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Cryptography and Secret Codes

or one reason that linear equations are cool.

Basic Cryptography Code

A B C D E F G H I J K L M N O P Q R S T U VW X Y ZD E F G H I J K L M N O P Q R S T U V W X Y Z A B C

• Romans would use this type of code

• Here “LINEAR” becomes “OLQHDU”

• What would “HTXDWLRQ” be?

Numerical Code

• We can do more if we change letters into numbers:

• So that “FWJH” becomes “6 23 10 8”• Can you come up with another example?

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Numerical Codes

• Or we can even modify this code!

• Here “FWJH” becomes “9 26 13 11”• Writing code like a table is hard though. • Can you think of a simple rule to describe this

code?


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Linear Codes

• The rule we came up with “Change it to a number and add 3” is a linear code

• It is more easily written with the linear equation y=x+3

• To encode “MUSTANGS” first change each letter to a number, and then apply the linear equation

• M 13 16;U 21 24; etc.

Linear codes• How would you write “MUSTANG ON” in the

linear code y=2x+5?• First turn it into numbers: “13 21 19 20 1 14 7

21 14”• Then apply the linear code: “31 47 43 45 7 33

19 47 33”• Now try with the linear code y=-3x+80• How might we undo this?


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Linear Block Codes

• Linear block codes use more than one equation to make a code. How might you do this?

• First take a message “CRYPTOGRAPHY” and break it up into blocks.

• Here we will use block size 2, so CR-YP-TO-GR-AP-HY are the blocks.

Linear Block Codes• You then need as many linear equations as your

block size.• We need 2 equations, so let’s use y=2x+5 and

y=-3x+80• To make a code for CRYPTOGRAPHY, first take

the first block ‘CR’ and encode it.• C 3 2*3+5=11; R 18 -3*18+80=26• So ‘CR’ becomes ‘11 26’. What is the rest of the

code?A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

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A Problem• Now encode “BOOT” using this block system.• It becomes ‘9 35 35 20’• The problem is that ‘O’ is encoded as ‘35’ for

both rules. This is called collision. • You can tell there is a double letter.• To see this, solve the equation 2x+5=-3x+80

for x.


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Review

• Codes: Assign a new symbol to each letter• Numerical Codes: Use numbers as the symbol• Linear Codes: Use linear equations to change

the numbers• Linear Block Codes: Use more than one linear

equation at a time• Collision: When the same letter is changed to

the same number in a linear block code

Now go make your own code!