TMAT 103

Supplemental Chapter

Cryptography

Cryptography

• Sending messages that cannot be read if stolen– Been in use for centuries (wars)– Used to transmit data securely over a network– Hundreds of techniques

• Utilizes mathematical functions and operations• Requires a key• Encode – uses corresponding function and key• Decode – uses inverse of the function and key

TMAT 103

Method 1

Sir Francis Bacon’s Code

Sir Francis Bacon’s Code

• Algorithm to encrypt a message using Bacon’s code1. For each letter, find the corresponding binary code

1. Determine the corresponding decimal number (a = 0, b = 1, c = 2, …, z = 25)

2. Convert the decimal number to binary

2. Concatenate the binary codes for all letters3. Create the encoded message

1. The message must have the same number of characters as 0s and 1s in the concatenated binary codes

2. Key: Choose how to distinguish 1s from 0sExample: Use uppercase for 1s and lowercase for 0s

Sir Francis Bacon’s Code

• Encode ‘DISCRETE’ using Bacon’s code

00011 01000 10010 00010 10001 00100 10011 00100OREGO NISAS TATEI NTHEU NITED STATE SOFAM ERICA

Message sent:

oreGOn Is a sTatE in thE uNiteD stAteS of AMerIca

• Example: Encode ‘TECHNOLOGY’

Sir Francis Bacon’s Code

• Reverse the process to decode• Example – decode the following

WilLiAm iS A FaMous auTHOR from tHE SixteeNth cEntuRY

Key: Lowercase stands for 0, uppercase stands for 1

TMAT 103

Method 2

Substitution Code

Substitution Code

• Algorithm to encrypt a message using the Substitution code1. Choose a key (a word with no repeated letters)

Example: DESTINY2. Correlate alphabet characters according to the key

3. Create the encoded message by substituting the corresponding letters

Substitution Code

• Example: Encode ‘DISCRETE’ using a substitution code with the key ‘DESTINY’

• Message sent:

TBPSOIQI

• Example: Encode ‘TECHNOLOGY’ using this scheme

Substitution Code

• Reverse the process to decode• Example: The following message was sent using a

substitution code

Decode the message:

D M W I M L C U L R F I R D E W E E H E L K

Key: BASKET

TMAT 103

Method 3

Transposition Code

Transposition Code

• Permutation Function

• Can also be written as a cycle

1 2 3 4 5 6 7 8 9 10

3 9 7 4 6 1 2 5 8 10

1,3,7,2,9,8,5,6

Transposition Code

• Algorithm to encrypt a message using the Transposition code1. Choose a key (a permutation function)

2. Change the order of the characters in the message according to the key

Transposition Code

• Example: Encode ‘DISCRETE’ using a Transposition code with the key (1, 3, 7, 8, 2, 6)

Message sent:

EEDCRIST

• Example: Encode ‘TECHNOLOGY’

Transposition Code

• Reverse the process to decode• Example – decode the following

OICLPCYIGULSOYRRVTTEA

Key: (16, 15, 6, 1, 3, 7, 11, 2) (8, 13, 4, 5, 19) (14, 10) (21, 20, 17)

TMAT 103

Method 4

Keyword Columnar Transposition

Keyword Columnar Transposition

• Algorithm to encrypt a message using the keyword columnar transposition1. Create a matrix (rows and columns) using the

letters in the key as the column labels

2. Fit the message into the matrix (left to right, top to bottom) – pad with Xs if necessary

3. Form the new message by taking the columns in alphabetical order by the column labels

Keyword Columnar Transposition

• Example: Use a keyword columnar transposition with the key ‘JONES’ to encrypt:

THE FIFTH GOBLET CONTAINS THE GOLD

Result:FGTAHDTFBONGEHETTLHTLNSOIOCIEX

Keyword Columnar Transposition

• Example: Use a keyword columnar transposition with the key ‘JONES’ to encrypt:

ALL DIFFICULT MATHEMATICAL PROBLEMS ARE CHARACTER BUILDING

Keyword Columnar Transposition

• Decoding a keyword columnar transposition – reverse the process– Algorithm to decrypt a message using the keyword

columnar transposition1. Create a matrix (rows and columns) using the letters in

the key as the column labels2. Count the characters in the message, and divide that

number by the number of characters in the key3. Starting from the beginning of the encrypted message,

group the letters according to the calculation in step 24. Fit the message into the matrix5. Read the message from the matrix (left to right, top to

bottom)

Keyword Columnar Transposition

• Reverse the process to decode• Example – decode the following

AAK7ENWHRSOER9SAETOELNNWIDYELXBO1DXKTI3R

Key: BASEBALL