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 MN O P Q R S T U VWX Y Z
D E F G H I J K L MN O P Q R S T U VWX 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:
A B C D E F G H I J K L MN O P Q R S T U V W X Y Z
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
• So that “FWJH” becomes “6 23 10 8”
• Can you come up with another example?
Numerical Codes
• Or we can even modify this code!
A B C D E F G H I J K L MN O P Q R S T U VW X Y Z
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
• 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?
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?
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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
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-GRAP-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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
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.
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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
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!