### Introduction to Cryptography: Understanding Symmetric and Asymmetric Encryption Mechanisms ###

1. Computer SecurityCryptography –an introduction

2. Encryption key KE key KD x plaintext y ciphertext original plaintext x . encryption decryption Eavesdropper

3. Encryption A cryptosystem involves • an encryption algorithm E, and a • a decryptionalgorithm D Both algorithms make use of a key. Let KEbe the encryption key and KD the decryption key. For symmetric cryptosystems the same key is used both encryption and decryption: KE = KD.

4. Encryption If P is the plaintext message, C the ciphertext, then for symmetric cryptosystems: C = E(K,P)and P = D(K,E(K,P)) = D(K,C) For an asymmetric cryptosystem C = E(KE,P)and P = D(KD,E(KE,P)) = D(KD,C)

5. Kerchoffs’ assumption The adversary knows all details of the encrypting function except the secret key

6. Symmetric key encryption There are two types of cipher systems: • Streamciphers, • Blockciphers.

7. Stream ciphers Encryption x = ISSOPMI y = wdhuvad Key KE

8. Block ciphers x = XNE OIG TPH YRK … y = .Key KE wdm . hut vap dgd … Encryption

9. Block ciphersAn overview of the DES Algorithm DES is an iterated block cipher with • 16 rounds, • block length 64 bits and • key length 56 bits

10. Iterating Block ciphers 1. Iterated block cipher Random (binary) key K  round keys:K1,..., KNr, 2. Round function g wr = g(wr-1, Kr), where wr-1is the previous state

11. Iterated cipher … Encryption operation: w0x (x =plaintext) w1 = g(w0, K1), w2 = g(w1, K2), wNr = g(wNr-1, KNr), ywNr(y =ciphertext)

12. Iterated cipher … For decryption we must have: g(.,K) must be invertible for all K Then decryption is the reverse of encryption (bottom-up)

13. Data Encryption Standard DES is a special type of iterated cipher called a Feistel cipher. Block length 64 bits Key length 56 bits Ciphertext length 64 bits

14. DES The round function is: g([Li-1,Ri-1 ]),Ki ) = (Li ,Ri), where Li = Ri-1 and Ri = Li-1 XOR f (Ri-1, Ki).

15. DES round encryption

16. DES inner function

17. DES computation path

18. Inner Function + A Round of DES 64 bit input 32 bit Rn 32 bit Ln Kn 32 bit Ln+1 32 bit Rn+1 64 bit output

19. Inner functionf Combine 32 bit input and 48 bit key into 32 bit output • Expand 32 bit input to 48 bits • XOR the 48 bit key with the expanded 48 bit input • Apply the S-boxes to the 48 bit input to produce 32 bit output • Permute the resulting 32 bits

20. S Boxes • There are 8 different S-Boxes,1 for each chunk • S-box process maps 6 bit input to 4 bit output • S box performs substitution on 4 bits • There are 8 possible substitutions in each S box • Inner 4 bits are fed into an S box • Outer 2 bits determine which substitution is used

21. DES: Initial and Final Permutations There is also an initial and a final permutation: the final permutation is the inverse of the initial permutation

22. Decrypting DES • DES (and all Feistel structures) is reversiblethrough a “reverse” encryption because: • No input data is mangled and passed to the output • The properties of XOR • S-boxes are not reversible (and don't need to be) • Everything needed (except the key) to produce the input to the n-1th step is available from the output of the nthstep. 4. The input to the nth step is the output of the n-1th step. 5. Work backwards to step 1.

23. Encrypt round n Decrypt round n+1 64 bit input 64 bit output 32 bit Rn 32 bit Rn Kn Kn 32 bit Ln 32 bit Ln Inner Function Inner Function + + 32 bit Ln+1 32 bit Rn+1 32 bit Ln+1 32 bit Rn+1 64 bit input 64 bit output

24. Attacks on DES • Brute force • Linear Cryptanalysis -- Known plaintext attack • Differential cryptanalysis • Chosen plaintext attack • Modify plaintext bits, observe change in ciphertext No dramatic improvement on brute force

25. Countering Attacks • Large keyspace combats brute force attack • Triple DES (say EDE mode, with usually 2 keys) • Use AES

26. Modes of operation Four basic modes of operation are available for block ciphers: • Electronic codebook mode: ECB • Cipher block chaining mode: CBC • Cipher feedback mode: CFB • Output feedback mode: OFB

27. Electronic Codebook mode, ECB Each plaintext xi is encrypted with the same key K: yi = eK(xi). So, the naïve use of a block cipher.

28. ECB x1 x2 x3 x4 DES DES DES DES y1 y2 y3 y4

29. Cipher Block Chaining mode, CBC Each cipher block yi-1 is xor-ed with the next plaintext xi : yi = eK(yi-1 XOR xi) before being encrypted to get the next plaintext yi. The chain is initialized with an initialization vector: y0 = IV with length, the block size.

30. CBC x1 x2 x3 x4 IV + + + + DES DES DES DES y1 y2 y3 y4

31. Cipher and Output feedback modes (CFB & OFB) CFB z0 = IV and recursively: zi = eK(yi-1) and yi = xi XOR zi OFB z0 = IV and recursively: zi = eK(zi-1) and yi = xi XOR zi

32. CFB mode x1 x2 IV eK + eK + eK y1 y2

33. OFB mode IV eK eK x1 x2 + + y1 y2

34. Double & Triple DES Double:C = E(k2,E(k1,m) Triple: C = E(k1,D(k2,E(k1,m)

35. AES Block length 128 bits. Key lengths 128 (or 192 or 256). The AES is an iterated cipher with Nr=10 (or 12 or 14) In each round we have: • Subkey mixing: State  Roundkey XOR State • A substitution: SubBytes(State) • A permutation:ShiftRows(State) & MixColumns(State)

36. Asymmetric key encryptionPublic Key Cryptography

37. Public Key Cryptography AliceBob Alice and Bob want to exchange a private key in public.

38. Public Key CryptographyThe Diffie-Hellman protocol Alicega mod pBob gb mod p The private key is:gab mod p where p is a prime and g is a generator of Zp

39. Finite Fields Theorem If p is a prime then Zp is a cyclic group. The generator of Zpis called a primitive element modulo p

40. Public Key CryptographyEncryption schemes Let • P be the set of all plaintext messages • C be the set of ciphertexts • K be the set of all keys

41. The RSA cryptosystem Let n = pq, where p and q are primes. Let P = C= Zn, and define K= {(n,p,q,e,d) : ed = 1 mod f(n) }. For each key K = (n,p,q,e,d), define c = eK(m) = me mod n and dK(c) = cd mod n, where (m,c)eZn. Public key = (n,e), Private key (n,d).

42. Check We have: ed = 1 mod f(n), so ed = 1 + tf(n). Therefore, dK(eK(m)) = (me)d = med = mtf(n)+1 = (mf(n))t m = 1.m = m mod n

43. Example p = 101, q = 113, n = 11413. f(n) = 100x112 = 11200 = 26527 For encryption use e = 3533. Then d = e-1 mod11200 = 6597. Bob publishes: n = 11413, e = 3533. Suppose Alice wants to encrypt: 9726. She computes 97263533 mod 11413 = 5761 To decrypt it Bob computes: 57616597 mod 11413 = 9726

44. Implementation • Generate two large primes: p,q • n pq and f(n)= (p-1)(q-1) • Choose random e: with 1<e< f(n) & gcd(e,f(n))=1 • d  e -1 modf(n) • The public key is (n,e) and the private key is (p,q,d)

45. Security of RSA • Relation to factoring. Recovering the plaintext m from an RSA ciphertext c is easy if factoring is possible. • The RSA problem Given (n,e) and c, compute: m such that me = c mod n

46. The ElGamal encryption scheme Let p be a prime and g e Zp a primitive element. Let P = Zp-1, C = Zp-1 x Zp-1 and K = {(p,g,x,y): y = gx modp }. • The values p,g,y are the public key. • x is the private key.

47. The ElGamal encryption scheme • Encryption Let me Zp-1 be a message. For K = {(p,g,x,y): y = gx modp }, and secret random number k e Zp-1, define: eK(m,k) = (s,t), where • s = gk modp • t = m yk modp • Decryption For s,t e Zp-1, define: dK(s,t) = t(sx)-1modp

48. The security of ElGamal • The Diffie-Hellman problem. Given a prime p,g e Zp-1, and x,y e Zp-1, find xlog gy modp. The security of the ElGamal encryption is reduced to the difficulty of breaking the Diffie-Hellman problem.

49. Digital Signatures

50. Public Key CryptographySignature schemes Let • P be the set of all messages • A be the set of signatures • K be the set of all keys