Introduction to Codes, Ciphers, and Cryptography

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Introduction to Codes, Ciphers, and Cryptography

• Michael A. Karls
• Ball State University

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Introduction

• Throughout history, people have had the need to
  send messages to other people in secret!
• Methods have been developed to disguise and break
  secret messages.

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Definitions and Terminology

• A code is a form of secret communication in which
  a word or phrase is replaced with a word, number,
  or symbol.
• For example, here is a simple code for an army.
• Attack at dawn ?Jupiter.
• The coast is clear ? Tippy-toe.
• Each commander and soldier would have a copy of
  the codes in some sort of codebook.

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Definitions and Terminology (cont.)

• A cipher is a form of secret communication in
  which letters are replaced with a letter, number,
  or symbol.
• One example of a cipher, attributed to Julius
  Caesar (1st Century B.C.), is outlined in the
  table below.
• Basically, the cipher alphabet is the plain
  alphabet shifted n spaces left.
• For example, if we take n 14, we get
• Plaintext attack at dawn
• Ciphertext OHHOQY OH ROKB
• Send ciphertext as OHH OQY OHR OKB.
• Recipient would need to know how message was
  created.

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Definitions and Terminology (cont.)

• Cryptography is the science of concealing the
  meaning of a message.
• It is also used to mean the science of anything
  connected with ciphersan alternative to
  cryptology, which is the science of secret
  writing.
• To encrypt a message, one conceals the meaning of
  the message via a code or cipher.
• Similar definitions hold for encode and encipher.
• To decrypt a message, one turns an ecrypted
  message back into the original message.
• Similar definitions hold for decode and decipher.

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Notes on Ciphers

• For any cipher there is an algorithm, which is a
  general encrypting method, and a key which
  specifies the exact details of the algorithm.
• For the Caesar cipher,
• Algorithmshift the alphabet.
• Keyhow many places to shift!
• Thus, there are 26 keys for this cipher.

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Notes on Ciphers (cont.)

• The key must remain secure!
• If the key is found, the code will be broken.
• If the algorithm is known, the code can still be
  secure!
• For a code or cipher, the greater the number of
  keys, the greater the security!

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Notes on Ciphers (cont.)

• The Caesar cipher is an example of a substitution
  cipher.
• This cipher is symmetric because sender and
  receiver must both know (have) the key.

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Ciphers via Modular Arithmetic

• Using modular arithmetic, we can make ciphers!
• For any non-negative integers a and n, we define
  a mod n to be the remainder when a is divided by
  n.
• For example,
• 18 mod 5 3, since 18 3 x 5 3
• 4 mod 7 4, since 4 0 x 7 4
• 28 mod 26 2, since 28 1 x 26 2
• 26 mod 13 0, since 26 2 x 13 0

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Ciphers via Modular Arithmetic (cont.)

• We can add and multiply numbers mod n too!
• For example, here is how to find (2715) mod 26
  and (27 x 15) mod 26.
• 2715 42 and 42 mod 26 16,
• so (2715) mod 26 16
• or
• 27 mod 26 1 and 15 mod 26 15,
• so (2715) mod 26 (115) mod 26 16 mod 26
  16
• 27 x 15 405 and 405 mod 26 15,
• so (27 x 15) mod 26 15
• or
• (27 x 15) mod 26 (1 x 15) mod 26 15 mod 26
  15

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Ciphers via Modular Arithmetic (cont.)

• To find (ab) mod n
• Add a to b, then find the resulting sum mod n,
• Or find a mod n, find b mod n, and add the
  results mod n.
• Multiplication works in a similar fashion!
• Now we are ready to make an additive cipher!

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Ciphers via Modular Arithmetic (cont.)

• First, assign 1, 2, , 26 mod 26 to a, b, , z.
• Next, choose a fixed integer m between 0 and 25.
• To get the ciphertext y from plaintext x, add m
  mod 26, i.e., y (xm) mod 26.
• As an example, here is additive cipher alphabet
  for m14.
• Notice that this is just the Caesar cipher we saw
  before!

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The RSA Encryption Scheme

• In 1977, MIT mathematicians Rivest, Shamir, and
  Adleman announced their invention of a public key
  cryptography scheme.
• In order to understand this scheme, commonly
  known as RSA, we need some definitions!

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Definition of Divisor

• Let a and b be integers, with b ? 0. We say that
  b divides a or b is a divisor of a if a bc for
  some integer c.
• Notation ba
• Example 1
• 324 since 24 3 x 8.
• Divisors of 12 are -12, -6, -4, -3, -2, -1,
  1, 2, 3, 4, 6, 12

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Definition of Greatest Common Divisor (GCD)

• Let a and b be integers, not both zero. The
  greatest common divisor (GCD) of a and b is the
  largest integer that divides both a and b.
• Notation (a,b)
• Example 2
• Divisors of 6 -6, -3, -2, -1, 1, 2, 3, 6
• Divisors of 8 -8, -4, -2, -1, 1, 2, 4, 8
• Thus, (6,8) 2
• Since the divisors of 7 are -7, -1, 1, 7,
  (7,8) 1.

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Definition of Relatively Prime

• Two integers whose GCD is 1 are said to be
  relatively prime.
• Example 3
• Since (7,8) 1, 7 and 8 are relatively prime.

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Definition of Prime Number

• A positive integer p is said to be prime if pgt1
  and the only positive divisors of p are 1 and p.
• Example 4
• 2, 3, and 7 are prime.
• 6, 8, 10, 100 are not prime (composite).

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RSA Scheme (with Alice and Bob!)

• Step 1
• Alice chooses two huge prime numbers p and q.
• Note Alice keeps p and q secret!

• Example 5
• p 47 and q 59.

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RSA Scheme (with Alice and Bob!) (cont.)

• Step 2
• Alice computes
• N p x q.
• Then she computes
• k (p-1)(q-1).
• Finally, she chooses an integer e such that
• 1lteltN and (e,k) 1.

• Example 5 (cont.)
• N 47 x 59 2773.
• k 46 x 58 2668.
• e 17.
• Choice of e is o.k, since 1lt17lt2773 and
  (17,2668) 1.

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RSA Scheme (with Alice and Bob!) (cont.)

• Step 3
• Alice computes
• d e-1 mod k.
• Alice publishes her public key N, e.
• Alice keeps secret her private key p, q, d, k.

• Example 5 (cont.)
• d 17-1 mod 2668 157.
• Alices public key
• N 2773 e 17.
• Alices private key
• p 47 q 59 d 157 k 2668.

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RSA Scheme (with Alice and Bob!) (cont.)

• Step 4
• Suppose Bob wants to send a message to Alice.
• To do so, he looks up Alices public key,
  converts the message into numbers MltN.

• Example 5 (cont.)
• Plaintext is HELLO
• HELLO ? HE LL O_
• Assign 00?space 01?A 02?B, , 26?Z (or use
  ASCII).

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RSA Scheme (with Alice and Bob!) (cont.)

• Step 4 (cont.)
• Next, for each plaintext number M, Bob computes
  
• C Me mod N (1)
• to get ciphertext number C.

• Example 5 (cont.)
• 080517 mod 2773 542.
• 121217 mod 2773 2345.
• 150017 mod 2773 2417.
• Encrypted message is 0542 2345 2417.

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RSA Scheme (with Alice and Bob!) (cont.)

• Step 5
• Bob emails Alice the encrypted message.
• To decrypt, Alice uses her private key and
  computes
• M Cd mod N (2)

• Example 5 (cont.)
• 0542157 mod 2773 805.
• 2345157 mod 2773 1212.
• 2417157 mod 2773 1500.
• Decrypted message is HE LL O_.

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References

• The Code Book by Simon Singh