3 Title : Classical Encryption 3.2 Cryptosystems and ... · In this section we examine some classical encryption techniques, based on substitution. A substitution technique is one

Information Security Lecture 3

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Lecture 3 Title : Classical Encryption

Lecture Outlines:

3.1 Symmetric Cipher Model

3.2 Cryptosystems and Cryptanalysis

3.3 Substitution Techniques

3.4 Transposition Techniques

Objectives:

After studying this lecture, you will be able to discuss

✓ Understand basic principle of symmetric cipher

✓ Encrypt and decrypt messages using simple substitution methods

✓ Understand the weakness of encryption methods

✓ Devise ways to strengthen the methods

✓ Devise cryptanalytic attacks on the methods

✓ Use transposition for encryption

✓ Understand the operation of rotor machines which is multiple

encryption

✓ Appreciate the strength and simplicity of steganography for hiding

messages


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3.1 Symmetric Cipher Model

A symmetric encryption scheme has five items (Figure 3.1). They are:

1. Plaintext: This is the original intelligible message or data that is fed into the

algorithm as input.

2. Encryption algorithm: The encryption algorithm performs various

substitutions and transformations on the plaintext.

3. Secret key: The secret key is also input to the encryption algorithm. The key is

a value independent of the plaintext and of the algorithm. The algorithm will

produce a different output depending on the specific key being used at the time.

The exact substitutions and transformations performed by the algorithm depend

on the key.

4. Cipher text: This is the scrambled message produced as output. It depends on

the plaintext and the secret key. For a given message, two different keys will

produce two different cipher texts. The cipher text is random stream of data

and, as it stands, is unintelligible.

5. Decryption algorithm: This is essentially the encryption algorithm run in

reverse. It takes the cipher text and the secret key and produces the original

plaintext.

Figure 3.1: Simplified Model of Conventional Encryption


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There are two requirements for secure use of conventional (classical)encryption:

1. We need a strong encryption algorithm. At a minimum, we would like the

algorithm to be such that an opponent who knows the algorithm and has access

to one or more cipher texts would be unable to decipher the cipher text or figure

out the key. The opponent should be unable to decrypt cipher text or discover the

key even if he or she has a number of cipher texts together with the plaintext that

produced each cipher text.

2. Sender and receiver must have obtained copies of the secret key in a secure

manner and must keep the key secure. If someone can discover the key and

knows the algorithm, all communication using this key is readable.

It is impractical keep the algorithm secret; we need to keep only the key secret. This

feature of symmetric encryption is what makes it feasible for widespread use. With

the use of symmetric encryption, the principal security problem is maintaining the

secrecy of the key. For this reason, key is sent to the receiver through a separate

secure channel. Alternatively, a trusted third party can generate the key and send

this to both source and destination.

Let us take a closer look at the essential elements of a symmetric encryption scheme,

using Figure 3.2. A source produces a message in plaintext,

X = [X1, X2, ... XM ]. The M elements of X are letters in some finite alphabet.

Traditionally, the alphabet usually consisted of the 26 capital letters. Nowadays, the

binary alphabet {0, 1} is typically used. For encryption, a key of the form K = [K1,

K2, ... KJ] is generated. If the key is generated at the message source, then it must

also be provided to the destination by means of some secure channel. Alternatively,

a third party could generate the key and securely deliver it to both source and

destination.

With the message X and the encryption key K as input, the encryption algorithm

forms the cipher text Y= [Y1, Y2, ..., YN]. We write this as Y= E(K, X).

This notation indicates that Y is produced by using encryption algorithm E as a

function of the plaintext X, with the specific function determined by the value of the

key K. The intended receiver, in possession of the key, is able to invert the

transformation using decryption algorithm and the secret key.


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We write this as X= D (K, Y). An opponent, observing Y but not having access to

K or X, may attempt to recover X or K or both X and K. It is assumed that the

opponent knows the encryption (E) and decryption (D)

algorithms.

Figure 3.2: Model of Conventional (Classical) Cryptosystem

3.2 Cryptosystems and Cryptanalysis

3.2.1 Cryptosystems

Cryptosystems (Cryptographic systems) are characterized along three independent

dimensions:

1. The type of operations used for transforming plaintext to cipher text: All

encryption algorithms are based on two general principles: substitution, in which each

element in the plaintext (bit, letter, group of bits or letters) is mapped into another

element, and transposition, in which elements in the plaintext are rearranged. The

fundamental requirement is that no information be lost (that is, that all operations are

reversible). Most systems, referred to as product systems, involve multiple stages of

substitutions and transpositions.

2. The number of keys used: If both sender and receiver use the same key, the system is

referred to as symmetric, single-key, secret-key, or conventional encryption. If the


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sender and receiver use different keys, the system is referred to as asymmetric, two-

key, or public-key encryption.

3. The way in which the plaintext is processed: A block cipher processes the input one

block of elements at a time, producing an output block for each input block. A stream

cipher processes the input elements continuously, producing output one element at a

time, as it goes along.

3.2.2 Cryptanalysis

Cryptanalytic attacks rely on the nature of the algorithm plus perhaps some

knowledge of the general characteristics of the plaintext or even some sample plain

text-cipher text pairs. This type of attack exploits the characteristics of the algorithm

to attempt to deduce a specific plaintext or to deduce the key being used.

Table 3.1 summarizes several various types of cryptanalytic attacks, based on the

amount of information known to the cryptanalyst:

Table 3.1: Types of attacks on encrypted messages

Type of Attack Known to Cryptanalyst

Cipher text Only • Encryption algorithm

• Cipher text

Known Plaintext

• Encryption algorithm

• Cipher text

• One or more plaintext–cipher text pairs formed with the secret key (Or the analyst may know that certain plaintext patterns will appear in a

message)

•

Chosen Plaintext


• Cipher text

• Plain text message chosen by cryptanalyst, together with its

corresponding cipher text generated with the secret key (The cryptanalyst can encrypt a large number of suitably chosen

plaintexts and try to use the resulting cipher texts to deduce the

key).


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Chosen Cipher text


• Cipher text

• Cipher text chosen by cryptanalyst, together with its

corresponding decrypted plaintext generated with the secret

key (The cryptanalyst can decrypt several string of symbols,

and tries to use the results to deduce the key)

Chosen Text


• Cipher text

• Plaintext message chosen by cryptanalyst, together with its

corresponding Cipher text generated with the secret key

• Cipher text chosen by cryptanalyst, together with its

corresponding decrypted plaintext generated with the secret

key

There are two other important definitions: An encryption scheme is

unconditionally secure if the cipher text generated by the scheme does not

contain enough information to determine uniquely the corresponding plaintext, no

matter how much cipher text is available. That is, no matter how much time an

opponent has, it is impossible for him or her to decrypt the cipher text, simply

because the required information is not there. With the exception of a scheme known

as the one-time pad (described later), there is no encryption algorithm that is

unconditionally secure.

Therefore, all that the users of an encryption algorithm can seek for is an algorithm

that meets one or both of the following criteria:

• The cost of breaking the cipher exceeds the value of the encrypted

information.

• The time required to break the cipher exceeds the useful lifetime of the

information.

An encryption scheme is said to be computationally secure if either of the

foregoing two criteria are met. All forms of cryptanalysis for symmetric encryption

schemes are designed to exploit the fact that traces of structure or pattern in the

plaintext may survive encryption and be distinguishable in the cipher text. This will

become clear as we examine various symmetric encryption schemes. We will see

that cryptanalysis for public-key schemes proceeds from a fundamentally different


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premise, namely, that the mathematical properties of the pair of keys may make it

possible for one of the two keys to be deduced from the other.

Brute-force attack

The attacker tries every possible key on a piece of cipher text until an

intelligible translation into plaintext is obtained. On average, half of all possible

keys must be tried to achieve success. Table 3.2 shows how much time is involved

for various key spaces. Results are shown for four binary key sizes. The 56-bit key

size is currently in use with the DES (Data Encryption Standard) algorithm, and the

168-bit key size is used for triple DES. The minimum key size specified for AES

(Advanced Encryption Standard) is 128 bits. Results are also shown for what are

called substitution codes that use a 26-character key (discussed later), in which all

possible permutations of the 26 characters serve as keys. For each key size, the

results are shown assuming that it takes 1 micro second to perform a single

decryption, which is a reasonable order of magnitude for today's machines. With the

use of massively parallel organizations of microprocessors, it may be possible to

achieve processing rates that are many orders of magnitude greater. The final

column of Table 3.2 considers the results for a system that can process 1 million

keys per microsecond. As you can see, at this performance level, DES can no longer

be considered computationally secure.

Table 3.2: Average time for brute force attack

Key Size (bits) Number of

Alternative

Keys

Time Required at the rate

of 1 Decryption/μs

Time

Required at

106

Decryptions/μs 32 2

32 = 4.3 x 10

9 2

31ms = 35.8 minutes

2.15 milliseconds

56 256

= 7.2 x 1016

255

ms = 1142 years 10.01 hours

128 2128

= 3.4 x 1038

2127

ms = 5.4 x 1024

years 5.4 x 1018

years

168 2168

= 3.7 x 1050

2167

ms = 5.9 x 1036

years 5.9 x 1030

years

26 characters

(permutation) 26! = 4 x 10

26 2 * 10

26ms = 6.4 x 10

12 years 6.4 x 106

years


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3.4 Substitution Techniques

In this section we examine some classical encryption techniques, based on

substitution. A substitution technique is one where letters of plain text are replaced

by other letters / numbers / symbols. If plain text is a bit pattern, then cipher is

another bit pattern of same length. However, substitutions ciphers are four basic

types :

• Monoalphapetic Substitution.

• Homophonic Substitution.

• Polyalphabetic Substitution.

• Polygram Substitution.

3.3.1 Monoalphapetic Substitution Ciphers

In simple substitution (monoalphabetic) ciphers, each character of the

plaintext is replaced with a corresponding character of ciphertext. A single one-to-

one mapping function (ƒ) from plaintext to ciphertext character is used to encrypt

the entire message using the same key (k); such that :

Ek(M)=f(m1)f(m2)….. f(mN)=C

Where N: is the length of the message.

M: is plaintext message given by M= (m1,m2 ...... mN).

C: is ciphertext message given by C= (c1, c2, ……….cN)

Several forms of f can be used in simple substitution, such as:

• Caesar cipher (Shifted alphabet)

f(mi) = (mi + k) mod n

Where k is the number of positions to be shifted, mi is a single character of the

alphabet, and n is the size of the alphabet.

If k = 3 then we can encrypt the following message as:

M = C O M P U T E R

C=Ek(M)= F R P S X W H U

• Affine :

f(mi)= (mi * k1+k0) mod n where k1 and n are relatively prime in order to

produce a complete set of residues.


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Relatively prime means that the greater common divisor (gcd) between k and n

equal to one (i.e. gcd(k,n)=1).

Simple substitution ciphers does not hide the underlying frequencies of the

different letters of the plaintext, and hence it can be easily broken.

3.3.2 Polyalphabetic Substitution Ciphers

A Polyalphabetic cipher means a sequence of monoalphabetic ciphers, which

are often referred to as its substitution alphabets or just alphabet. In another

meaning; it is made of multiple simple substitutions. The sequence of the

substituting alphabet may have fixed length (d) and is denoted as its period. Given

a period d, cipher alphabet (C1, ..., C2, and fi: A Ci be a mapping from a

plaintext A to its ciphertext C, and M =m1,…,md,md+1,…,m2d,... is enciphered by

repeating the sequence of mapping f1,…,fd every d characters.

Ek(M)=f1(m1),…,fd(md) , f1(md+1),….,fd(m2d)

For d=1 ,the cipher is monoalphabetic.

Several forms of f can be used in polyaphabetic substitution, such as Vigenere

and Beaufort ciphers.

• Vigenere cipher

It is a popular form of periodic substitution ciphers. The key is specified by a

sequence of letters, K= k1,k2,…,kd , then Vigenere cipher system is defined as:

fi (mi) = (mi +kj) mod n for j=1,2, ...., d

Example:

M= GOOD MORNING and Key=AB . '

M: G O O D M O R N I N G

K: A B A B A B A B A B A

C: H Q P F N Q S P J P H

The strength of the Vigenere that cipher letters of same plain text letter is usually

different. Letter frequency information is hidden. However not all knowledge of

plain text is lost. Suppose the attacker knows its Mono alphabetic or Vigenere

cipher. If Mono alphabetic cipher is used statistical properties of characters in


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Letter Homophones

D 17 19 34 4 56 60 67 83

G 08 22 53 65 88 90

O 03 44 76

cipher will break the code. If the opponent discovers it is not a Mono alphabetic,

he knows its Vigenere. If identical patterns in cipher text are discovered, then

length of the keyword is distance between the patterns or a factor of this.

3.3.3 Homophonic Substitution Cipher

Homophonic substitution ciphers maps each character (a) of the plaintext

alphabet into a set of ciphertext elements f(a) called homophone. Beale and

High-order are examples of homophonic ciphers.

• Beale Cipher: A plaintext message M=m1 m2... .... is encrypted as C = c1 c1 ... ...... where

ci is picked at random from the set of homophones f(mi).

Example: M=G O O D

C = 08 03 44 17

Homophonic substitution ciphers are more complicated than simple substitution

ciphers, but still do not obscure all of the statistical properties of the plaintext

language.

3.3.4 Polygram Substitution Cipher

Polygram cipher systems are ciphers in which group of letters are

encrypted together, and includes enciphering large blocks of letters. Therefore,

permits arbitrary substitution for groups of characters. For example the plaintext

group "ABC" could be encrypted to "RTQ", "ABB" could be encrypted to "SLL",

and so on. Examples of such ciphers are Playfair and Hill ciphers.

• PlayFair Cipher:

Playfair cipher is a diagram substitution cipher, the key is given by a 5*5

matrix of 25 letters ( j was not used ), as described in figure 3.3. Each pair of

plaintext letters are encrypted according to the following rules:

1. If m1 and m2 are in the same row, then c1 and c2 are to the right of m1 and m2,


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respectively. The first column is considered to the right of the last column.

2. If m1 and m2 are in the same column, then c1 and c2 are below m1 and m2

respectively. The first row is considered to be below the last row.

3. If m1 and m2 are in different rows and columns, then c1 and c2 are the other two

corners of the rectangle.

4. If m1=m2 a null letter is inserted into the plaintext between m1 and m2 to eliminate

the double.

5. If the plaintext has an odd number of characters, a null letter is appended to the end

of the plaintext.

H

A

R

P

S I C O D B

E M

F N

G Q

K T

L U

V W X Y Z

Figure 3.3 Key for Playfair cipher

Example:

M = CO MP UT ER

Ek(M) = OD TH MU GH

3.5 Transposition Techniques

This kind of mapping is achieved by performing some sort of permutation on

the plaintext letters. This technique is referred to as a transposition cipher. Rail

fence and Row Transposition ciphers are example of such cipher.

3.4.1 Rail fence

is simplest of such cipher, in which the plaintext is written down as a sequence

of diagonals and then read off as a sequence of rows.

Example: Plaintext = meet at the school house

To encipher this message with a rail fence of depth 2, we write the message as

follows:

M E A T E C O L O S

E T T H S H O H U E


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The encrypted message (cipher text) is MEATECOLOSETTHSHOHUE

3.4.2 Row Transposition Cipher

A more complex scheme is to write the message in a rectangle, row by row,

and read the message off, column by column, but permute the order of the columns.

The order of columns then becomes the key of the algorithm.

Example: Plaintext = meet at the school house

Key = 4 3 1 2 5 6 7

PT = M E E T A T T

H E S C H O O

L H O U S E

CT = ESOTCUEEHMHLAHSTOETO

A pure transposition cipher is easily recognized because it has the same letter

frequencies as the original plaintext. The transposition cipher can be made

significantly more secure by performing more than one stage of transposition. The

result is more complex permutation that is not easily reconstructed.

3 Title : Classical Encryption 3.2 Cryptosystems and ... · In this section we examine some classical encryption techniques, based on substitution. A substitution technique is one

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