CSCI 391: Practical Cryptology

1
CSCI 391 Practical Cryptology

• Substitution Monoalphabetic
• Ciphers

2
Julius Caesar Cipher

• The letters of the alphabet are coded as
• A B C D ... Z
• 0 1 2 3 ... 25
• Caesar Cipher
• One of the simplest examples of a substitution or
  shift cipher.
• Entire alphabet is shifted (or rotated) by 3
  letters. The last three letters are shifted to
  the first three letters of the alphabet.
• Used by Julius Caesar to communicate with his
  army
• Caesar is considered to be one of the first
  persons to have ever employed encryption for the
  sake of securing messages

3
Caesar Cipher

• Caesar decided that shifting each letter in the
  message would be his standard algorithm
• Caesar simply replaced each letter in a message
  with the letter that is three places further down
  the alphabet - encryption

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
E F G H I J K L M N O P Q R S T U V W X Y Z A B C
A
D
4
Caesar Cipher

• Ciphertext may be deciphered or decrypted by
  replacing each letter by the third previous
  letter.
• Example
• Plaintext dog
• Ciphertext GRJ
• Example
• Ciphertext BDQD
• Plaintext yana

5
Caesar Cipher

• Remember we think of each letter as
  corresponding to a number from 0 to 25
• To encrypt, we map numbers according to
• C ( P 3 )(mod 26)
• To decrypt, we map numbers according to
• P (C 3) (mod 26)

6
General Shift Cipher

• To encrypt C (P K) (mod 26), K is the KEY
• To decrypt P (C - K) (mod 26), K is the SAME
  KEY
• The sender and receiver of the messages agree in
  advance upon a key a shared secret
• Brute Force Attack the naive but determined
  adversary to start trying every possible
  shifting, and wait to see which message seemed to
  make sense

7
Shift Cipher Attack Example

• RCC FW XRLC ZJ UZMZUVU ZEKF KYIVV GRIKJ
• Shift back by 01 qbb ev wqkb yi tylytut ydje
  jxhuu fqhji (P C 1 )(mod 26)), a is encrypted
  by B, b is encrypted by C, etc.)
• Shift back by 02 paa du vpja xh sxkxsts xcid
  iwgtt epgih (P (C 2) (mod 26)), a is
  encrypted by C, b is encrypted by D, etc)
• .
• Shift back by 17 all of gaul is divided into
  three parts

8
Modular Arithmetic

• Division Principle Definition
• Let m be a positive integer and let b be any
  integer.
• Then there is exactly one pair of integers q and
  r satisfying 0 r lt m, such that b q m r
• q is called quotient, q b/m
• r is called a remainder, r b m
• examples
• 17 35 2 (b 17, m5, q 3, r 2) , 12
  34 0,
• -8 -33 1 (b -8, m 2, q -3, r 1)

9
Modular Arithmetic

• If b is a positive number, the following simple
  rule can be applied
• If r b m, then m r -b m
• Examples
• 17 5 2 and -17 5 5 2 3, (-17 -45
  3)
• 8 3 2 and -8 3 3 2 1 (-8 -33
  1)
• Practice
• -24 5 ?
• -13 2

10
Modular Arithmetic

• Let m be a positive integer (the modulus of our
  arithmetic).
• We say that two integers a and b are congruent
  modulo m if b - a is evenly divisible by m and we
  write a b (mod m).
• Examples
• 3 3 (mod 10), - 6 4 (mod 10)
•  

11
Affine Cipher
Let m be a positive integer (the modulus of our
arithmetic). We say that two integers a and b are
congruent modulo m if b-a is evenly divisible by
m and we write a b (mod m). Examples 3 3
(mod 10), - 6 4 (mod 10)

• To encrypt C (A?P B) (mod 26)
• A and B are KEYS.
• A is relatively prime to 26
• 0 B 25
• To decrypt P A-1 ? (C - B) (mod 26)
• A-1 is multiplicative inverse of A mod 26
• There are 12 choices for A, and 26 for B, giving
  a total of 1226 312 transformations of this
  type.
• Decimation Cipher C A ? P (mod 26) (case B
  0)

12
Multiplicative Inverse
Let m be a positive integer (the modulus of our
arithmetic). We say that two integers a and b are
congruent modulo m if b-a is evenly divisible by
m and we write a b (mod m). Examples 3 3
(mod 10), - 6 4 (mod 10)

• Multiplicative inverse of an integer A modulo M
  is an integer D such that A?D 1(mod M)
• Solution exists if and only if (A, M) 1, means
  A and M are relatively prime.
• We denote multiplicative inverse of A by A-1
• Examples
• 2-1 3 (mod 5)
• 3 is a multiplicative inverse of 2 (mod 5)
• 5-1 21 (mod 26)
• 21 is a multiplicative inverse of 5 (mod 26)