Cryptographya 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

▫ 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

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

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

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

Transpostition

•Examples:▫Greek Skytale▫Rail Fence Cipher▫Route Transposition Cipher

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.

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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

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

X Y Z A B C D E F G H I J K L M N O P Q R S T U V W

)26(mod)( nppE

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

25)(0and,250,1)26,(,251 where

)26(mod)(

pEbaa

bappE

250and,1)26,(,251 where

)26(mod)(

edd

edccD

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:

)26(mod75

43

)26(mod75and)26(mod43where

)(

2

1

2

1

212211

2121

P

P

C

C

PPCPPC

CCPPE

)26(mod321

227

35

471

M

One-time Pad and Polyalphabetic Cipher•One time pad:

•Polyalphabetic Cipher:

)26(mod

,,,,;,,,, 321321

iii

nn

KPC

KKKKKPPPPP

)(mod where)26(mod with replaced be it will

message theofletter th theis if ;,,,, 321

kijKpc

ickkkkk

j

m

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:

)(mod)( nPPEC j

))1)(1((mod1 qpjk

)(mod)( nCCDP k

THE END!

•Any questions?