30.1

Chapter 30

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Cryptography

30.2

30-1 INTRODUCTION

Let us introduce the issues involved in cryptography. First, we need to define some terms; then we give some taxonomies.

Definitions�Two Categories

Topics discussed in this section:

30.3

Figure 30.1 Cryptography components

30.4

Figure 30.2 Categories of cryptography

30.5

Figure 30.3 Symmetric-key cryptography

30.6

In symmetric-key cryptography, the same key is used by the sender

(for encryption) �and the receiver (for decryption).

The key is shared.

30.7

Figure 30.4 Asymmetric-key cryptography

30.8

Figure 30.5 Keys used in cryptography

30.9

Figure 30.6 Comparison between two categories of cryptography

30.10

30-2 SYMMETRIC-KEY CRYPTOGRAPHY

Symmetric-key cryptography started thousands of years ago when people needed to exchange secrets (for example, in a war). We still mainly use symmetric-key cryptography in our network security.

Traditional Ciphers�Simple Modern Ciphers�Modern Round Ciphers�Mode of Operation

Topics discussed in this section:

30.11

Figure 30.7 Traditional ciphers

30.12

A substitution cipher replaces one symbol with another.

30.13

The following shows a plaintext and its corresponding ciphertext. Is the cipher monoalphabetic?

Example 30.1

Solution

The cipher is probably monoalphabetic because both occurrences of L’s are encrypted as O’s.

30.14

The following shows a plaintext and its corresponding ciphertext. Is the cipher monoalphabetic?

Example 30.2

Solution

The cipher is not monoalphabetic because each occurrence of L is encrypted by a different character. The first L is encrypted as N; the second as Z.

30.15

The shift cipher is sometimes referred to as the Caesar cipher.

30.16

Use the shift cipher with key = 15 to encrypt the message “HELLO.”

Solution

We encrypt one character at a time. Each character is shifted 15 characters down. Letter H is encrypted to W. Letter E is encrypted to T. The first L is encrypted to A. The second L is also encrypted to A. And O is encrypted to D. The cipher text is WTAAD.

Example 30.3

30.17

Use the shift cipher with key = 15 to decrypt the message “WTAAD.”

Solution

We decrypt one character at a time. Each character is shifted 15 characters up. Letter W is decrypted to H. Letter T is decrypted to E. The first A is decrypted to L. The second A is decrypted to L. And, finally, D is decrypted to O. The plaintext is HELLO.

Example 30.4

30.18

A transposition cipher reorders (permutes) symbols in a block of symbols.

30.19

Figure 30.8 Transposition cipher

30.20

Encrypt the message “HELLO MY DEAR,” using the key shown in Figure 30.8.

Solution

We first remove the spaces in the message. We then divide the text into blocks of four characters. We add a bogus character Z at the end of the third block. The result is HELL OMYD EARZ. We create a three-block ciphertext ELHLMDOYAZER.

Example 30.5

30.21

Using Example 30.5, decrypt the message “ELHLMDOYAZER”.

Solution

The result is HELL OMYD EARZ. After removing the bogus character and combining the characters, we get the original message “HELLO MY DEAR.”

Example 30.6

30.22

Figure 30.9 XOR cipher

30.23

Figure 30.10 Rotation cipher

30.24

Figure 30.11 S-box

30.25

Figure 30.12 P-boxes: straight, expansion, and compression

30.26

Figure 30.13 DES

30.27

Figure 30.14 One round in DES ciphers

30.28

Figure 30.15 DES function

30.29

Figure 30.16 Triple DES

30.30

Table 30.1 AES configuration

30.31

AES has three different configurations with respect to the number of rounds and key size.

30.32

Figure 30.17 AES

30.33

Figure 30.18 Structure of each round

30.34

Figure 30.19 Modes of operation for block ciphers

30.35

Figure 30.20 ECB mode

30.36

Figure 30.21 CBC mode

30.37

Figure 30.22 CFB mode

30.38

Figure 30.23 OFB mode

30.39

30-3 ASYMMETRIC-KEY CRYPTOGRAPHY

An asymmetric-key (or public-key) cipher uses two keys: one private and one public. We discuss two algorithms: RSA and Diffie-Hellman.

RSA�Diffie-Hellman

Topics discussed in this section:

30.40

Figure 30.24 RSA

30.41

In RSA, e and n are announced to the public; d and Φ are kept secret.

30.42

Bob chooses 7 and 11 as p and q and calculates �n = 7 · 11 = 77. The value of Φ = (7 − 1) (11 − 1) or 60. Now he chooses two keys, e and d. If he chooses e to be 13, then d is 37. Now imagine Alice sends the plaintext 5 to Bob. She uses the public key 13 to encrypt 5.

Example 30.7

30.43

Example 30.7 (continued)

Bob receives the ciphertext 26 and uses the private key 37 to decipher the ciphertext:

The plaintext 5 sent by Alice is received as plaintext 5 by Bob.

30.44

Jennifer creates a pair of keys for herself. She chooses �p = 397 and q = 401. She calculates n = 159,197 and �Φ = 396 · 400 = 158,400. She then chooses e = 343 and �d = 12,007. Show how Ted can send a message to Jennifer if he knows e and n.

Example 30.8

30.45

Solution

Suppose Ted wants to send the message “NO” to Jennifer. He changes each character to a number (from 00 to 25) with each character coded as two digits. He then concatenates the two coded characters and gets a four-digit number. The plaintext is 1314. Ted then uses e and n to encrypt the message. The ciphertext is 1314³⁴³ = 33,677 mod 159,197. Jennifer receives the message 33,677 and uses the decryption key d to decipher it as 33,677^12,007 = 1314 mod 159,197. Jennifer then decodes 1314 as the message “NO”. Figure 30.25 shows the process.

Example 30.8 (continuted)

30.46

Figure 30.25 Example 30.8

30.47

Let us give a realistic example. We randomly chose an integer of 512 bits. The integer p is a 159-digit number.

Example 30.9

The integer q is a 160-digit number.

30.48

We calculate n. It has 309 digits:

Example 30.9 (continued)

We calculate Φ. It has 309 digits:

30.49

We choose e = 35,535. We then find d.

Example 30.9 (continued)

Alice wants to send the message “THIS IS A TEST” which can be changed to a numeric value by using the 00–26 encoding scheme (26 is the space character).

30.50

The ciphertext calculated by Alice is C = P^e, which is.

Example 30.9 (continued)

Bob can recover the plaintext from the ciphertext by using P = C^d, which is

The recovered plaintext is THIS IS A TEST after decoding.

30.51

The symmetric (shared) key in the �Diffie-Hellman protocol is

K = g^xy mod p.

30.52

Let us give a trivial example to make the procedure clear. Our example uses small numbers, but note that in a real situation, the numbers are very large. Assume g = 7 and � p = 23. The steps are as follows:

1. Alice chooses x = 3 and calculates R₁ = 7³ mod 23 = 21.

2. Bob chooses y = 6 and calculates R₂ = 7⁶ mod 23 = 4.

3. Alice sends the number 21 to Bob.

4. Bob sends the number 4 to Alice.

5. Alice calculates the symmetric key K = 4³ mod 23 = 18.

6. Bob calculates the symmetric key K = 21⁶ mod 23 = 18.

The value of K is the same for both Alice and Bob; �g^xy mod p = 7¹⁸ mod 23 = 18.

Example 30.10

30.53

Figure 30.27 Diffie-Hellman idea

30.54

Figure 30.28 Man-in-the-middle attack