1. a connection between language and mathematics
Cryptography
a connection between language and mathematics

2. Introduction
Cryptography: the procedures, processes, methods, etc., of making and using secret writing, as codes or ciphers
crypto-: “hidden” or “secret”; -graphy: a process or form of drawing, writing, representing, recording, describing
Cryptanalysis: the procedures, processes, methods, etc., used to translate or interpret secret writings, as codes and ciphers, for which the key is unknown
Cryptology: the science that includes cryptography and cryptanalysis

3. Brief History First hint of cryptography Julius Caesar (100-44 B.C.)
Egyptian (1900 B.C.) funeral incriptions
Julius Caesar ( B.C.)
First military use of code? or was it the Greeks with the skytale.
Francois Viete ( )
Deciphered Spanish Code of more than 400 characters
Mary, Queen of Scots (beheaded in 1587)
Plotted to overthrow Queen Elizabeth I
John Wallis ( )
Deciphered code during English Civil War
World War I
British cryptologists deciphered the Zimmermann Telegram in 1917
World War II
Cryptanalysis allows numerically inferior Amercian navy to defeat the Japanese at the Battle of the Ccoral Sea and in the Battle of Midway Island
Zimmerman – alliance between germany and mexico against america

4. Side Note on Literature
Sir Arthur Conan Doyle – Sherlock Holmes
“The Adventure of the Gloria Scott
Null Cipher
“The Adventure of the Dancing Men”
Substitution Cipher
Edgar Allen Poe
“The Gold Bug”

5. Some vocabulary Enciphering: the process of encoding a message
Deciphering: the process of decoding a message
Literal plain text: original message
Numerical plain text: numerical equivalent of the literal plaintext
Literal cipher text: encoded message in literal form
Numerical plain text: encoded message in numerical form

6. A word about steganography…
The practice of hiding messages, so that the presence of the message itself is hidden, often by writing them in places where they may not be found.
stegano-: “covered” or “protected”
Examples:
Histaiaeus, a Greek general, would tattoo his servants’ shaved heads
Romans would sew a message in the sole of a sandal
Null Cipher
Cardano Grille
Girolamo cardono 1556

7. Two basic transformations
Transposition: letters of the plain text are jumbled or disarranged
Generally considered harder to break
For example, take the phrase “Math history is super fun” which has 21 letters. That means there are 21! ways to rearrange the letters.
Substitution: letters of the plain text are substituted by other letters, numbers, or symbols.
Generally considered easier to use
Transposition: letters retain their identities but lose their position; Substitution: letters retain their position but lose there identities.

8. Transpostition Examples: Greek Skytale Rail Fence Cipher
Route Transposition Cipher
Rail fence: civil war resembles a rail fence

9. Code or cipher? In general, “code” is distinguished from “cipher”
A code consists of thousands of words, phrases, letters, and syllables with codewords or codenumbers that replace plain text.
A cipher uses the basic unit length of one letter, sometimes a letter pair, but rarely larger groups of letters.
Ciphers operate on plaintext units of regular length; codes operate on plaintext units of variable length 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 00 o1 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

10. Transition to Math... Caeser Cipher Rot-n Cipher
Shifting the alphabet 3 places
Rot-n Cipher
“rot” for “rotation”
Let p (plaintext) be a unit of numerical plain text 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

11. Linear Cipher
Let p be a two digit unit of numerical plain text, we can encipher using the key:
The inverse transform of E(p) is the decryption key, where c is a two digit unit of numerical cipher text:
Since there are 12 possible values of d and 26 possible values of e, there is 12*26=312 possible decryption keys
Example: S=18, 7x+3, 7(18)+3=129==25 (mod 26)
y=7x+3, y-3=7x … -3 (mod 26)=23
y+23=7x, (y+23)/7=x; 7*15 mod(26)=1 “Extended Euclidean Algorithm”
15(y+23)=15y+345; 345 (mod 26)=7
D(x)=15x+7
15(25)+7=382==18 (mod 26)

12. A word about Cryptanalysis…
Exhaustive cryptanalysis: trying all possible decryption keys until the right one is found.
Consider a character cipher consisting of a permutation of the alphabet. There would be 26! possible decryption keys.
Frequency analysis: comparing the frequency of characters in a cipher to the relative frequency of letters used in the English language.
Letters of the English language in order of relative frequency: E T A O I N S R H D L C U M F P G W Y B V K X J Q Z

13. Block or Matrix Ciphers
A diagraph, or two character block cipher, might be encoded using the following encryption key:
Designate M as the encryption matrix, then we need M-1 (mod 26) for decryption:
Encription: C==MP (mod 26); Decryption: P==M^-1*C (mod 26)

14. One-time Pad and Polyalphabetic Cipher

15. Public-Key Encryption
Allows the encryption key to be public.
Relies on the computational infeasibility of factoring large numbers, which keeps the decryption key secret.
Let n=pq, where p and q are prime numbers. Let j be an integer such that 2<j<(p-1)(q-1) and (j, (p- 1)(q-1))=1.
Encryption key:
Let k be the multiplicative inverse of j (mod (p-1)(q- 1)), that is
Decryption key:

16. THE END!
Any questions?