Classical Cryptography

1. Introduction: Some Simple Cryptosystems

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Outline [1] Introduction: Some Simple Cryptosystems

<1> The Shift Cipher <2> The Substitution Cipher <3> The Affine Cipher <4> The Vigenère Cipher <5> The Hill Cipher <6> The Permutation Cipher <7> Stream Ciphers

[2] Cryptanalysis <1> Cryptanalysis of the Affine Cipher <2> Cryptanalysis of the Substitution Cipher <3> Cryptanalysis of the Vigenère Cipher <4> Cryptanalysis of the Hill Cipher <5> Cryptanalysis of the LFSR Stream Cipher

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Introduction: Some Simple Cryptosystems

[1] Introduction

Alice encrypter decrypter

Bob

Oscar

secure channel

key source

x x

K

y

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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

CP :Ke

PC:Kd

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Introduction: Some Simple Cryptosystems

Definition 1.2: a and b are integers, m is a positive integer

congruence: a≡b (mod m) if m divides b-a

Zm: the set {0,1,…,m-1}

with 2 operations + and ☓ 10+20=4 in Z26 (10+20 mod 26=4) 10 20=18 in Z☓ 26 (10 20 mod 26=18)☓

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Introduction: Some Simple Cryptosystems

<1> Shift Cipher Cryptosystem 1.1: Shift Cipher

P = C =K = Z26

K, x, y ∈Z26

eK(x)=(x+K) 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 K=11 Plaintext: student Ciphertext: DEFOPZE

plaintexts t u d e n t

18

19

20

3 413

19

+K 3 4 514

15

25

4

ciphertext D E F O P Z E

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Introduction: Some Simple Cryptosystems

<2> Substitution Cipher Cryptosystem 1.2: Substitution Cipher

P=C=Z26

K: all possible permutations of the 26 symbols For each ∈K

e(x)=(x) d(y)=-1(y)

where -1 is the inverse permutation to

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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

e(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

e(x) S F L R C V M U E K J D I

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Introduction: Some Simple Cryptosystems <3> Affine Cipher

Theorem 1.1: ax≡b (mod m) has a unique solution x∈Zm for every b∈Zm iff gcd(a,m)=1

Definition 1.3: Suppose a≥1 and m≥2 are integers a and m are relatively prime if gcd(a,m)=1 (m): the number of integers in Zm that are relativel

y prime to m

Theorem 1.2: Suppose

n

i

eiipm

1

n

i

ei

ei

ii ppm1

1)()(

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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-1≡a-1a≡1 (mod m)

Cryptosystem 1.3: Affine Cipher P = C = Z26

K={(a,b) ∈Z26 Z☓ 26 : gcd(a,26)=1} For K=(a,b)∈K ; x, y∈Z26

eK(x)=(ax+b) 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)=(7x+3) mod 26

dK(y)=15(y-3) mod 26

plaintexts t u d e n t

18 1920

3 4 13 19

eK(x) 25 613

24 5 16 6

ciphertext Z G N Y F Q G

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Introduction: Some Simple Cryptosystems

<4> 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)=(x1+k1,x2+k2,…,xm+km) dK(y1,y2,…,ym)=(y1-k1,y2-k2,…,ym-km)

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Introduction: Some Simple Cryptosystems

e.g.: Suppose m=4 and K=(2,8,15,7) Plaintext: student Ciphertext: UBJKGVI

plaintexts t u d e n t

18

19

20

3 413

19

+K 2 815

7 2 815

ciphertext 20

1 910

621

8

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Introduction: Some Simple Cryptosystems

<5> 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 m=1: det A=a1,1

m>1: for any fixed i

A*=(a*i,j): the adjoint matrix of A

a*i,j=(-1)i+jdet Aj,i

m

jjiji

ji AaA1

,, det)1( det

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Introduction: Some Simple Cryptosystems

Theorem 1.3: Suppose K=(ki,j) is an m m invertible mat☓rix over Zn

K-1=(det K)-1K*

e.g.:

det K=11 7-8 3 mod 26=1☓ ☓

K-1=(det K)-1K*=

73

811K

73

811 32,1 K

1123

187*K

1123

187

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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 Z☓ 26} 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)

3425

16223

171521

,

1198

21143

125101KK

91922

1198

21143

12510

3146

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Introduction: Some Simple Cryptosystems

<6> 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)

e(x1,…,xm)=(x(1),…,x(m))

where -1 is the inverse permutation to

),...,(),...,()()1(1 11 mm yyyyd

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Introduction: Some Simple Cryptosystems

e.g.: Suppose m=6 Plaintext: CYBERFORMULA Ciphertext: BRCFEYMLOAUR

x 1 2 3 4 5 6

(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 <7> 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

CP :Ke

PC:Kd

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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)=(x+z) mod 26 dz(y)=(y-z) mod 26 Keystream z1z2… = k1k2..km k1k2..km k1k2..km …

1 if

1 if

miz

mikz

mi

ii

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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

1

01

m

jjjkc

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Introduction: Some Simple Cryptosystems

e.g.: in Figure 1.2,suppose K=(1,0,0,0) c0=1, c1=1, c2=0, c3=0 The keystream is

100010011010111…

k1 k2 k3 k4

+Figure 1.2

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Introduction: Some Simple Cryptosystems

Non-synchronous stream cipher: Each keystream element zi depends on previous pl

aintext or ciphertext elements

Cryptosystem 1.7: Autokey Cipher P = C = K = L = Z26

z1=K, zi=xi-1 for all i>1 For x, y, z ∈Z26

ez(x)=(x+z) mod 26 dz(y)=(y-z) mod 26

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Introduction: Some Simple Cryptosystems

e.g.: Suppose K=8 Plaintext: student Ciphertext: ALNXHRG

plaintexts t u d e n t

18

19

20

3 413

19

keystream 818

19

20

3 413

ciphertext0

11

13

23

717

6

A L N X H R G