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Bits, binary numbers, the XOR cipher, perfect security and one-time pads. For MAT 140: Introduction to the Mathematical Sciences, at Franklin College.
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CryptologyDay 3: First look at computer ciphers
MAT 140: Introduction to the Mathematical Sciences19 September 2008
Robert Talbert, PhDAssociate Professor of Mathematics and Computing Science rtalbert@franklincollege.edu
Recap of Day 2
• Integer congruence modulo n
•Using integer congruence to make and break shift ciphers
• Idea: Representing text characters as numbers and then using math to manipulate them
Unicode
Worldwide standard for representing characters from the keyboard and various languages
in numerical form
Bits and bytes
Computers store and process information in binary states,
represented by 0 (off) or 1 (on).
Bit = Binary Digit = 0 or 1.Byte = 8 bits, e.g. 01101110
Every positive integer can be represented as a string of bits...
Decimal representation
Base 10 or “decimal” representation
461 = 400 + 30 + 1= 4! 102 + 3! 101 + 1! 100
Binary representation
461 = 256 + 128 + 64 + 8 + 4 + 1= 1! 28 + 1! 27 + 1! 26 + 0! 25 + 0! 24
+1! 23 + 1! 22 + 0! 21 + 1! 20
Binary representation
461 = 256 + 128 + 64 + 8 + 4 + 1= 1! 28 + 1! 27 + 1! 26 + 0! 25 + 0! 24
+1! 23 + 1! 22 + 0! 21 + 1! 20
Binary representation
461 = 256 + 128 + 64 + 8 + 4 + 1= 1! 28 + 1! 27 + 1! 26 + 0! 25 + 0! 24
+1! 23 + 1! 22 + 0! 21 + 1! 20
461 in base 10 = 111001101 in base 2Binary representation
Binary representation
461 = 256 + 128 + 64 + 8 + 4 + 1= 1! 28 + 1! 27 + 1! 26 + 0! 25 + 0! 24
+1! 23 + 1! 22 + 0! 21 + 1! 20
461 in base 10 = 111001101 in base 2Binary representation
The number 461 is a 9-bit integer.Number of bits in an integer n:
!log2 n" + 1
Binary representation of text
B
Binary representation of text
B 66Unicode
Binary representation of text
B 66Unicode
1000010Convert decimal
to binary
Binary representation of text
B 66Unicode
1000010Convert decimal
to binary01000010
Pad with extra 0
Binary representation of text
B 66Unicode
1000010Convert decimal
to binary01000010
Pad with extra 0
8-bit ASCII representation of “B”
Adding bits and bitstrings
“XOR”
Adding bits and bitstrings
0 ⊕ 0 = 0
“XOR”
Adding bits and bitstrings
0 ⊕ 0 = 0
1 ⊕ 0 = 1
“XOR”
Adding bits and bitstrings
0 ⊕ 0 = 0
1 ⊕ 0 = 1
0 ⊕ 1 = 1
“XOR”
Adding bits and bitstrings
0 ⊕ 0 = 0
1 ⊕ 0 = 1
0 ⊕ 1 = 1
1 ⊕ 1 = 0
“XOR”
Adding bits and bitstrings
0 ⊕ 0 = 0
1 ⊕ 0 = 1
0 ⊕ 1 = 1
1 ⊕ 1 = 0
“XOR”
110011 ⊕ 101010 110011⊕ 101010
011001
Any bit XOR-ed to itself = 0
Simple XOR cipher
[Choose bitstring for key]
11011010
11011010
Simple XOR cipher
[Choose bitstring for key]
11011010
11011010
DOG
Simple XOR cipher
[Choose bitstring for key]
11011010
11011010
DOG 01000100 01001111 01000111
Simple XOR cipher
[Choose bitstring for key]
11011010
11011010
DOG 01000100 01001111 01000111 ⊕11011010 11011010 11011010
Simple XOR cipher
[Choose bitstring for key]
11011010
11011010
DOG 01000100 01001111 01000111 ⊕11011010 11011010 11011010
10011110 10010101 10011101
Simple XOR cipher
[Choose bitstring for key]
11011010
11011010
DOG 01000100 01001111 01000111 ⊕11011010 11011010 11011010
10011110 10010101 10011101
쇦뷵쇝10011110 10010101 10011101
Simple XOR cipher
[Choose bitstring for key]
11011010
11011010
DOG 01000100 01001111 01000111 ⊕11011010 11011010 11011010
10011110 10010101 10011101
쇦뷵쇝10011110 10010101 10011101
⊕11011010 11011010 11011010
Simple XOR cipher
[Choose bitstring for key]
11011010
11011010
DOG 01000100 01001111 01000111 ⊕11011010 11011010 11011010
10011110 10010101 10011101
01000100 01001111 01000111
쇦뷵쇝10011110 10010101 10011101
⊕11011010 11011010 11011010
Simple XOR cipher
[Choose bitstring for key]
11011010
11011010
DOG 01000100 01001111 01000111 ⊕11011010 11011010 11011010
10011110 10010101 10011101
01000100 01001111 01000111 DOG
쇦뷵쇝10011110 10010101 10011101
⊕11011010 11011010 11011010
Activity: Experiments with the XOR cipher
When key length ≥ message length
Plaintext ⊕ key = ciphertextPlaintext ⊕ key ⊕ Plaintext = ciphertext ⊕ Plaintext
key = ciphertext ⊕ plaintext
When key length ≥ message length
11001100 01010011 11111111Ciphertext
Plaintext ⊕ key = ciphertextPlaintext ⊕ key ⊕ Plaintext = ciphertext ⊕ Plaintext
key = ciphertext ⊕ plaintext
When key length ≥ message length
11001100 01010011 11111111Ciphertext
Plaintext ⊕ key = ciphertextPlaintext ⊕ key ⊕ Plaintext = ciphertext ⊕ Plaintext
key = ciphertext ⊕ plaintext
01000011 01000001 01010100Plaintext (CAT)
When key length ≥ message length
11001100 01010011 11111111Ciphertext
Plaintext ⊕ key = ciphertextPlaintext ⊕ key ⊕ Plaintext = ciphertext ⊕ Plaintext
key = ciphertext ⊕ plaintext
10001111 00010010 1010101101000011 01000001 01010100Plaintext (CAT)
When key length ≥ message length
11001100 01010011 11111111Ciphertext
Plaintext ⊕ key = ciphertextPlaintext ⊕ key ⊕ Plaintext = ciphertext ⊕ Plaintext
key = ciphertext ⊕ plaintext
10001111 00010010 10101011
11001100 01010011 11111111Ciphertext
01000011 01000001 01010100Plaintext (CAT)
When key length ≥ message length
11001100 01010011 11111111Ciphertext
Plaintext ⊕ key = ciphertextPlaintext ⊕ key ⊕ Plaintext = ciphertext ⊕ Plaintext
key = ciphertext ⊕ plaintext
10001111 00010010 10101011
11001100 01010011 11111111Ciphertext
01000011 01000001 01010100Plaintext (CAT)
01000100 01001111 01000111Plaintext (DOG)
When key length ≥ message length
11001100 01010011 11111111Ciphertext
Plaintext ⊕ key = ciphertextPlaintext ⊕ key ⊕ Plaintext = ciphertext ⊕ Plaintext
key = ciphertext ⊕ plaintext
10001111 00010010 10101011
11001100 01010011 11111111Ciphertext
10001000 00011100 10111000
01000011 01000001 01010100Plaintext (CAT)
01000100 01001111 01000111Plaintext (DOG)
One-time pads and perfect security
When key length ≥ message length in the XOR cipher: No information about the real plaintext is given by the ciphertext.
= Perfect security
One-time pads and perfect security
When key length ≥ message length in the XOR cipher: No information about the real plaintext is given by the ciphertext.
= Perfect security
Given a plaintext ASCII message: Alice and Bob choose a random
string of bits at least as long as the message.
Encrypt using XOR. Burn the key.
= One-time pad
Next time
•The key distribution problem
•Public-key cryptography and RSA
•Wrapping up: Current cutting edge of cryptology; careers in cryptology and information security; cryptology at Franklin College
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