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Cryptography

Cryptography. a connection between language and mathematics. Introduction. Cryptography : the procedures, processes, methods, etc., of making and using secret writing, as codes or ciphers

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Cryptography

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  1. 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 • 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

  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” • Substitution Cipher

  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

  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

  8. Transpostition • Examples: • Greek Skytale • Rail Fence Cipher • Route Transposition Cipher

  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.

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

  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

  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:

  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:

  14. One-time Pad and Polyalphabetic Cipher • One time pad: • 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?

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