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Introduction to Codes, Ciphers, and Cryptography

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Title: Introduction to Codes, Ciphers, and Cryptography


1
Introduction to Codes, Ciphers, and Cryptography
  • Michael A. Karls
  • Ball State University

2
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.

3
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.

4
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.

5
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.

6
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.

7
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!

8
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.

9
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

10
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

11
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!

12
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!

13
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!

14
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

15
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.

16
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.

17
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).

18
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.

19
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.

20
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.

21
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).

22
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.

23
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_.

24
References
  • The Code Book by Simon Singh
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