no proof that this must be the case. The
important points in practice are:
1. public-keycryptographyfacilitates efﬁcient signatures (particularlynon-repudiation)
and key mangement; and
2. symmetric-key cryptography is efﬁcient for encryption and some data integrity ap-
plications.
1.53 Remark (key sizes: symmetric key vs. private key) Private keys in public-key systems
must be larger (e.g., 1024 bits for RSA) than secret keys in symmetric-key systems (e.g., 64
or 128 bits) because whereas (for secure algorithms) the most efﬁcient attack on symmetrickey
systems is an exhaustivekey search, all known public-keysystems are subject to “shortcut”
attacks (e.g., factoring)more efﬁcient than exhaustivesearch. Consequently,for equivalent
security, symmetric keys have bitlengths considerablysmaller than that of private keys
in public-key systems, e.g., by a factor of 10 or more.
6It is, of course, arguable that some public-key schemes which are based on hard mathematical problems have
a long history since these problems have been studied for many years. Although this may be true, one must be
wary that the mathematics was not studied with this application in mind.
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.9 Hash functions 33
1.9 Hash functions
One of the fundamental primitives in modern cryptography is the cryptographic hash function,
often informallycalled a one-wayhash function. A simpliﬁed deﬁnition for the present
discussion follows.
1.54 Deﬁnition A hash function is a computationally efﬁcient function mapping binary strings
of arbitrary length to binary strings of some ﬁxed length, called hash-values.
For a hash function which outputs n-bit hash-values (e.g., n = 128 or 160) and has desirable
properties, the probability that a randomly chosen string gets mapped to a particular
n-bit hash-value (image) is 2−n
. The basic idea is that a hash-value serves as a compact
representative of an input string. To be of cryptographic use, a hash function h is typically
chosen such that it is computationally infeasible to ﬁnd two distinct inputs which hash to a
common value (i.e., two colliding inputs x and y such that h(x) = h(y)), and that given
a speciﬁc hash-value y, it is computationally infeasible to ﬁnd an input (pre-image) x such
that h(x) = y.
The most common cryptographic uses of hash functions are with digital signatures and
for data integrity. With digital signatures, a long message is usually hashed (using a publicly
available hash function) and only the hash-value is signed. The party receiving the
message then hashes the received message, and veriﬁes that the received signature is correct
for this hash-value. This saves both time and space compared to signing the message
directly, which would typically involve splitting the message into appropriate-sized blocks
and signing each block individually. Note here that the inability to ﬁnd two messages with
the same hash-value is a security requirement, since otherwise, the signature on one message
hash-value would be the same as that on another, allowing a signer to sign one message
and at a later point in time claim to have signed another.
Hash functions may be used for data integrity as follows. The hash-value corresponding
to a particular input is computed at some point in time. The integrity of this hash-value
is protected in some manner. At a subsequent point in time, to verify that the input data
has not been altered, the hash-value is recomputed using the input at hand, and compared
for equality with the original hash-value. Speciﬁc applications include virus protection and
software distribution.
A third application of hash functions is their use in protocols involving a priori commitments,
including some digital signature schemes and identiﬁcation protocols (e.g., see
Chapter 10).
Hash functions as discussed above are typically publicly known and involve no secret
keys. When used to detect whether the message input has been altered, they are called modiﬁcation
detection codes (MDCs). Related to these are hash functions which involve a secret
key, and provide data origin authentication (§9.76) as well as data integrity; these are called
message authentication codes (MACs).
1.10 Protocols and mechanisms
1.55 Deﬁnition A cryptographic protocol (protocol) is a distributed algorithm deﬁned by a sequence
of steps precisely specifying the actions required of two or more entities to achieve
a speciﬁc security objective.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
34 Ch. 1 Overview of Cryptography
1.56 Remark (protocol vs. mechanism) As opposed to a protocol, a mechanism is a more general
term encompassing protocols, algorithms (specifying the steps followed by a single entity),
and non-cryptographictechniques (e.g., hardware protection and procedural controls)
to achieve speciﬁc security objectives.
Protocols play a major role in cryptography and are essential in meeting cryptographic
goals as discussed in §1.2. Encryption schemes, digital signatures, hash functions, and random
number generation are among the primitives which may be utilized to build a protocol.
1.57 Example (a simple key agreement protocol) Alice and Bob have chosen a symmetric-key
encryption scheme to use in communicating over an unsecured channel. To encrypt information
they require a key. The communication protocol is the following:
1. Bob constructs a public-keyencryptionscheme and sends his public key to Alice over
the channel.
2. Alice generates a key for the symmetric-key encryption scheme.
3. Alice encrypts the key using Bob’s public key and sends the encrypted key to Bob.
4. Bob decrypts using his private key and recovers the symmetric (secret) key.
5. Alice and Bob begin communicating with privacy by using the symmetric-key system
and the common secret key.
This protocol uses basic functions to attempt to realize private communications on an unsecured
channel. The basic primitives are the symmetric-key and the public-key encryption
schemes. The protocol has shortcomings including the impersonation attack of §1.8.2, but
it does convey the idea of a protocol.
Often the role of public-key encryption in privacy communications is exactly the one
suggested by this protocol – public-key encryption is used as a means to exchange keys
for subsequent use in symmetric-key encryption, motivated by performance differences between
symmetric-key and public-key encryption.
Protocol and mechanism failure
1.58 Deﬁnition A protocolfailure or mechanism failure occurswhen a mechanismfails to meet
the goals for which it was intended, in a manner whereby an adversary gains advantage
not by breaking an underlying primitive such as an encryption algorithm directly, but by
manipulating the protocol or mechanism itself.
1.59 Example (mechanism failure) Alice and Bob are communicating using a stream cipher.
Messages which they encrypt are known to have a special form: the ﬁrst twenty bits carry
information which represents a monetary amount. An active adversary can simply XOR an
appropriate bitstring into the ﬁrst twenty bits of ciphertext and change the amount. While
the adversary has not been able to read the underlying message, she has been able to alter
the transmission. The encryption has not been compromised but the protocol has failed to
perform adequately; the inherent assumption that encryption provides data integrity is in-
correct.
1.60 Example (forward search attack) Suppose that in an electronic bank transaction the 32bit
ﬁeld which records the value of the transaction is to be encrypted using a public-key
scheme. This simple protocol is intended to provide privacy of the value ﬁeld – but does
it? An adversary could easily take all 232
possible entries that could be plaintext in this ﬁeld
and encrypt them using the public encryption function. (Remember that by the very nature
of public-key encryption this function must be available to the adversary.) By comparing
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.11 Key establishment, management, and certiﬁcation 35
each of the 232
ciphertexts with the one which is actually encrypted in the transaction, the
adversary can determine the plaintext. Here the public-key encryption function is not compromised,
but rather the way it is used. A closely related attack which applies directly to
authentication for access control purposes is the dictionary attack (see §10.2.2).
1.61 Remark (causes of protocol failure) Protocols and mechanisms may fail for a number of
reasons, including:
1. weaknesses in a particular cryptographic primitive which may be ampliﬁed by the
protocol or mechanism;
2. claimed or assumed security guarantees which are overstated or not clearly understood;
and
3. the oversight of some principle applicable to a broad class of primitives such as en-
cryption.
Example 1.59 illustrates item 2 if the stream cipher is the one-time pad, and also item 1.
Example 1.60 illustrates item 3. See also §1.8.2.
1.62 Remark (protocol design) When designing cryptographic protocols and mechanisms, the
following two steps are essential:
1. identify all assumptions in the protocol or mechanism design; and
2. for each assumption, determine the effect on the security objective if that assumption
is violated.
1.11 Key establishment, management, and
certiﬁcation
This section gives a brief introduction to methodology for ensuring the secure distribution
of keys for cryptographic purposes.
1.63 Deﬁnition Key establishment is any process whereby a shared secret key becomes available
to two or more parties, for subsequent cryptographic use.
1.64 Deﬁnition Key management is the set of processes and mechanisms which support key
establishment and the maintenanceof ongoingkeying relationshipsbetween parties, including
replacing older keys with new keys as necessary.
Key establishment can be broadly subdivided into key agreement and key transport.
Many and various protocols have been proposed to provide key establishment. Chapter 12
describes a numberof these in detail. For the purposeof this chapteronlya brief overviewof
issues related to key management will be given. Simple architectures based on symmetrickey
and public-key cryptography along with the concept of certiﬁcation will be addressed.
As noted in §1.5, a major issue when using symmetric-key techniques is the establishment
of pairwise secret keys. This becomes more evident when considering a network of
entities, any two of which may wish to communicate. Figure 1.15 illustrates a network consisting
of 6 entities. The arrowed edges indicate the 15 possible two-party communications
which could take place. Since each pair of entities wish to communicate, this small network
requires the secure exchange of 6
2 = 15 key pairs. In a network with n entities, the
number of secure key exchanges required is n
2 = n(n−1)
2 .
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
36 Ch. 1 Overview of Cryptography
A1 A2
A5
A3A6
A4
Figure 1.15: Keying relationships in a simple 6-party network.
The network diagram depicted in Figure 1.15 is simply the amalgamation of 15 twoparty
communications as depicted in Figure 1.7. In practice, networks are very large and
the key management problem is a crucial issue. There are a number of ways to handle this
problem. Two simplistic methods are discussed; one based on symmetric-key and the other
on public-key techniques.
1.11.1 Key management through symmetric-key techniques
One solution which employs symmetric-key techniques involves an entity in the network
which is trusted by all other entities. As in §1.8.3, this entity is referred to as a trusted third
party (TTP). Each entity Ai shares a distinct symmetric key ki with the TTP. These keys are
assumed to have been distributed over a secured channel. If two entities subsequently wish
to communicate, the TTP generates a key k (sometimes called a session key) and sends it
encrypted under each of the ﬁxed keys as depicted in Figure 1.16 for entities A1 and A5.
A1
Ek(m)
Ek5
(k)
A5
TTP
A6
k5
k
Ek1
(k)
k1
k6
A2
A3
A4
k2
k3
k4
source
key
Figure 1.16: Key management using a trusted third party (TTP).
Advantages of this approach include:
1. It is easy to add and remove entities from the network.
2. Each entity needs to store only one long-term secret key.
Disadvantages include:
1. All communications require initial interaction with the TTP.
2. The TTP must store n long-term secret keys.
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.11 Key establishment, management, and certiﬁcation 37
3. The TTP has the ability to read all messages.
4. If the TTP is compromised, all communications are insecure.
1.11.2 Key management through public-key techniques
There are a number of ways to address the key management problem through public-key
techniques. Chapter 13 describes many of these in detail. For the purpose of this chapter a
very simple model is considered.
Each entity in the network has a public/private encryption key pair. The public key
along with the identity of the entity is stored in a central repository called a public ﬁle. If
an entity A1 wishes to send encrypted messages to entity A6, A1 retrieves the public key
e6 of A6 from the public ﬁle, encrypts the message using this key, and sends the ciphertext
to A6. Figure 1.17 depicts such a network.
private key d5
c
private key d6
private key d1
c = Ee6 (m)
Public ﬁle
e6
A1 : e1
A2 : e2
A3 : e3
A4 : e4
A5 : e5
A6 : e6
private key d2
private key d3
A5 A4
private key d4
A3A6
A1 A2
m = Dd6
(c)
Figure 1.17: Key management using public-key techniques.
Advantages of this approach include:
1. No trusted third party is required.
2. The public ﬁle could reside with each entity.
3. Only n public keys need to be stored to allow secure communications between any
pair of entities, assuming the only attack is that by a passive adversary.
The key managementproblem becomes more difﬁcult when one must take into account
an adversary who is active (i.e. an adversary who can alter the public ﬁle containing public
keys). Figure 1.18 illustrates how an active adversary could compromise the key management
scheme given above. (This is directly analogous to the attack in §1.8.2.) In the ﬁgure,
the adversary alters the public ﬁle by replacing the public key e6 of entity A6 by the adversary’s
public key e∗
. Any message encrypted for A6 using the public key from the public
ﬁle can be decrypted by only the adversary. Having decrypted and read the message, the
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
38 Ch. 1 Overview of Cryptography
adversary can now encrypt it using the public key of A6 and forward the ciphertext to A6.
A1 however believes that only A6 can decrypt the ciphertext c.
c
A1
e∗
A1 : e1
A2 : e2
A3 : e3
A4 : e4
A5 : e5
A6 : e6 e∗
Dd6 (c ) = mEe6 (m) = c c
Adversary
private key
d6
Public ﬁle
Ee∗ (m) = c
private key
A6
d∗
Dd∗ (c) = m
Figure 1.18: An impersonation of A6 by an active adversary with public key e∗
.
To prevent this type of attack, the entities may use a TTP to certify the public key of
each entity. The TTP has a private signing algorithm ST and a veriﬁcation algorithm VT
(see §1.6) assumed to be known by all entities. The TTP carefully veriﬁes the identity of
each entity, and signs a message consisting of an identiﬁer and the entity’s authentic public
key. This is a simple example of a certiﬁcate, binding the identity of an entity to its public
key (see §1.11.3). Figure 1.19 illustrates the network under these conditions. A1 uses the
public key of A6 only if the certiﬁcate signature veriﬁes successfully.
e6, s6
A1, e1, ST (A1 e1) = s1
A2, e2, ST (A2 e2) = s2
A3, e3, ST (A3 e3) = s3
A4, e4, ST (A4 e4) = s4
A5, e5, ST (A5 e5) = s5
A6, e6, ST (A6 e6) = s6
A1
Ee6 (m)
VT (A6 e6, s6)
Public ﬁle
Dd6 (c) = m
d6
c =
A6
veriﬁcation
private key
Figure 1.19: Authentication of public keys by a TTP. denotes concatenation.
Advantages of using a TTP to maintain the integrity of the public ﬁle include:
1. It prevents an active adversary from impersonation on the network.
2. The TTP cannot monitor communications. Entities need trust the TTP only to bind
identities to public keys properly.
3. Per-communication interaction with the public ﬁle can be eliminated if entities store
certiﬁcates locally.
Even with a TTP, some concerns still remain:
1. If the signing key of the TTP is compromised, all communications become insecure.
2. All trust is placed with one entity.
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.12 Pseudorandom numbers and sequences 39
1.11.3 Trusted third parties and public-key certiﬁcates
A trusted third party has been used in §1.8.3 and again here in §1.11. The trust placed on
this entity varies with the way it is used, and hence motivates the following classiﬁcation.
1.65 Deﬁnition A TTP is said to be unconditionally trusted if it is trusted on all matters. For
example, it may have access to the secret and private keys of users, as well as be charged
with the association of public keys to identiﬁers.
1.66 Deﬁnition A TTP is said to be functionally trusted if the entity is assumed to be honest
and fair but it does not have access to the secret or private keys of users.
§1.11.1 provides a scenario which employs an unconditionally trusted TTP. §1.11.2
uses a functionally trusted TTP to maintain the integrity of the public ﬁle. A functionally
trusted TTP could be used to register or certify users and contents of documents or, as in
§1.8.3, as a judge.
Public-key certiﬁcates
The distribution of public keys is generally easier than that of symmetric keys, since secrecy
is not required. However, the integrity (authenticity)of public keys is critical (recall §1.8.2).
A public-key certiﬁcate consists of a data part and a signature part. The data part consists
of the name of an entity, the public key correspondingto that entity, possibly additional
relevant information (e.g., the entity’s street or network address, a validity period for the
public key, and various other attributes). The signature part consists of the signature of a
TTP over the data part.
In order for an entity B to verify the authenticity of the public key of an entity A, B
must have an authentic copy of the public signature veriﬁcation function of the TTP. For
simplicity, assume that the authenticity of this veriﬁcation function is providedto B by noncryptographic
means, for example by B obtaining it from the TTP in person. B can then
carry out the following steps:
1. Acquire the public-key certiﬁcate of A over some unsecured channel, either from a
central database of certiﬁcates, from A directly, or otherwise.
2. Use the TTP’s veriﬁcation function to verify the TTP’s signature on A’s certiﬁcate.
3. If this signature veriﬁes correctly, accept the public key in the certiﬁcate as A’s authentic
public key; otherwise, assume the public key is invalid.
Before creating a public-key certiﬁcate for A, the TTP must take appropriate measures
to verify the identity of A and the fact that the public key to be certiﬁcated actually belongs
to A. One method is to require that A appear before the TTP with a conventional passport
as proof of identity, and obtain A’s public key from A in person along with evidence that
A knows the corresponding private key. Once the TTP creates a certiﬁcate for a party, the
trust that all other entities have in the authenticity of the TTP’s public key can be used transitively
to gain trust in the authenticity of that party’s public key, through acquisition and
veriﬁcation of the certiﬁcate.
1.12 Pseudorandom numbers and sequences
Random number generation is an important primitive in many cryptographic mechanisms.
For example, keys for encryption transformationsneed to be generated in a manner which is
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
40 Ch. 1 Overview of Cryptography
unpredictable to an adversary. Generating a random key typically involves the selection of
random numbers or bit sequences. Random number generation presents challenging issues.
A brief introduction is given here with details left to Chapter 5.
Often in cryptographic applications, one of the following steps must be performed:
(i) From a ﬁnite set of n elements (e.g., {1, 2, . . . , n}), select an element at random.
(ii) From the set of all sequences (strings) of length m over some ﬁnite alphabet A of n
symbols, select a sequence at random.
(iii) Generate a random sequence (string) of symbols of length m over a set of n symbols.
It is not clear what exactly it means to select at random or generate at random. Calling a
number random without a context makes little sense. Is the number 23 a random number?
No, but if 49 identical balls labeled with a number from 1 to 49 are in a container, and this
container mixes the balls uniformly, drops one ball out, and this ball happens to be labeled
with the number 23, then one would say that 23 was generated randomly from a uniform
distribution. The probability that 23 drops out is 1 in 49 or 1
49 .
If the number on the ball which was dropped from the container is recorded and the ball
is placed back in the container and the process repeated 6 times, then a random sequence
of length 6 deﬁned on the alphabet A = {1, 2, . . . , 49} will have been generated. What is
the chance that the sequence 17, 45, 1, 7, 23, 35occurs? Since each element in the sequence
has probability 1
49 of occuring, the probability of the sequence 17, 45, 1, 7, 23, 35occurring
is
1
49
×
1
49
×
1
49
×
1
49
×
1
49
×
1
49
=
1
13841287201
.
There are precisely 13841287201 sequences of length 6 over the alphabet A. If each of
these sequences is written on one of 13841287201balls and they are placed in the container
(ﬁrst removing the original 49 balls) then the chance that the sequence given above drops
out is the same as if it were generated one ball at a time. Hence, (ii) and (iii) above are
essentially the same statements.
Finding good methods to generate random sequences is difﬁcult.
1.67 Example (random sequence generator) To generate a random sequence of 0’s and 1’s, a
coin could be tossed with a head landing up recorded as a 1 and a tail as a 0. It is assumed
that the coin is unbiased, which means that the probabilityof a 1 on a given toss is exactly 1
2 .
This will depend on how well the coin is made and how the toss is performed. This method
would be of little value in a system where random sequences must be generated quickly
and often. It has no practical value other than to serve as an example of the idea of random
number generation.
1.68 Example (random sequence generator) A noise diode may be used to produce random
binary sequences. This is reasonable if one has some way to be convinced that the probability
that a 1 will be produced on any given trial is 1
2 . Should this assumption be false, the
sequence generated would not have been selected from a uniform distribution and so not
all sequences of a given length would be equally likely. The only way to get some feeling
for the reliability of this type of random source is to carry out statistical tests on its output.
These are considered in Chapter 5. If the diode is a source of a uniform distribution on the
set of all binary sequences of a given length, it provides an effective way to generate random
sequences.
Since most true sources of random sequences (if there is such a thing) come from physical
means, they tend to be either costly or slow in their generation. To overcome these
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.13 Classes of attacks and security models 41
problems, methods have been devised to construct pseudorandom sequences in a deterministic
manner from a shorter random sequence called a seed. The pseudorandom sequences
appear to be generated by a truly random source to anyone not knowing the method of generation.
Often the generation algorithm is known to all, but the seed is unknown except by
the entity generating the sequence. A plethora of algorithms has been developedto generate
pseudorandom bit sequences of various types. Many of these are completely unsuitable for
cryptographic purposes and one must be cautious of claims by creators of such algorithms
as to the random nature of the output.
1.13 Classes of attacks and security models
Over the years, many different types of attacks on cryptographic primitives and protocols
have been identiﬁed. The discussion here limits consideration to attacks on encryption and
protocols. Attacks on other cryptographic primitives will be given in appropriate chapters.
In §1.11the roles of an active and a passive adversarywere discussed. The attacks these
adversaries can mount may be classiﬁed as follows:.
1. A passive attack is one where the adversary only monitors the communication channel.
A passive attacker only threatens conﬁdentiality of data.
2. An active attack is one where the adversary attempts to delete, add, or in some other
way alter the transmission on the channel. An active attacker threatens data integrity
and authentication as well as conﬁdentiality.
A passive attack can be further subdivided into more specialized attacks for deducing
plaintext from ciphertext, as outlined in §1.13.1.
1.13.1 Attacks on encryption schemes
The objective of the following attacks is to systematically recover plaintext from ciphertext,
or even more drastically, to deduce the decryption key.
1. A ciphertext-only attack is one where the adversary (or cryptanalyst) tries to deduce
the decryption key or plaintext by only observing ciphertext. Any encryption scheme
vulnerable to this type of attack is considered to be completely insecure.
2. A known-plaintext attack is one where the adversary has a quantity of plaintext and
corresponding ciphertext. This type of attack is typically only marginally more difﬁcult
to mount.
3. A chosen-plaintext attack is one where the adversary chooses plaintext and is then
given corresponding ciphertext. Subsequently, the adversary uses any information
deduced in order to recover plaintext corresponding to previously unseen ciphertext.
4. An adaptive chosen-plaintext attack is a chosen-plaintext attack wherein the choice
of plaintext may depend on the ciphertext received from previous requests.
5. A chosen-ciphertext attack is one where the adversary selects the ciphertext and is
then given the corresponding plaintext. One way to mount such an attack is for the
adversary to gain access to the equipment used for decryption (but not the decryption
key, which may be securely embedded in the equipment). The objective is then to
be able, without access to such equipment, to deduce the plaintext from (different)
ciphertext.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
42 Ch. 1 Overview of Cryptography
6. An adaptive chosen-ciphertext attack is a chosen-ciphertext attack where the choice
of ciphertext may depend on the plaintext received from previous requests.
Most of these attacks also apply to digital signature schemes and message authentication
codes. In this case, the objective of the attacker is to forge messages or MACs, as discussed
in Chapters 11 and 9, respectively.
1.13.2 Attacks on protocols
The following is a partial list of attacks which might be mounted on various protocols. Until
a protocol is proven to provide the service intended, the list of possible attacks can never
be said to be complete.
1. known-key attack. In this attack an adversary obtains some keys used previously and
then uses this information to determine new keys.
2. replay. In this attack an adversary records a communication session and replays the
entire session, or a portion thereof, at some later point in time.
3. impersonation. Here an adversary assumes the identity of one of the legitimate parties
in a network.
4. dictionary. This is usually an attack against passwords. Typically, a password is
stored in a computer ﬁle as the image of an unkeyed hash function. When a user
logs on and enters a password, it is hashed and the image is compared to the stored
value. An adversary can take a list of probable passwords, hash all entries in this list,
and then compare this to the list of true encrypted passwords with the hope of ﬁnding
matches.
5. forward search. This attack is similar in spirit to the dictionary attack and is used to
decrypt messages. An example of this method was cited in Example 1.60.
6. interleaving attack. This type of attack usually involves some form of impersonation
in an authentication protocol (see §12.9.1).
1.13.3 Models for evaluating security
The security of cryptographic primitives and protocols can be evaluated under several different
models. The most practical security metrics are computational, provable, and ad hoc
methodology, although the latter is often dangerous. The conﬁdence level in the amount
of security provided by a primitive or protocol based on computational or ad hoc security
increases with time and investigation of the scheme. However, time is not enough if few
people have given the method careful analysis.
(i) Unconditional security
The most stringent measure is an information-theoretic measure – whether or not a system
has unconditional security. An adversary is assumed to have unlimited computational
resources, and the question is whether or not there is enough information available to defeat
the system. Unconditional security for encryption systems is called perfect secrecy.
For perfect secrecy, the uncertainty in the plaintext, after observing the ciphertext, must be
equal to the a priori uncertainty about the plaintext – observation of the ciphertext provides
no information whatsoever to an adversary.
A necessary condition for a symmetric-key encryption scheme to be unconditionally
secure is that the key be at least as long as the message. The one-time pad (§1.5.4) is an example
of an unconditionally secure encryption algorithm. In general, encryption schemes
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.13 Classes of attacks and security models 43
do not offer perfect secrecy, and each ciphertext character observed decreases the theoretical
uncertainty in the plaintext and the encryption key. Public-key encryption schemes cannot
be unconditionally secure since, given a ciphertext c, the plaintext can in principle be
recovered by encrypting all possible plaintexts until c is obtained.
(ii) Complexity-theoretic security
An appropriate model of computation is deﬁned and adversaries are modeled as having
polynomial computational power. (They mount attacks involving time and space polynomial
in the size of appropriatesecurity parameters.) A proofof security relative to the model
is then constructed. An objective is to design a cryptographic method based on the weakest
assumptions possible anticipating a powerful adversary. Asymptotic analysis and usually
also worst-case analysis is used and so care must be exercised to determine when proofs
have practical signiﬁcance. In contrast, polynomial attacks which are feasible under the
model might, in practice, still be computationally infeasible.
Security analysis of this type, although not of practical value in all cases, may nonetheless
pave the way to a better overall understanding of security. Complexity-theoretic analysis
is invaluable for formulating fundamental principles and conﬁrming intuition. This is
like many other sciences, whose practical techniques are discovered early in the development,
well before a theoretical basis and understanding is attained.
(iii) Provable security
A cryptographic method is said to be provably secure if the difﬁculty of defeating it can be
shown to be essentially as difﬁcult as solving a well-known and supposedly difﬁcult (typically
number-theoretic) problem, such as integer factorization or the computation of discrete
logarithms. Thus, “provable” here means provable subject to assumptions.
This approach is considered by some to be as good a practical analysis technique as
exists. Provable security may be considered part of a special sub-class of the larger class of
computational security considered next.
(iv) Computational security
This measures the amount of computational effort required, by the best currently-known
methods, to defeat a system; it must be assumed here that the system has been well-studied
to determine which attacks are relevant. A proposed technique is said to be computationally
secure if the perceived level of computation required to defeat it (using the best attack
known) exceeds, by a comfortable margin, the computational resources of the hypothesized
adversary.
Often methods in this class are related to hard problems but, unlike for provable security,
no proof of equivalence is known. Most of the best known public-key and symmetrickey
schemes in current use are in this class. This class is sometimes also called practical
security.
(v) Ad hoc security
This approach consists of any variety of convincing arguments that every successful attack
requiresa resourcelevel (e.g., time and space) greaterthan the ﬁxed resources of a perceived
adversary. Cryptographic primitives and protocols which survive such analysis are said to
have heuristic security, with security here typically in the computational sense.
Primitives and protocols are usually designed to counter standard attacks such as those
given in §1.13. While perhaps the most commonlyused approach(especially for protocols),
it is, in some ways, the least satisfying. Claims of security generally remain questionable
and unforeseen attacks remain a threat.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
44 Ch. 1 Overview of Cryptography
1.13.4 Perspective for computational security
To evaluate the security of cryptographic schemes, certain quantities are often considered.
1.69 Deﬁnition The work factor Wd is the minimum amount of work (measured in appropriate
units such as elementary operations or clock cycles) required to compute the private key d
given the public key e, or, in the case of symmetric-key schemes, to determine the secret
key k. More speciﬁcally, one may consider the work required under a ciphertext-onlyattack
given n ciphertexts, denoted Wd(n).
If Wd is t years, then for sufﬁcientlylarget the cryptographicscheme is, for all practical
purposes, a secure system. To date no public-key system has been found where one can
prove a sufﬁciently large lower bound on the work factor Wd. The best that is possible to
date is to rely on the following as a basis for security.
1.70 Deﬁnition The historical work factor Wd is the minimum amount of work required to
compute the private key d from the public key e using the best known algorithms at a given
point in time.
The historical work factor Wd varies with time as algorithms and technology improve.
It corresponds to computational security, whereas Wd corresponds to the true security level,
although this typically cannot be determined.
How large is large?
§1.4 described how the designer of an encryption system tries to create a scheme for which
the best approach to breaking it is through exhaustive search of the key space. The key
space must then be large enough to make an exhaustive search completely infeasible. An
important question then is “How large is large?”. In order to gain some perspective on the
magnitude of numbers, Table 1.2 lists various items along with an associated magnitude.
Reference Magnitude
Seconds in a year ≈ 3 × 107
Age of our solar system (years) ≈ 6 × 109
Seconds since creation of solar system ≈ 2 × 1017
Clock cycles per year, 50 MHz computer ≈ 1.6 × 1015
Binary strings of length 64 264
≈ 1.8 × 1019
Binary strings of length 128 2128
≈ 3.4 × 1038
Binary strings of length 256 2256
≈ 1.2 × 1077
Number of 75-digit prime numbers ≈ 5.2 × 1072
Electrons in the universe ≈ 8.37 × 1077
Table 1.2: Reference numbers comparing relative magnitudes.
Some powers of 10 are referred to by preﬁxes. For example, high-speed modern computers
are now being rated in terms of teraﬂops where a teraﬂop is 1012
ﬂoating point operations
per second. Table 1.3 provides a list of commonly used preﬁxes.
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.14 Notes and further references 45
Preﬁx Symbol Magnitude
exa E 1018
peta P 1015
tera T 1012
giga G 109
mega M 106
kilo k 103
hecto h 102
deca da 10
Preﬁx Symbol Magnitude
deci d 10−1
centi c 10−2
milli m 10−3
micro µ 10−6
nano n 10−9
pico p 10−12
femto f 10−15
atto a 10−18
Table 1.3: Preﬁxes used for various powers of 10.
1.14 Notes and further references
§1.1
Kahn [648] gives a thorough, comprehensive, and non-technical history of cryptography,
published in 1967. Feistel [387] provides an early exposition of block cipher ideas. The
original speciﬁcation of DES is the 1977 U.S. Federal Information Processing Standards
Publication 46 [396]. Public-key cryptography was introduced by Difﬁe and Hellman
[345]. The ﬁrst concrete realization of a public-key encryption scheme was the knapsack
scheme by Merkle and Hellman [857]. The RSA public-key encryption and signature scheme
is due to Rivest, Shamir, and Adleman [1060], while the ElGamal public-key encryption
and signature schemes are due to ElGamal [368]. The two digital signature standards,
ISO/IEC 9796 [596] and the Digital Signature Standard [406], are discussed extensively in
Chapter 11.
Cryptography has used specialized areas of mathematics such as number theory to realize
very practical mechanisms such as public-keyencryptionand digital signatures. Such usage
was not conceived as possible a mere twenty years ago. The famous mathematician, Hardy
[539], went as far as to boast about its lack of utility:
“ . . . both Gauss and lesser mathematicians may be justiﬁed in rejoicing that
there is one science at any rate, and that their own, whose very remoteness from
ordinary human activities should keep it gentle and clean.”
§1.2
This section was inspired by the foreword to the book Contemporary Cryptology, The Science
of Information Integrity, edited by Simmons [1143]. The handwritten signature came
into the British legal system in the seventeenth century as a means to provide various functions
associated with information security. See Chapter 9 of Meyer and Matyas [859] for
details.
This book only considers cryptography as it applies to information in digital form. Chapter
9 of Beker and Piper [84] provides an introduction to the encryption of analogue signals,
in particular, speech. Although in many cases physical means are employed to facilitate
privacy, cryptography plays the major role. Physical means of providing privacy include
ﬁber optic communication links, spread spectrum technology, TEMPEST techniques, and
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
46 Ch. 1 Overview of Cryptography
tamper-resistant hardware. Steganography is that branch of information privacy which attempts
to obscure the existence of data through such devices as invisible inks, secret compartments,
the use of subliminal channels, and the like. Kahn [648] provides an historical
account of various steganographic techniques.
Excellent introductions to cryptography can be found in the articles by Difﬁe and Hellman
[347], Massey [786], and Rivest [1054]. A concise and elegant way to describe cryptography
was given by Rivest [1054]: Cryptography is about communication in the presence of
adversaries. The taxonomy of cryptographic primitives (Figure 1.1) was derived from the
classiﬁcation given by Bosselaers, Govaerts, and Vandewalle [175].
§1.3
The theory of functions is fundamental in modern mathematics. The term range is often
used in place of image of a function. The latter, being more descriptive, is preferred. An
alternate term for one-to-one is injective; an alternate term for onto is surjective.
One-way functions were introduced by Difﬁe and Hellman [345]. A more extensive history
is given on page 377. Trapdoor one-way functions were ﬁrst postulated by Difﬁe and Hellman
[345] and independently by Merkle [850] as a means to obtain public-key encryption
schemes; several candidates are given in Chapter 8.
§1.4
The basic concepts of cryptography are treated quite differently by various authors, some
being more technical than others. Brassard [192] provides a concise, lucid, and technically
accurate account. Schneier [1094] gives a less technical but very accessible introduction.
Salomaa [1089], Stinson [1178], and Rivest [1054] present more mathematical approaches.
Davies and Price [308] provide a very readable presentation suitable for the practitioner.
The comparison of an encryption scheme to a resettable combination lock is from Difﬁe
and Hellman [347]. Kerckhoffs’ desiderata [668] were originally stated in French. The
translation stated here is given in Kahn [648]. Shannon [1121] also gives desiderata for
encryption schemes.
§1.5
Symmetric-key encryption has a very long history, as recorded by Kahn [648]. Most systems
invented prior to the 1970s are now of historical interest only. Chapter 2 of Denning
[326] is also a good source for many of the more well known schemes such as the Caesar
cipher, Vigen`ere and Beaufort ciphers, rotor machines (Enigma and Hagelin), running key
ciphers, and so on; see also Davies and Price [308] and Konheim [705]. Beker and Piper
[84] give an indepth treatment, including cryptanalysis of several of the classical systems
used in World War II. Shannon’s paper [1121] is considered the seminal work on secure
communications. It is also an excellent source for descriptions of various well-known historical
symmetric-key ciphers.
Simple substitution and transposition ciphers are the focus of §1.5. Hill ciphers [557], a
class of substitution ciphers which substitute blocks using matrix methods, are covered in
Example 7.52. The idea of confusion and diffusion (Remark 1.36) was introduced by Shannon
[1121].
Kahn [648] gives 1917 as the date when Vernam discovered the cipher which bears Vernam’s
name, however, Vernam did not publish the result until 1926 [1222]; see page 274
for further discussion. Massey [786] states that reliable sources have suggested that the
Moscow-Washington hot-line (channel for very high level communications) is no longer
secured with a one-time pad, which has been replaced by a symmetric-key cipher requiring
a much shorter key. This change would indicate that conﬁdence and understanding in the
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.
§1.14 Notes and further references 47
ability to construct very strong symmetric-key encryption schemes exists. The one-time
pad seems to have been used extensively by Russian agents operating in foreign countries.
The highest ranking Russian agent ever captured in the United States was Rudolph Abel.
When apprehended in 1957 he had in his possession a booklet the size of a postage stamp
(17
8 × 7
8 × 7
8 inches) containing a one-time key; see Kahn [648, p.664].
§1.6
The concept of a digital signature was introduced by Difﬁe and Hellman [345] and independentlyby
Merkle [850]. The ﬁrst practical realizationof a digital signaturescheme appeared
in the paper by Rivest, Shamir, and Adleman [1060]. Rabin [1022] (see also [1023]) also
claims to have independently discovered RSA but did not publish the result.
Most introductory sources for digital signatures stress digital signatures with message recovery
coming from a public-key encryption system. Mitchell, Piper, and Wild [882] give
a good general treatment of the subject. Stinson [1178] provides a similar elementary but
general introduction. Chapter 11 generalizes the deﬁnition of a digital signature by allowing
randomization. The scheme described in §1.8 is referred to as deterministic. Many other
types of digital signatures with speciﬁc properties have been created, such as blind signatures,
undeniable signatures, and failstop signatures (see Chapter 11).
§1.7
Much effort has been devoted to developing a theory of authentication. At the forefront of
this is Simmons [1144], whose contributions are nicely summarized by Massey [786]. For
a more concrete example of the necessity for authentication without secrecy, see the article
by Simmons [1146].
§1.8
1976 marked a major turning point in the history of cryptography. In several papers that
year, Difﬁe and Hellman introduced the idea of public-key cryptography and gave concrete
examples of how such a scheme might be realized. The ﬁrst paper on public-key cryptography
was “Multiuser cryptographic techniques” by Difﬁe and Hellman [344], presented
at the National Computer Conference in June of 1976. Although the authors were not satisﬁed
with the examples they cited, the concept was made clear. In their landmark paper,
Difﬁe and Hellman [345] provided a more comprehensive account of public-key cryptography
and described the ﬁrst viable method to realize this elegant concept. Another good
source for the early history and development of the subject is Difﬁe [343]. Nechvatal [922]
also provides a broad survey of public-key cryptography.
Merkle [849, 850] independently discovered public-key cryptography, illustrating how this
concept could be realized by giving an elegant and ingenious example now commonly referred
to as the Merkle puzzle scheme. Simmons [1144, p.412] notes the ﬁrst reported application
of public-key cryptography was ﬁelded by Sandia National Laboratories (U.S.) in
1978.
§1.9
Much of the early work on cryptographic hash functions was done by Merkle [850]. The
most comprehensive current treatment of the subject is by Preneel [1004].
§1.10
A large number of successful cryptanalytic attacks on systems claiming security are due to
protocol failure. An overview of this area is given by Moore [899], including classiﬁcations
of protocol failures and design principles.
Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.
48 Ch. 1 Overview of Cryptography
§1.11
One approach to distributing public-keys is the so-called Merkle channel (see Simmons
[1144, p.387]). Merkle proposed that public keys be distributed over so many independent
public channels (newspaper, radio, television, etc.) that it would be improbable for an adversary
to compromise all of them.
In 1979 Kohnfelder [702] suggested the idea of using public-key certiﬁcates to facilitate
the distribution of public keys over unsecured channels, such that their authenticity can be
veriﬁed. Essentially the same idea, but by on-line requests, was proposed by Needham and
Schroeder (ses Wilkes [1244]).
A provablysecure key agreementprotocol has been proposedwhose security is based on the
Heisenberg uncertainty principle of quantum physics. The security of so-called quantum
cryptography does not rely upon any complexity-theoretic assumptions. For further details
on quantum cryptography, consult Chapter 6 of Brassard [192], and Bennett, Brassard, and
Ekert [115].
§1.12
For an introductionand detailed treatment of many pseudorandomsequence generators, see
Knuth [692]. Knuth cites an example of a complex scheme to generate random numbers
which on closer analysis is shown to produce numbers which are far from random, and concludes:
...random numbers should not be generated with a method chosen at random.
§1.13
The seminal work of Shannon [1121] on secure communications, published in 1949, remains
as one of the best introductions to both practice and theory, clearly presenting many
of the fundamentalideas includingredundancy,entropy, and unicity distance. Various models
under which security may be examined are considered by Rueppel [1081], Simmons
[1144], and Preneel [1003], among others; see also Goldwasser [476].
c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.