cryptography a connection between language and mathematics
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Cryptographya connection between language and mathematics
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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
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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
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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
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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
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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
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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
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Transpostition
•Examples:▫Greek Skytale▫Rail Fence Cipher▫Route Transposition Cipher
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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.
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
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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
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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
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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
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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
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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
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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
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THE END!
•Any questions?