Functions, Relations, and M atrices
Mathematical Structures for Computer Science
Chapter 4
Copyright 2006 W.H. Freeman & Co.
MSCS Slides
The Mighty Mod Function
The Mighty Mod Function
q
q
We define the modulo n function as follows: If x = qn + r, 0 r n, then x mod n = r. In other words, x mod n is the nonnegative remainder (also called the residue of x modulo n) when x is divided by the positive integer n.
Section 4.5
The Mighty Mod Function
2
Hashing
q
q
q
q
A hash function h: S T, where the domain S is a set of text strings or integer values and the codomain T is the set of integers {0, 1, ... , t - 1}, where t is some relatively small positive integer. If the domain S consists of text strings, we can imagine them encoded in some way into integer values. For example, an algorithm as simple as converting each individual letter of a text string into its position in the alphabet and adding up the resulting list. The function h therefore maps a potentially large set of values S into a relatively small window of integer values T. Consequently h isn't likely to be a one-toone function.
The Mighty Mod Function 3
Section 4.5
Hashing
q q
q
q
A hash function is often used as part of a search algorithm. In a search using a hash function, n elements are stored in an array (a one-dimensional table) called a hash table, where the array is indexed from 0 through t - 1; the table is of size t. The element x is passed as the argument to the hash function, and the resulting h(x) value gives the array index at which the element is then stored. Later, when a search is carried out, the target value is run through the same hash function, giving an index location in the hash table, which is where to look for the matching stored element.
Section 4.5
The Mighty Mod Function
4
Hashing
q
q
q
However, because the hash function is not one-to-one, things are not quite that simple. Different values may hash to the same array index, producing a collision. Here are several collision resolution algorithms available. One is called linear probing.
s
Keep going in the array and store element x in the next available empty slot.
q
Another method, called chaining, builds a linked list for each array index.
Section 4.5
The Mighty Mod Function
5
Hashing
q
Following are the desirable properties of a hash function:
1.
2.
Given an argument value x, h(x) can be computed quickly. The number of collisions will be reduced because h(x) does a good job of distributing values throughout the hash table.
q
q
Use of a modulo function as the hash function accomplishes goal 1. Goal 2 is harder to achieve, but distribution seems to work better on the average if the table size (the modulo value) is a prime number.
Section 4.5
The Mighty Mod Function
6
Hashing
q
q
q
q
The average number of comparisons required to search for an element using hashing depends on the ratio of n to the total table size t. If this ratio is low, then (using linear probing) there are lots of empty slots, so you won't have to look very far to find a place to insert a new element into the table. If this ratio is low and chaining is used, the average length of any linked list you may have to (sequentially) search for a target element should be short. This ratio n/t is called the load factor of the hash table.
Section 4.5
The Mighty Mod Function
7
Computer Security
q
q
q
q
The mod function plays a part in many aspects of security. Military information, financial information, and company proprietary information that must be transmitted securely uses some encoding/decoding scheme. The original information (called the plaintext) is encrypted using an encryption key, resulting in coded text called the ciphertext. The ciphertext is transmitted, and when it is received, it can be decoded using the corresponding decryption key. Encryption and decryption are inverse functions in the sense that:
s
decryption(encryption(plaintext)) = plaintext
The Mighty Mod Function 8
Section 4.5
Computer Security
q
q
q
Cryptography is the study of various encryption/decryption schemes. Military use of cryptographic techniques can be traced back to Julius Caesar, who sent messages to his generals in the field using a scheme now known as the Caesar cipher. Let us assume that plaintext messages only use the 26 capital letters of the alphabet, that spaces between words are suppressed, and that each letter is first mapped to its corresponding position in the alphabet.
Section 4.5
The Mighty Mod Function
9
Computer Security
q
q
q
q
We'll denote this mapping as the bijection g: {A, ... , Z} {0, ... , 25}. Then a positive integer key value k is chosen that shifts each number k positions to the right with a "wrap-around" back to the beginning if needed (this is the mod function). Finally, the function g-1 is applied to translate the resulting number back into a letter. The encoding function is given by:
s
f(p) = g-1([g(p) + k] mod 26) f -1(c) = g-1([g(c) - k] mod 26)
q
The decoding function is:
s
Section 4.5
The Mighty Mod Function
10
Computer Security
q
q
q
The Caesar cipher is a simple substitution cipher, meaning that each plaintext character is coded consistently into the same single ciphertext character. Encryption techniques where a single plaintext character contributes to several ciphertext characters introduce diffusion. The advantage to diffusion is that it hides the frequency statistics of individual letters, making analysis of an intercepted ciphertext message much more difficult. DES (Data Encryption Standard) is an internationally standard encryption algorithm developed in 1976. DES was developed to safeguard the security of digital information, so we may consider the plaintext to be a string of bits.
The Mighty Mod Function 11
Section 4.5
Computer Security
q q
q q
q
DES is a block cipher. A block of 64 plaintext bits is encoded as a unit using a 56-bit key. This results in a block of 64 ciphertext bits. Changing one bit in the plaintext or one bit in the key changes about half the resulting 64 ciphertext bits, so DES exhibits high diffusion. Because the DES algorithm is well known, the only "secret" part is the 56-bit key that is used.
Section 4.5
The Mighty Mod Function
12
Computer Security
q
q q
q
AES (Advanced Encryption Standard) is also a block encryption scheme, but it uses a key length of 128 bits or more. AES also uses a form of the Euclidean algorithm. Disadvantage of both DES and AES is that they are symmetric encryption (also called private key encryption) schemes. The same key is used to both encode and decode the message. In a private key encryption scheme, both the sender and receiver must know the key. The problem of securely transmitting a message turns into the problem of securely transmitting the key to be used for the encryption and decryption.
Section 4.5
The Migh...