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Here are a few basic rules of thumb to consider when doing transformations They can be illustrated considering the square region shown in Fig 96 In that section, we saw how to shift the position of the square by adding a constant, that is, we wrote down a linear transformation of the form w = z + a Let s review the other types of transformations that are possible A transformation of the type w = az will expand the region if a > 1 and will shrink the region if a < 1 Consider the rst case with w = 2 z This expands the square region from 0 x 1, 0 y 1 to 0 x 2, 0 y 2 This is illustrated in Fig 917

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If you want to take some data and make figuring out what it means difficult for other people, you need a cipher A cipher is a series of complex and hard to reverse mathematics called an algorithm you run on a string of ones and zeroes to make a new set of seemingly meaningless ones and zeroes A cipher and the methods used to implement that cipher is commonly called the complete algorithm (I know that s a mouthful of new terms check the note for details) NOTE The terms cipher, algorithm, and complete algorithm lend themselves to a lot of confusion, especially because most people in the IT industry use them interchangeably Here s the scoop A cipher is a general term for a way to encrypt data The algorithm is the mathematical formula that underlies the cipher The complete algorithm is both the cipher and the implementation of that cipher The problem with the terms is compounded by the lack of a third, distinct term Most people drop the word complete from complete algorithm, for example, thus the meanings of the three terms become muddied Let s say we have a string of ones and zeroes that looks like this:

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14:01:22" "11/20/1999 14:00:04" "11/20/1999 14:00:32"

B 1

01001101010010010100101101000101

Services" "Telnet" "Oracle" "IP" "IP"

This string may not mean much to you, but if it was part of an HTTP packet, your Web browser would instantly know that this is Unicode that is, numbers representing letters and other characters and convert it into text:

D 1

Server41" "[132189304]" "[1321435783]"

So let s create a cipher to encrypt this cleartext All binary encryption requires some interesting binary math You could do something really simple such as add 1 to every value (and ignore carrying the 1): 0+1 = 1 and 1+1 = 0

The transformation w = sin( z / a) maps the region in Fig 915 to the region in Fig 916 with corresponding points indicated

10110010101101101011010010111010

#Pkts (A->B)

No big deal; that just reversed the values Any decent hacker would see the pattern and break this code in about three seconds Let s try something harder to break by bringing in a second value (a key) of any eight binary numbers (let s use 10101010 for this example) and doing some math to every eight binary values using this algorithm:

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Figure 917 For review, a transformation of the form w = az expands the region of interest if | a | >1

#Bytes(A->B)

0 0 1 1

#Pkts(B->A)

Now suppose that w = (1/ 2) z This shrinks the square, as shown in Fig 918 To rotate the region by an angle , we use a transformation of the form w = ei z (94)

0 1 0 1

#Bytes(B->A)

0 1 1 0

For our square, this rotates the square by in the counterclockwise direction assuming that > 0 This is illustrated in Fig 919

"Novell_ Server34" 922 "[132157 18224]" 95 "Novell_ Server34" 88 "[13215 244152]" 29 "[13240 21561]" 675

This is known as a binary XOR (eXclusive OR) Line up the key against the first eight values in the cleartext:

In this section, we consider a transformation of the type: Tz = az + b cz + d ad bc 0 (95)

10101010 01001101010010010100101101000101 11100111

79,377 38,130 6,394 9,354 337,690

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