An Introduction to Cryptography
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- Jemimah Todd
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1 An Introduction to
2 Terminology is the study of secret writing. This is the only branch of mathematics to be designated by the U.S. government as export-controlled. Cryptographic knowledge is considered to be war materials! While we won t head off into TOP SECRET territory we will have a bit of fun working out how to make (and to break) good secret codes.
3 the enigma Terminology
4 WACs Terminology
5 or Cryptology? Terminology means secret writing
6 or Cryptology? Terminology means secret writing Cryptology means the study of secrets
7 or Cryptology? Terminology means secret writing Cryptology means the study of secrets practically speaking, they are synonyms...
8 Cast of Characters Terminology Alice (the sender) wants to send a message to...
9 Cast of Characters Terminology Alice (the sender) wants to send a message to... Bob (the recipient) but they are afraid that...
10 Cast of Characters Terminology Alice (the sender) wants to send a message to... Bob (the recipient) but they are afraid that... Eve (the eavesdropper) will snoop on them and learn their secrets.
11 Components Terminology plaintext is the message that Alice wants to send.
12 Components Terminology plaintext is the message that Alice wants to send. ciphertext is the scrambled/unreadable thing she actually sends.
13 Components Terminology plaintext is the message that Alice wants to send. ciphertext is the scrambled/unreadable thing she actually sends. encryption is the process of converting plaintext to ciphertext.
14 Components Terminology plaintext is the message that Alice wants to send. ciphertext is the scrambled/unreadable thing she actually sends. encryption is the process of converting plaintext to ciphertext. decryption is the reverse process.
15 Components Terminology plaintext is the message that Alice wants to send. ciphertext is the scrambled/unreadable thing she actually sends. encryption is the process of converting plaintext to ciphertext. decryption is the reverse process. is what Eve has to do in order to break the code.
16 A cryptosystem Terminology The term cryptosystem is used to describe any systematic way to do encryption and decryption of messages.
17 A cryptosystem Terminology The term cryptosystem is used to describe any systematic way to do encryption and decryption of messages. Usually a key must also be chosen (in advance) by Alice and Bob.
18 A cryptosystem Terminology The term cryptosystem is used to describe any systematic way to do encryption and decryption of messages. Usually a key must also be chosen (in advance) by Alice and Bob. If Eve knows the cryptosystem, she can attempt a brute force attack try every possible key...
19 security through obscurity Terminology People used to believe that a really Byzantine cryptosystem for which Eve couldn t even guess what the possible keys could be would allow Alice and Bob to communicate securely.
20 security through obscurity Terminology People used to believe that a really Byzantine cryptosystem for which Eve couldn t even guess what the possible keys could be would allow Alice and Bob to communicate securely. Arguably, this is why Japan and Germany lost World War II.
21 security through obscurity Terminology People used to believe that a really Byzantine cryptosystem for which Eve couldn t even guess what the possible keys could be would allow Alice and Bob to communicate securely. Arguably, this is why Japan and Germany lost World War II. Shannon s Maxim: The enemy knows the system (Claude Shannon ( )
22 cribs Terminology If you have some idea what an encrypted message may be about, this allows you to make a list of cribs. Cribs are words or phrases that may be part of the plaintext.
23 steganography the Caesar shift how can they decode it if they don t even know there is a message?
24 steganography the Caesar shift how can they decode it if they don t even know there is a message? Tatoo a message on a slave s shaved head, then let their hair grow back.
25 steganography the Caesar shift how can they decode it if they don t even know there is a message? Tatoo a message on a slave s shaved head, then let their hair grow back. Yesterday, Oliver used a relatively easy secret message analyzer really terrific!
26 steganography the Caesar shift how can they decode it if they don t even know there is a message? Tatoo a message on a slave s shaved head, then let their hair grow back. Yesterday, Oliver used a relatively easy secret message analyzer really terrific! It has been claimed that the Al Qaeda network hides messages in the low bits of pixels in internet porn.
27 Roman cryptography steganography the Caesar shift Supposedly, Julius Caesar invented a simple cryptosystem: shift each letter in a message 3 units up in the alphabet:
28 Roman cryptography steganography the Caesar shift Supposedly, Julius Caesar invented a simple cryptosystem: shift each letter in a message 3 units up in the alphabet: So, Attack at dawn on Friday would be encrypted as
29 steganography the Caesar shift Roman cryptography Supposedly, Julius Caesar invented a simple cryptosystem: shift each letter in a message 3 units up in the alphabet: So, Attack at dawn on Friday would be encrypted as Dwwdfn dw gdzq rq iulgdb
30 Activity I steganography the Caesar shift Write a creative message (not too long please... ) and encrypt it with the Caesar shift. Trade with another group and decrypt their message.
31 shift ciphers some mathematical preliminaries The Caesar shift is the basis of the simple cryptosystem known as the shift cipher. The key in a shift cipher is the amount of shifting that we will do to encode a message. (For the original Caesar shift the key is k = 3.)
32 its not that many possibilities some mathematical preliminaries There are only 25 possible different amounts of shifting that one can do.
33 its not that many possibilities some mathematical preliminaries There are only 25 possible different amounts of shifting that one can do. We say there are 25 elements in the keyspace
34 its not that many possibilities some mathematical preliminaries There are only 25 possible different amounts of shifting that one can do. We say there are 25 elements in the keyspace It s not really that hard to just try all the possibilities and see if any of them look intelligible.
35 Activity II some mathematical preliminaries Pick a key this should be a relatively small integer. Lets keep things in the range -5 to 5. Write a creative message and encrypt it with the shift cipher using your key. Trade with another group and decrypt their message.
36 too easy some mathematical preliminaries To get a useful cryptosystem we will need to develop a scheme where there are many more keys!
37 clock arithmetic some mathematical preliminaries If it is 9:00 o clock, what time will it be in 5 hours?
38 clock arithmetic some mathematical preliminaries If it is 9:00 o clock, what time will it be in 5 hours? So 9+5 = 2.
39 clock arithmetic some mathematical preliminaries If it is 9:00 o clock, what time will it be in 5 hours? So 9+5 = 2. Hmmmm...
40 clock arithmetic some mathematical preliminaries If it is 9:00 o clock, what time will it be in 5 hours? So 9+5 = 2. Hmmmm... Suppose it s zero o clock. (You can continue to call it 12 if you want, but 0 is really more sensible.)
41 clock arithmetic some mathematical preliminaries If it is 9:00 o clock, what time will it be in 5 hours? So 9+5 = 2. Hmmmm... Suppose it s zero o clock. (You can continue to call it 12 if you want, but 0 is really more sensible.) What time will it be after 5 five hour time periods go by?
42 clock arithmetic some mathematical preliminaries If it is 9:00 o clock, what time will it be in 5 hours? So 9+5 = 2. Hmmmm... Suppose it s zero o clock. (You can continue to call it 12 if you want, but 0 is really more sensible.) What time will it be after 5 five hour time periods go by? So 5 5 = 1.
43 some practice some mathematical preliminaries =
44 some practice some mathematical preliminaries = 1
45 some practice some mathematical preliminaries = =
46 some practice some mathematical preliminaries = = 35
47 some practice some mathematical preliminaries = = 35 = 36 1
48 some practice some mathematical preliminaries = = 35 = 36 1 = 11
49 some practice some mathematical preliminaries = = 35 = 36 1 = =
50 some practice some mathematical preliminaries = = 35 = 36 1 = = 121
51 some practice some mathematical preliminaries = = 35 = 36 1 = = 121 = 1
52 some practice some mathematical preliminaries = = 35 = 36 1 = = 121 = =
53 some practice some mathematical preliminaries = = 35 = 36 1 = = 121 = = 48 = 0
54 clocks on other planets some mathematical preliminaries A clock with 7 hours on it:
55 mod 7 operations some mathematical preliminaries =
56 mod 7 operations some mathematical preliminaries = 2
57 mod 7 operations some mathematical preliminaries = =
58 mod 7 operations some mathematical preliminaries = = 3
59 mod 7 operations some mathematical preliminaries = = =
60 mod 7 operations some mathematical preliminaries = = = 5
61 mod 7 operations some mathematical preliminaries = = = 5 Notice that the zero product property holds, since 7 is prime.
62 some mathematical preliminaries mod 26 operations and the alphabet Each letter of the alphabet can be thought of as a number from 0 to 25. (A=0, B=1, C=2, etc.)
63 some mathematical preliminaries mod 26 operations and the alphabet Each letter of the alphabet can be thought of as a number from 0 to 25. (A=0, B=1, C=2, etc.) The Caesar shift can now be described mathematically: x x + 3 (mod 26) The general shift cipher with key k is: x x + k (mod 26)
64 trouble in paradise some mathematical preliminaries Sadly, arithmetic mod 26 is not so nice. Twenty-six is not prime and the zero product property fails in mod 26.
65 trouble in paradise some mathematical preliminaries Sadly, arithmetic mod 26 is not so nice. Twenty-six is not prime and the zero product property fails in mod 26. The problem we are worried about is whether a given operation can be inverted. Adding (i.e. doing a shift) is always invertible (just shift the other way).
66 trouble in paradise some mathematical preliminaries Sadly, arithmetic mod 26 is not so nice. Twenty-six is not prime and the zero product property fails in mod 26. The problem we are worried about is whether a given operation can be inverted. Adding (i.e. doing a shift) is always invertible (just shift the other way). Multiplying, on the other hand...
67 trouble in paradise some mathematical preliminaries Sadly, arithmetic mod 26 is not so nice. Twenty-six is not prime and the zero product property fails in mod 26. The problem we are worried about is whether a given operation can be inverted. Adding (i.e. doing a shift) is always invertible (just shift the other way). Multiplying, on the other hand... The trouble arises because 26 = 2 13 so if we avoid numbers that have either 2 or 13 as factors life will be good.
68 the affine cipher some mathematical preliminaries In the affine cipher we encrypt using the map x mx + b (mod 26).
69 the affine cipher some mathematical preliminaries In the affine cipher we encrypt using the map x mx + b (mod 26). A key is now a pair of things: m and b.
70 the affine cipher some mathematical preliminaries In the affine cipher we encrypt using the map x mx + b (mod 26). A key is now a pair of things: m and b. The choices for m are limited to odd numbers other than 13 in the range 0 to 25. For b we can use anything in that range.
71 the affine cipher some mathematical preliminaries In the affine cipher we encrypt using the map x mx + b (mod 26). A key is now a pair of things: m and b. The choices for m are limited to odd numbers other than 13 in the range 0 to 25. For b we can use anything in that range. The keyspace contains = 312 elements.
72 the affine cipher some mathematical preliminaries In the affine cipher we encrypt using the map x mx + b (mod 26). A key is now a pair of things: m and b. The choices for m are limited to odd numbers other than 13 in the range 0 to 25. For b we can use anything in that range. The keyspace contains = 312 elements. Suddenly a brute force approach is looking less enticing.
73 Activity III some mathematical preliminaries The following message was encrypted with an affine cipher where the key was m = 3 and b = 2. I also took out spaces and punctuation (as is typical of encrypted ciphertext). What does it say?
74 Activity III some mathematical preliminaries The following message was encrypted with an affine cipher where the key was m = 3 and b = 2. I also took out spaces and punctuation (as is typical of encrypted ciphertext). What does it say? CBOQOXCNAPURKPWOH
75 duoncfzimepghvrxtjsbwlyqak the substitution cipher the Vigenère cipher the one time pad
76 duoncfzimepghvrxtjsbwlyqak the substitution cipher the Vigenère cipher the one time pad You can make a cipher by choosing an essentially random encoding for each letter of the alphabet
77 duoncfzimepghvrxtjsbwlyqak the substitution cipher the Vigenère cipher the one time pad You can make a cipher by choosing an essentially random encoding for each letter of the alphabet There are as many keys as there are possible permutations of 26 things.
78 duoncfzimepghvrxtjsbwlyqak the substitution cipher the Vigenère cipher the one time pad You can make a cipher by choosing an essentially random encoding for each letter of the alphabet There are as many keys as there are possible permutations of 26 things. 26! =
79 duoncfzimepghvrxtjsbwlyqak the substitution cipher the Vigenère cipher the one time pad You can make a cipher by choosing an essentially random encoding for each letter of the alphabet There are as many keys as there are possible permutations of 26 things. 26! = Kind of a lot of those keys would be weak, in the sense that too many letters would be encrypted as themselves. But even if we restrict to only those permutations where every letter gets moved there are plenty of keys.
80 frequency analysis the substitution cipher the Vigenère cipher the one time pad If you have a chunk of ciphertext and one-fifth of the symbols are Q s I ll bet you can guess what letter Q represents.
81 frequency analysis the substitution cipher the Vigenère cipher the one time pad If you have a chunk of ciphertext and one-fifth of the symbols are Q s I ll bet you can guess what letter Q represents. RSTLN and E
82 frequency analysis the substitution cipher the Vigenère cipher the one time pad If you have a chunk of ciphertext and one-fifth of the symbols are Q s I ll bet you can guess what letter Q represents. RSTLN and E With a sufficiently large sample of ciphertext we can use an analysis of the frequency that the symbols occur to guess (accurately) about what the decryptions of certain symbols are.
83 The undecipherable cipher the substitution cipher the Vigenère cipher the one time pad
84 The undecipherable cipher the substitution cipher the Vigenère cipher the one time pad Vigenère actually invented an even better type of cipher, but through a misattribution his name is associated with this scheme, so he s just stuck with it.
85 The undecipherable cipher the substitution cipher the Vigenère cipher the one time pad Vigenère actually invented an even better type of cipher, but through a misattribution his name is associated with this scheme, so he s just stuck with it. In this cipher we return to simply shifting the symbols of our plaintext up in the alphabet, but each symbol is shifted by a different amount.
86 The undecipherable cipher the substitution cipher the Vigenère cipher the one time pad Vigenère actually invented an even better type of cipher, but through a misattribution his name is associated with this scheme, so he s just stuck with it. In this cipher we return to simply shifting the symbols of our plaintext up in the alphabet, but each symbol is shifted by a different amount. Each time we run into an E it will get shifted to some other letter but a different one each time! Frequency analysis will no longer work.
87 the substitution cipher the Vigenère cipher the one time pad lemonlemonlemonlemonlemonlem
88 the substitution cipher the Vigenère cipher the one time pad lemonlemonlemonlemonlemonlem The original implementations of the Vigenère cipher involved using a word or a short phrase as the key.
89 the substitution cipher the Vigenère cipher the one time pad lemonlemonlemonlemonlemonlem The original implementations of the Vigenère cipher involved using a word or a short phrase as the key. The key s letters tell you how much to shift.
90 the substitution cipher the Vigenère cipher the one time pad lemonlemonlemonlemonlemonlem The original implementations of the Vigenère cipher involved using a word or a short phrase as the key. The key s letters tell you how much to shift. The key would be repeated as often as necessary so as to produce shift amounts for all of the letters in the plaintext.
91 really?!? the substitution cipher the Vigenère cipher the one time pad
92 really?!? the substitution cipher the Vigenère cipher the one time pad Cryptanalysts eventually showed how to break Vigenere ciphers.
93 really?!? the substitution cipher the Vigenère cipher the one time pad Cryptanalysts eventually showed how to break Vigenere ciphers. If you can figure out the length of the key you can break a Vigenere cipher into a bunch of parallel shift ciphers
94 really?!? the substitution cipher the Vigenère cipher the one time pad Cryptanalysts eventually showed how to break Vigenere ciphers. If you can figure out the length of the key you can break a Vigenere cipher into a bunch of parallel shift ciphers Each of those is easy to break seperately.
95 provably secure the substitution cipher the Vigenère cipher the one time pad
96 provably secure the substitution cipher the Vigenère cipher the one time pad Even if you make the key for a Vigenere cipher so long that there are never repeats, it is possible to break them with some very high-powered statistical analysis.
97 provably secure the substitution cipher the Vigenère cipher the one time pad Even if you make the key for a Vigenere cipher so long that there are never repeats, it is possible to break them with some very high-powered statistical analysis. However, if we make the key for a Vigenere-type cipher be an arbitrarily long random sequence of letters we will have secure communication.
98 provably secure the substitution cipher the Vigenère cipher the one time pad Even if you make the key for a Vigenere cipher so long that there are never repeats, it is possible to break them with some very high-powered statistical analysis. However, if we make the key for a Vigenere-type cipher be an arbitrarily long random sequence of letters we will have secure communication. This is called a one time pad
99 provably secure the substitution cipher the Vigenère cipher the one time pad Even if you make the key for a Vigenere cipher so long that there are never repeats, it is possible to break them with some very high-powered statistical analysis. However, if we make the key for a Vigenere-type cipher be an arbitrarily long random sequence of letters we will have secure communication. This is called a one time pad There are certain places where you do not want to be caught with an arbitrarily long random sequence of letters about your person.
100 thanks! the substitution cipher the Vigenère cipher the one time pad Thank for coming, I hope you had fun!
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