MAT 1000 Mathematics in Today's World Winter 2015

Last Time We looked at the number of issues a parity-check sums system can detect

Today We take a look at cryptography.

Cryptography Cryptography: The study of methods to make and break secret codes. The process of coding information to prevent unauthorized use is called encryption U

What do we do with Secret messages? Encrypt: Change the letters of the message so that it cannot be read Decrypt: Take an encrypted message and return the letters back to normal

Types of Ciphers Caesar Cipher Decimation Cipher Linear Cipher Vigenere Cipher

Caesar Cipher Used by Julius Caesar to send messages to the troops Shifts the letters of the alphabet by s units (Caesar used a shift of s=3) ABCDEFGHIJKLMNOPQRSTUVWXYZ DEFGHIJKLMNOPQRSTUVWXYZABC

To Encrypt Write the Alphabet down Under each letter write the letter of the alphabet that is s units later in the alphabet. Notice: If s=3, A will have a D below it since D is 3 letters after A. Take your message and change each letter to the associated letter below it (from above)

To Decrypt Use the grid from encryption and replace each letter with the letter above it Notice that A would encrypt as D and D would decrypt as A

Example

Modular Arithmetic Notice that s is just a number Each letter of the alphabet can also be given a number…A:0 B:1 C:2…Z:25 A shift of 3 would take A to D, but 0+3=3 which is the number associated to D. We could add the shift to each of the associated numbers to encrypt a Caesar shift Problem: What do we do with Y? 24+3=27

Modular Arithmetic Before, we just repeated the Alphabet 26 would go to A, 27 to B, 28 to C… Notice that we now have that A is associated to 0 and 26, B to1 and 27, C to 2 and 28 0 and 26 have remainder 0 when dividing by 26 (number of letters in the alphabet) 1 and 27 have remainder 1 when dividing by 26 2 and 28 have remainder 2

Modular Arithmetic Like the days of the week, we see that some things are cyclic To identify numbers, we compare their remainders when dividing by some number n We say that this is working mod n There are 7 days in the week, so we could work mod 7 15, 22, 29, 36 all have the same remainder when dividing by 7 (they have remainder 1) so we say that 15, 22, 29 and 36 are all the same mod 7. Notice that 15, 22, 29, 36 days from now it will be Tuesday…so they really are same The remainder measures how far past an even number of weeks 15, 22, 29, 36 went.

Caesar Shift For the Caesar shift we could work mod 26 Assign to each letter their respective numbers: A:0 F:5 K:10 P:15 U:20 Z:25 B:1 G:6 L:11 Q:16 V:21 C:2 H:7 M:12 R:17 W:22 D:2 I:8 N:13 S:18 X:23 E:4 J:9 O:14 T:19 Y:24

Caesar Shift To encrypt with a shift of s add s to the number associated to each letter and replace with the letter associated to the sum (mod 25). To decrypt: subtract s Notice, with decryption we can get negatives. The remainder must always be positive…To find the actual remainder for a negative number n: let r be the mod 26 representative of |n|, then the actual remainder is 26-r

Examples