CSCI 391 Practical Cryptology Substitution Monoalphabetic Ciphers Julius

CSCI 391: Practical Cryptology Substitution Monoalphabetic Ciphers

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

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 A 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 J K L M N O P Q R S T U V WX Y Z A B C

Caesar 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

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)

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

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

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 < m, such that b = q* m + r q is called quotient, q = b/m r is called a remainder, r = b % m examples: 17 = 3*5 + 2 (b = 17, m=5, q = 3, r = 2) , 12 = 3*4 + 0, -8 = -3*3 + 1 (b = -8, m = 2, q = -3, r = 1)

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 = -4*5 + 3) 8 % 3 = 2 and -8 % 3 = 3 – 2 = 1 (-8 = -3*3 + 1) Practice: -24 % 5 = ? -13 % 2 =

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)

we write a ≡ b (mod m). Examples: 3 ≡ 3 (mod 10), - 6 ≡ 4 (mod 10) Affine Cipher 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 12*26 = 312 transformations of this type. Decimation Cipher: C = A P (mod 26) (case B = 0)

we write a ≡ b (mod m). Examples: 3 ≡ 3 (mod 10), - 6 ≡ 4 (mod 10) Multiplicative Inverse 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)