a connection between language and mathematics Cryptography a connection between language and mathematics

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

Brief History First hint of cryptography Julius Caesar (100-44 B.C.) Egyptian (1900 B.C.) funeral incriptions Julius Caesar (100-44 B.C.) First military use of code? or was it the Greeks with the skytale. Francois Viete (1540-1603) Deciphered Spanish Code of more than 400 characters Mary, Queen of Scots (beheaded in 1587) Plotted to overthrow Queen Elizabeth I John Wallis (1616-1703) 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

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”

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

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

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.

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

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

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

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)

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

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)

One-time Pad and Polyalphabetic Cipher

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:

THE END! Any questions?