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Classical Cryptography

Classical Cryptography. 1. Introduction: Some Simple Cryptosystems. Outline. [1] Introduction: Some Simple Cryptosystems <1> The Shift Cipher <2> The Substitution Cipher <3> The Affine Cipher <4> The Vigen è re Cipher <5> The Hill Cipher <6> The Permutation Cipher <7> Stream Ciphers

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Classical Cryptography

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  1. Classical Cryptography 1. Introduction: Some Simple Cryptosystems

  2. 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

  3. Oscar x y x Alice encrypter decrypter Bob secure channel K key source Introduction: Some Simple Cryptosystems • [1] Introduction

  4. 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

  5. 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 Z26 (10☓20 mod 26=18)

  6. 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

  7. Introduction: Some Simple Cryptosystems • eg.: Suppose K=11 • Plaintext: student • Ciphertext: DEFOPZE

  8. 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 p∈K • ep(x)=p(x) • dp(y)=p-1(y) where p-1 is the inverse permutation to p

  9. Introduction: Some Simple Cryptosystems • eg.: • Plaintext: student • Ciphertext: VMUSHSM

  10. 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 • f(m): the number of integers in Zm that are relatively prime to m • Theorem 1.2: Suppose

  11. 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☓Z26 : 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

  12. 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

  13. 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)

  14. Introduction: Some Simple Cryptosystems • e.g.: Suppose m=4 and K=(2,8,15,7) • Plaintext: student • Ciphertext: UBJKGVI

  15. 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

  16. 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 K=11☓7-8☓3 mod 26=1 • K-1=(det K)-1K*=

  17. 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

  18. Introduction: Some Simple Cryptosystems • e.g.: • Plaintext: GOD (6 14 3) • Ciphertext: WTJ (22 19 9)

  19. 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) p • ep(x1,…,xm)=(xp(1),…,xp(m)) where p-1 is the inverse permutation to p

  20. Introduction: Some Simple Cryptosystems • e.g.: Suppose m=6 • Plaintext: CYBERFORMULA • Ciphertext: BRCFEYMLOAUR

  21. 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) …

  22. 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…

  23. 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

  24. 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 …

  25. 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

  26. + k1 k2 k3 k4 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… Figure 1.2

  27. 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 • 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

  28. Introduction: Some Simple Cryptosystems • e.g.: Suppose K=8 • Plaintext: student • Ciphertext: ALNXHRG

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