1. Cryptography and Secret Codes or one reason that linear equations are cool.

2. Basic Cryptography Code ABCDEFGHIJKLMNOPQRSTUVWXYZ DEFGHIJKLMNOPQRSTUVWXYZABC Romans would use this type of code Here “LINEAR” becomes “OLQHDU” What would “HTXDWLRQ” be?

3. 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? ABCDEFGHIJKLMNOPQRSTUVWXYZ 1234567891011121314151617181920212223242526

4. 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? ABCDEFGHIJKLMNOPQRSTUVWXYZ 4567891011121314151617181920212223242526272829

5. 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.

6. 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? ABCDEFGHIJKLMNOPQRSTUVWXYZ 1234567891011121314151617181920212223242526

7. 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.

8. 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? ABCDEFGHIJKLMNOPQRSTUVWXYZ 1234567891011121314151617181920212223242526

9. ABCDEFGHIJKLMNOPQRSTUVWXYZ 1234567891011121314151617181920212223242526 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. ABCDEFGHIJKLMNOPQRSTUVWXYZ 1234567891011121314151617181920212223242526

10. 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

11. Now go make your own code!