# Basic terminology and concepts

The scientific study of any discipline must be built upon rigorous definitions arising from fundamental concepts. What follows is a Ust 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 encoded in 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 /С is called a*key.* - • Each element e e
*K,*uniquely determines a bijection from*M*to*C,*denoted by*E,.. E,*is called an*encryption function*or an*encryption transformation.*Note that*E*must be a bijection if the process is to be reversed and a unique plaintext message recovered for each distinct ciphertext._{e }^{1} - • For each
*d*e*1C, Dd*denotes a bijection from*C*to*M*(i.e.,*D,t: C*—»*M). Da*is called a*decryption function*or*decryption transformation.* - • The process of applying the transformation
*E*to a message_{e}*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.

'More generally is obtained if *E,.* is simply defined as a 1 — 1 transformation from *M* to *C.* That is to say, E_{e} is a bijection from *M* to Im(E_{e}) where Im(E_{e}) is a subset of *C.*

- • An
*encryption scheme*consists of a set*{E*e_{e}: e*K.)*of encryption transformations and a corresponding set*{Dd: d*e /С} of decryption transformations with the property that for each e €*K.*there is a unique key*d**e**1C*such that*Dd = E~*that is,^{l}*D,{(E*for all m €_{e}(m)) = 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*1C,*a set of encryption transformations {*E*_{(}.*: e**e**1C},*and a corresponding set of decryption transformations {Д/:*d**e**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 pah (e, *d).* At a subsequent point in time, if Alice wishes to send a message *m* e *M* to Bob, she computes c = *E _{e}(m*) and transmits this to Bob. Upon receiving

*c,*Bob computes

*D,/(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 =* {mi, m2, m3} and *С* = {с!,С2,сз}. 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 *E*,, 1 < *i <* 6. Alice and Bob agree on a trans-

**Figure 1.5: ***Schematic of a simple encryption scheme.*

formation, say *E.* To encrypt the message mi, Alice computes Ej (mi) = c_{3} and sends сз to Bob. Bob decrypts c_{3} by reversing the arrows on the diagram for *E* and observing that c_{3} points to mi.

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 algorithms. □

Figure 1.6 provides a simple model of a two-party communication using encryption.

**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-party communication which 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 information security 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 channel*is one from which an adversary does not have the ability to reorder, delete, insert, or read.

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 communicating parties, and a dedicated communication link, to name a few.

Security

A fundamental premise in cryptography is that the sets *M.C. К, {E _{e}: *

*e*e /С}, {Д/:

*d e*

*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. Histoiy 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 thud 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 number of keys (i.e., the size of the key space) should be large enough to make this approach computationally infeasible. It is the objective of a designer of an encryption scheme that this be the best approach to break the system.

Frequently cited in the literature are *Kercklioffs’ desiderata,* a set of requirements for cipher systems. They are given here essentially as Kercklioffs 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 requirements was articulated in 1883 and, for the 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> sen-ice* 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.

- •
*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.