Classical Cryptography

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Classical Cryptography

• 1. Introduction Some Simple Cryptosystems

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Outline

• 1 Introduction Some Simple Cryptosystems
• lt1gt The Shift Cipher
• lt2gt The Substitution Cipher
• lt3gt The Affine Cipher
• lt4gt The Vigenère Cipher
• lt5gt The Hill Cipher
• lt6gt The Permutation Cipher
• lt7gt Stream Ciphers
• 2 Cryptanalysis
• lt1gt Cryptanalysis of the Affine Cipher
• lt2gt Cryptanalysis of the Substitution Cipher
• lt3gt Cryptanalysis of the Vigenère Cipher
• lt4gt Cryptanalysis of the Hill Cipher
• lt5gt Cryptanalysis of the LFSR Stream Cipher

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Introduction Some Simple Cryptosystems

• 1 Introduction

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Introduction Some Simple Cryptosystems

• Definition 1.1 A cryptosystem is a five-tuple
  (P,C,K,E,D) satisfies
• P is a finite set of possible plaintexts
• C is a finite set of possible ciphertexts
• K, the keyspace, is a finite set of possible keys
• For each K?K, there is an encryption rule eK?E
  and a corresponding decryption rule dK?D
• dK(eK(x))x for every plaintext x?P

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Introduction Some Simple Cryptosystems

• Definition 1.2 a and b are integers,
• m is a positive integer
• congruence ab (mod m) if m divides b-a
• Zm the set 0,1,,m-1
• with 2 operations and ?
• 10204 in Z26 (1020 mod 264)
• 10?2018 in Z26 (10?20 mod 2618)

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Introduction Some Simple Cryptosystems

• lt1gt Shift Cipher
• Cryptosystem 1.1 Shift Cipher
• P C K Z26
• K, x, y ?Z26
• eK(x)(xK) mod 26
• dK(y)(y-K) mod 26

A B C D E F G H I J K L M
0 1 2 3 4 5 6 7 8 9 10 11 12
N O P Q R S T U V W X Y Z
13 14 15 16 17 18 19 20 21 22 23 24 25
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Introduction Some Simple Cryptosystems

• eg. Suppose K11
• Plaintext student
• Ciphertext DEFOPZE

plaintext s t u d e n t
plaintext 18 19 20 3 4 13 19
K 3 4 5 14 15 25 4
ciphertext D E F O P Z E
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Introduction Some Simple Cryptosystems

• lt2gt Substitution Cipher
• Cryptosystem 1.2 Substitution Cipher
• PCZ26
• K all possible permutations of the 26 symbols
• For each p?K
• ep(x)p(x)
• dp(y)p-1(y)
• where p-1 is the inverse permutation to p

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Introduction Some Simple Cryptosystems

• eg.
• Plaintext student
• Ciphertext VMUSHSM

x a b C d e f g h i j k l m
ep(x) X N Y A H P O G Z Q W B T
x n o p q r s t u v w x y z
ep(x) S F L R C V M U E K J D I
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Introduction Some Simple Cryptosystems

• lt3gt Affine Cipher
• Theorem 1.1 axb (mod m) has a unique solution
  x?Zm for every b?Zm iff gcd(a,m)1
• Definition 1.3 Suppose a1 and m2 are integers
• a and m are relatively prime if gcd(a,m)1
• f(m) the number of integers in Zm that are
  relatively prime to m
• Theorem 1.2 Suppose

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Introduction Some Simple Cryptosystems

• Definition 1.4 Suppose a?Zm
• a-1 mod m
• the multiplicative inverse of a modulo m
• aa-1a-1a1 (mod m)
• Cryptosystem 1.3 Affine Cipher
• P C Z26
• K(a,b) ?Z26?Z26 gcd(a,26)1
• For K(a,b)?K x, y?Z26
• eK(x)(axb) mod 26
• dK(y)a-1(y-b) mod 26

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Introduction Some Simple Cryptosystems

• e.g. Suppose K(7,3)
• 7-1 mod 26 15
• Plaintext student
• Ciphertext ZGNYFQG

eK(x)(7x3) mod 26
dK(y)15(y-3) mod 26
plaintext s t u d e n t
plaintext 18 19 20 3 4 13 19
eK(x) 25 6 13 24 5 16 6
ciphertext Z G N Y F Q G
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Introduction Some Simple Cryptosystems

• lt4gt Vigenère Cipher
• Cryptosystem 1.4 Vigenère Cipher
• m a positive integer
• P C K (Z26)m
• For a key K(k1,k2,,km)
• eK(x1,x2,,xm)(x1k1,x2k2,,xmkm)
• dK(y1,y2,,ym)(y1-k1,y2-k2,,ym-km)

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Introduction Some Simple Cryptosystems

• e.g. Suppose m4 and K(2,8,15,7)
• Plaintext student
• Ciphertext UBJKGVI

plaintext s t u d e n t
plaintext 18 19 20 3 4 13 19
K 2 8 15 7 2 8 15
ciphertext 20 1 9 10 6 21 8
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Introduction Some Simple Cryptosystems

• lt5gt Hill Cipher
• Definition 1.5 Suppose A(ai,j) is an m?m matrix
• Ai,j the matrix obtained from A by deleting the
  ith row and the jth column
• det A the determinant of A
• m1 det Aa1,1
• mgt1 for any fixed i
• A(ai,j) the adjoint matrix of A
• ai,j(-1)ijdet Aj,i

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Introduction Some Simple Cryptosystems

• Theorem 1.3 Suppose K(ki,j) is an m?m
  invertible matrix over Zn
• K-1(det K)-1K
• e.g.
• det K11?7-8?3
  mod 261
• K-1(det K)-1K

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Introduction Some Simple Cryptosystems

• Cryptosystem 1.5 Hill Cipher
• M 2 is an integer
• P C (Z26)m
• K m?m invertible matrices over Z26
• For a key K
• eK(x)xK
• dK(y)yK-1
• where K-1 is the inverse of K

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Introduction Some Simple Cryptosystems

• e.g.
• Plaintext GOD (6 14 3)
• Ciphertext WTJ (22 19 9)

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Introduction Some Simple Cryptosystems

• lt6gt Permutation Cipher
• Cryptosystem 1.6 Permutation Cipher
• m is a positive integer
• P C (Z26)m
• K consist of all permutations of 1,,m
• For a key(a permutation) p
• ep(x1,,xm)(xp(1),,xp(m))
• where p-1 is the inverse permutation to p

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Introduction Some Simple Cryptosystems

• e.g. Suppose m6
• Plaintext CYBERFORMULA
• Ciphertext BRCFEYMLOAUR

x 1 2 3 4 5 6
p(x) 3 5 1 6 4 2
plaintext C Y B E R F O R M U L A
ciphertext B R C F E Y M L O A U R
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Introduction Some Simple Cryptosystems

• lt7gt Stream Ciphers
• Block ciphers
• Plaintext string x x1x2 (each xi is a
  plaintext)
• Ciphertext string y y1y2 eK(x1)eK(x2)
• Stream ciphers
• Plaintext string x x1x2
• Generate a keystream (by using some K) z z1z2
  
• Ciphertext string y y1y2 ez1(x1)ez2(x2)

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Introduction Some Simple Cryptosystems

• Definition 1.6 A synchronous stream cipher is a
  tuple (P,C,K,L,E,D) with a function g
• P a finite set of possible plaintexts
• C a finite set of possible ciphertexts
• K a finite set of possible keys
• L a finite set called the keystream alphabet
• g the keystream generator
• Input K
• g generates an infinite string z1z2

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Introduction Some Simple Cryptosystems

• Definition 1.6 (cont.)
• For each z?L, there is an encryption rule ez?E
  and a corresponding decryption rule dZ?D
• dz(ez(x))x for every plaintext x?P

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Introduction Some Simple Cryptosystems

• Vigenère Cipher can be defined as a synchronous
  stream cipher
• K (Z26)m
• P C L Z26
• ez(x)(xz) mod 26
• dz(y)(y-z) mod 26
• Keystream z1z2
• k1k2..km k1k2..km k1k2..km
  

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Introduction Some Simple Cryptosystems

• Keystream can be produced efficiently in hardware
  using a LFSR (Linear Feedback Shift Register)
• k1 would be tapped as the next keystream bet
• k2,km would each be shifted 1 stage to the left
• The new value of km would be
• this is linear feedback (see Figure 1.2)
• This system is modulo 2

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Introduction Some Simple Cryptosystems

• e.g. in Figure 1.2,suppose K(1,0,0,0)
• c01, c11, c20, c30
• The keystream is
• 100010011010111

Figure 1.2
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Introduction Some Simple Cryptosystems

• Non-synchronous stream cipher
• Each keystream element zi depends on previous
  plaintext or ciphertext elements
• Cryptosystem 1.7 Autokey Cipher
• P C K L Z26
• z1K, zixi-1 for all igt1
• For x, y, z ?Z26
• ez(x)(xz) mod 26
• dz(y)(y-z) mod 26

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Introduction Some Simple Cryptosystems

• e.g. Suppose K8
• Plaintext student
• Ciphertext ALNXHRG

plaintext s t u d e n t
plaintext 18 19 20 3 4 13 19
keystream 8 18 19 20 3 4 13
ciphertext 0 11 13 23 7 17 6
ciphertext A L N X H R G