PUBLIC-KEY CRYPTOGRAPHY AND RSA – Chapter 9

1. PUBLIC-KEY CRYPTOGRAPHY AND RSA – Chapter 9 • Principles • Applications • Requirements • RSA Algorithm • Description • Security

2. Historically – Symmetric-Key (one key) substitution (confusion) permutation (diffusion) More Recently – Asymmetric-Key (two keys) PUBLIC-KEY CRYPTOGRAPHY (PKC) – A New Idea

3. PKC more secure than symmetric encryp. WRONG!! • PKC more useful than symmetric encryp. • WRONG!! – PKC costly • PKC doesn’t need complicated protocol • WRONG!! MISCONCEPTIONS PKC vs Symmetric Encryption

4. Key Management • Signature PKC - USES

5. Plaintext – input to encryp. algorithm • output from decryp. algorithm • Encryp. Algorithm – acts on plaintext • - controlled by public or private key • Public and Private Key • - one for encryption • - one for decryption • Ciphertext – output from encryp. algorithm • input to decryp. algorithm • Decryp. Algorithm – acts on ciphertext • - controlled by public or private key PKC – SIX INGREDIENTS

6. Each user generates two related keys • - PUBLIC and PRIVATE • 2. Each user makes: • public key  PUBLIC • private key  PRIVATE • access  ALL public keys • 3. BOB: Encr(plaintext,PUBLICAlice) ciphertext ALICE • 4. ALICE: Decr(ciphertext,PRIVATEAlice) PKC – STEPS

7. PKC for a) ENCRYPTION b) AUTHENTICATION

8. At ANY TIME, ANY Private/Public key pair can be changed. Public key should be made public IMMEDIATELY KEYS EASILY UPDATED

9. Symmetric-Key: One SECRET KEY Asymmetric-Key (PKC): One PRIVATE KEY One PUBLIC KEY CIPHER TERMINOLOGY

10. CONFIDENTIALITY

11. AUTHENTICATION (source)(Integrity/Signature)

12. CONFIDENTIALITY and AUTHENTICATION

13. Encryp./Decryp. • Sender encrypts with RECIPIENT’S PUBLIC key. • Applied to ALL of message. • Digital Signature • Sender signs with SENDER’S PRIVATE key. • Applied to ALL or PART of message. • Key Exchange • Uses one or more PRIVATE keys. • Several approaches APPLICATIONS OF PKC

14. APPLICATIONS OF PKC Table 9.2

15. Every value has an inverse • Y = F(X)  X = F-1(Y) • Y = F(X) - easy • X = F-1(Y) - infeasible • easy – polynomial time (poly in message length) • infeasible - > poly time (e.g. exp. in message length) ONE-WAY FUNCTION

16. Y = fk(X) - easy if k and X known X = fk-1(Y) - easy if k and Y known X = fk-1(Y) - infeasible if only Y known TRAP-DOOR ONE-WAY FUNCTION (e.g. PKC)

17. Brute-Force Attack  Use LARGE keys But, PKC COMPLEXITY GROWS fast with key size So, PKC TOO COMPLEX encryp/decryp PKC only for key management and signature PKC – THE PROBLEM OF KEY SIZE

18. PKC: 1960’s (NSA) 1970 Ellis – CESG 1976 Diffie and Hellman RSA: 1973 Cocks – CESG 1977 Rivest, Shamir, Adleman - MIT RSA ALGORITHM

19. Plaintext and Ciphertext integers between 0 and n-1 i.e. k bits, 2k < n <2k+1 Encryption: C = Me mod n Decryption: M = Cd mod n = (Me)d mod n = Med mod n RSA

20. Sender knows n,e Receiver knows n,d  PUBLIC key, KU = {e,n}  PRIVATE key, KR = {d} RSA (continued)

21. PKC REQUIREMENTS OF RSA 1. There exists e,d,n s.t. Med = M mod n 2. Easy to calculate Me and Cd given {M,e} or {C,d}, resp. 3. Infeasible to find d given {e,n}

22. p = 17, q = 11 n = p.q = 187 mod p = 17, {1,6,62,63,64,65,66,67,68,69,610,611,612,613,614,615} = {1,6,2,12,4,7,8,14,16,11,15,5,13,10,9,3} Mod p = 11 {1,2,4,8,5,10,9,7,3,6} EXAMPLE

23. 57 = (6,2), 572 = (2,4), 573 = (12,8), 574 = (4,5) EXAMPLE

24. We want number, g, between 1 and 186 s.t. g mod 17 = 6, g mod 11 = 2 Use CRT: g = 154.6 + 34.2 mod 187 = 57 EXAMPLE Chinese Remainder Theorem

25. EXAMPLE RSA COMPUTATION

26. Brute-Force Attacks • – try all possible private keys. • Mathematical Attacks • - all equivalent to factoring n. • Timing Attacks • - depend on running time of • decryption algorithm. SECURITY OF RSA

27. Table 9.3 Progress in Factorisation

28. MIPS-years NEEDED TO FACTOR

29. For Decryption: • Constant exponentiation time • Random delay • Blinding • Generate random r • C’ = Cre • M’ = C’d • M = M’r-1 TIMING ATTACKS ON RSA - countermeasures