1 / 41

CPS 214 Computer Networks and Distributed Systems

CPS 214 Computer Networks and Distributed Systems. Cryptography Basics RSA SSL SSH Kerberos. Plaintext. E key1 (M) = C. Encryption. Key 1. Cyphertext. Decryption. D key2 (C) = M. Key 2. Original Plaintext. Basic Definitions. Private Key or Symmetric : Key 1 = Key 2

shyla
Download Presentation

CPS 214 Computer Networks and Distributed Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CPS 214 Computer Networks and Distributed Systems Cryptography Basics RSA SSL SSH Kerberos 15-853

  2. Plaintext Ekey1(M) = C Encryption Key1 Cyphertext Decryption Dkey2(C) = M Key2 Original Plaintext Basic Definitions • Private Key or Symmetric: Key1 = Key2 • Public Key or Asymmetric: Key1  Key2Key1 or Key2 is public depending on the protocol 15-853

  3. What does it mean to be secure? • Unconditionally Secure: Encrypted message cannot be decoded without the key • Shannon showed in 1943 that key must be as long as the message to be unconditionally secure – this is based on information theory • A one time pad – xor a random key with a message (Used in 2nd world war) • Security based on computational cost: it is computationally “infeasible” to decode a message without the key. • No (probabilistic) polynomial time algorithm can decode the message. 15-853

  4. Primitives: One-Way Functions • (Informally): A function • Y = f(x) • is one-way if it is easy to compute y from x but “hard” to compute x from y • Building block of most cryptographic protocols • And, the security of most protocols rely on their existence. • Unfortunately, not known to exist. This is true even if we assume P  NP. 15-853

  5. One-way functions: possible definition • F(x) is polynomial time • F-1(x) is NP-hard • What is wrong with this definition? 15-853

  6. One-way functions:better definition • For most y no single PPT (probabilistic polynomial time) algorithm can compute x • Roughly: at most a fraction 1/|x|k instances x are easy for any k and as |x| ->  • This definition can be used to make the probability of hitting an easy instance arbitrarily small. 15-853

  7. Some examples (conjectures) • Factoring:x = (u,v)y = f(u,v) = u*v If u and v are prime it is hard to generate them from y. • Discrete Log:y = gx mod p where p is prime and g is a “generator” (i.e., g1, g2, g3, … generates all values < p). • DES with fixed message:y = DESx(m) This would assume a family of DES functions of increasing key size (for asymptotics) 15-853

  8. One-way functions in private-key protocols • y = ciphertext m = plaintext k= key • Is • y = Ek(m) (i.e. f = Ek) • a one-way function with respect to y and m? • What do one-way functions have to do with private-key protocols? 15-853

  9. One-way functions in private-key protocols • y = ciphertext m = plaintext k= key • How about • y = Ek(m) = E(k,m) = Em(k) (i.e. f = Em) • should this be a one-way function? • In a known-plaintext attack we know a (y,m) pair. • The m along with E defines f • Em(k) needs to be easy • Em-1(y) should be hard • Otherwise we could extract the key k. 15-853

  10. One-way functions in public-key protocols • y = ciphertext m = plaintext k = public key • Consider: y = Ek(m) (i.e., f = Ek) • We know k and thus f • Ek(m) needs to be easy • Ek-1(y) should be hard • Otherwise we could decrypt y. • But what about the intended recipient, who should be able to decrypt y? 15-853

  11. One-Way Trapdoor Functions • A one-way function with a “trapdoor” • The trapdoor is a key that makes it easy to invert the function y = f(x) • Example: RSA (conjecture) y = xe mod n Where n = pq (p, q are prime) p or q or d (where ed = 1 mod (p-1)(q-1)) can be used as trapdoors • In public-key algorithms f(x) = public key (e.g., e and n in RSA) Trapdoor = private key (e.g., d in RSA) 15-853

  12. One-way Hash Functions • Y = h(x) where • y is a fixed length independent of the size of x. In general this means h is not invertible since it is many to one. • Calculating y from x is easy • Calculating any x such that y = h(x) give y is hard • Used in digital signatures and other protocols. 15-853

  13. Protocols: Digital Signatures • Goals: • Convince recipient that message was actually sent by a trusted source • Do not allow repudiation, i.e., that’s not my signature. • Do not allow tampering with the message without invalidating the signature • Item 2 turns out to be really hard to do 15-853

  14. Dk1(m) Alice Bob Using Public Keys K1 = Alice’s private key Bob decrypts it with her public key More Efficiently Dk1(h(m)) + m Alice Bob h(m) is a one-way hash of m 15-853

  15. Key Exchange Private Key method Trent Eka(k) Ekb(k) Generates k Alice Bob Public Key method Ek1(k) Alice Bob Generates k k1 = Bob’s public key Or we can use a direct protocol, such as Diffie-Hellman (discussed later) 15-853

  16. Plaintext Ek(M) = C Encryption Key1 Cyphertext Decryption Dk(C) = M Key1 Original Plaintext Private Key Algorithms What granularity of the message does Ek encrypt? 15-853

  17. Private Key Algorithms • Block Ciphers: blocks of bits at a time • DES (Data Encryption Standard)Banks, linux passwords (almost), SSL, kerberos, … • Blowfish (SSL as option) • IDEA (used in PGP, SSL as option) • Rijndael (AES) – the new standard • Stream Ciphers: one bit (or a few bits) at a time • RC4 (SSL as option) • PKZip • Sober, Leviathan, Panama, … 15-853

  18. Private Key: Block Ciphers • Encrypt one block at a time (e.g. 64 bits) • ci = f(k,mi) mi = f’(k,ci) • Keys and blocks are often about the same size. • Equal message blocks will encrypt to equal codeblocks • Why is this a problem? • Various ways to avoid this: • E.g. ci = f(k,ci-1 mi) “Cipher block chaining” (CBC) • Why could this still be a problem? Solution: attach random block to the front of the message 15-853

  19. Iterated Block Ciphers m key • Consists of n rounds • R = the “round” function • si = state after round i • ki = the ith round key k1 R s1 k2 R s2 . . . . . . kn R c 15-853

  20. Iterated Block Ciphers: Decryption m key • Run the rounds in reverse. • Requires that R has an inverse. k1 R-1 s1 k2 R-1 s2 . . . . . . kn R-1 c 15-853

  21. Feistel Networks • If function is not invertible rounds can still be made invertible. Requires 2 rounds to mix all bits. high bits low bits R R-1 ki ki F F XOR XOR Forwards Backwards • Used by DES (the Data Encryption Standard) 15-853

  22. Product Ciphers • Each round has two components: • Substitution on smaller blocksDecorrelate input and output: “confusion” • Permutation across the smaller blocksMix the bits: “diffusion” • Substitution-Permutation Product Cipher • Avalanche Effect: 1 bit of input should affect all output bits, ideally evenly, and for all settings of other in bits 15-853

  23. Rijndael • Selected by AES (Advanced Encryption Standard, part of NIST) as the new private-key encryption standard. • Based on an open “competition”. • Competition started Sept. 1997. • Narrowed to 5 Sept. 1999 • MARS by IBM, RC6 by RSA, Twofish by Counterplane, Serpent, and Rijndael • Rijndael selected Oct. 2000. • Official Oct. 2001? (AES page on Rijndael) • Designed by Rijmen and Daemen (Dutch) 15-853

  24. Public Key Cryptosystems • Introduced by Diffie and Hellman in 1976. Plaintext • Public Key systems • K1 = public key • K2 = private key Ek(M) = C Encryption K1 Cyphertext • Digital signatures • K1 = private key • K2 = public key Decryption Dk(C) = M K2 Original Plaintext Typically used as part of a more complicated protocol. 15-853

  25. One-way trapdoor functions • Both Public-Key and Digital signatures make use of one-way trapdoor functions. • Public Key: • Encode: c = f(m) • Decode: m = f-1(c) using trapdoor • Digital Signatures: • Sign: c = f-1(m) using trapdoor • Verify: m = f(c) 15-853

  26. Example of SSL (3.0) • SSL (Secure Socket Layer) is the standard for the web (https). • Protocol (somewhat simplified): Bob -> amazon.com • B->A: clienthello: protocol version, acceptable ciphers • A->B: serverhello: cipher, session ID, |amazon.com|verisign • B->A: key exchange, {masterkey}amazon’s public key • A->B: server finish: ([amazon,prev-messages,masterkey])key1 • B->A: client finish: ([bob,prev-messages,masterkey])key2 • A->B: server message: (message1,[message1])key1 • B->A: client message: (message2,[message2])key2 • |h|issuer = Certificate • = Issuer, <h,h’s public key, time stamp>issuer’s private key • <…>private key= Digital signature {…}public key= Public-key encryption • [..] = Secure Hash (…)key = Private-key encryption • key1 and key2 are derived frommasterkeyand session ID hand-shake data 15-853

  27. Diffie-Hellman Key Exchange • A group (G,*) and a primitive element (generator) g is made public. • Alice picks a, and sends ga to Bob • Bob picks b and sends gb to Alice • The shared key is gab • Note this is easy for Alice or Bob to compute, but assuming discrete logs are hard is hard for anyone else to compute. • Can someone see a problem with this protocol? 15-853

  28. ga gc Alice Mallory Bob gd gb Key1 = gad Key1 = gcb Person-in-the-middle attack Mallory gets to listen to everything. 15-853

  29. RSA • Invented by Rivest, Shamir and Adleman in 1978 • Based on difficulty of factoring. • Used to hide the size of a group Zn* since: • . • Factoring has not been reduced to RSA • an algorithm that generates m from c does not give an efficient algorithm for factoring • On the other hand, factoring has been reduced to finding the private-key. • there is an efficient algorithm for factoring given one that can find the private key. 15-853

  30. RSA Public-key Cryptosystem • What we need: • p and q, primes of approximately the same size • n = pq(n) = (p-1)(q-1) • e  Z (n)* • d = inv. of e in Z (n)* i.e., d = e-1 mod (n) Public Key: (e,n) Private Key: d Encode: m  Zn E(m) = me mod n Decode: D(c) = cd mod n 15-853

  31. RSA continued • Why it works: • D(c) = cd mod n • = med mod n • = m1 + k(p-1)(q-1) mod n • = m1 + k (n) mod n • = m(m (n))k mod n • = m (because (n) = 0 mod (n)) • Why is this argument not quite sound? What if m  Zn* then m(n) 1 mod n Answer 1: Not hard to show that it still works. Answer 2: jackpot – you’ve factored n 15-853

  32. RSA computations • To generate the keys, we need to • Find two primes p and q. Generate candidates and use primality testing to filter them. • Find e-1 mod (p-1)(q-1). Use Euclid’s algorithm. Takes time log2(n) • To encode and decode • Take me or cd. Use the power method.Takes time log(e) log2(n) and log(d) log2(n) . • In practice e is selected to be small so that encoding is fast. 15-853

  33. Security of RSA • Warning: • Do not use this or any other algorithm naively! • Possible security holes: • Need to use “safe” primes p and q. In particular p-1 and q-1 should have large prime factors. • p and q should not have the same number of digits. Can use a middle attack starting at sqrt(n). • e cannot be too small • Don’t use same n for different e’s. • You should always “pad” 15-853

  34. RSA Performance • Performance: (600Mhz PIII) (from: ssh toolkit): 15-853

  35. RSA in the “Real World” • Part of many standards: PKCS, ITU X.509, ANSI X9.31, IEEE P1363 • Used by: SSL, PEM, PGP, Entrust, … • The standards specify many details on the implementation, e.g. • e should be selected to be small, but not too small • “multi prime” versions make use of n = pqr…this makes it cheaper to decode especially in parallel (uses Chinese remainder theorem). 15-853

  36. Factoring in the Real World • Quadratic Sieve (QS): • Used in 1994 to factor a 129 digit (428-bit) number. 1600 Machines, 8 months. • Number field Sieve (NFS): • Used in 1999 to factor 155 digit (512-bit) number. 35 CPU years. At least 4x faster than QS • Used in 2003-2005 to factor 200 digits (663 bits) 75 CPU years ($20K prize) 15-853

  37. SSH v2 • Server has a permanent “host” public-private key pair (RSA or DSA) . Client warns if public host key changes. • Diffie-Hellman used to exchange session key. • Server selects g and p and sends to client. • Client and server create DH private keys. Client sends public DH key. • Server sends public DH key and signs hash of DH shared secret and other 12 other values with its private “host” key. • Symmetric encryption using 3DES, Blowfish, AES, or Arcfour begins. • User can authenticate by sending password or using public-private key pair. • If using keys, server sends “challenge” signed with users public key for user to decode with private key. 15-853

  38. Kerberos • A key-serving system based on Private-Keys (DES). • Assumptions • Built on top of TCP/IP networks • Many “clients” (typically users, but perhaps software) • Many “servers” (e.g. file servers, compute servers, print servers, …) • User machines and servers are potentially insecure without compromising the whole system • A kerberos server must be secure. 15-853

  39. Kerberos (kinit) Kerberos Authentication Server • Request ticket-granting-ticket (TGT) • <TGT> • Request server-ticket (ST) • <ST> • Request service Ticket Granting Server (TGS) 2 1 3 4 Service Server Client 5 15-853

  40. Kerberos V Message Formats • C = client S = server K = key or session key • T = timestamp V = time range • TGS = Ticket Granting Service A = Net Address Ticket Granting Ticket: TC,TGS = TGS,{C,A,V,KC,TGS}KTGS Server Ticket: TC,S = S, {C,A,V,KC,S}KS Authenticator: AC,S = {C,T}KC,S • Client to Kerberos: C,TGS • Kerberos to Client: {KC,TGS}KC,TC,TGS • Client to TGS: {TC,TGS , S},AC,TGS • TGS to Client: {KC,S}KC,TGS, TC,S • Client to Server: AC,S, TC,S Possibly repeat 15-853

  41. Kerberos Notes • All machines have to have synchronized clocks • Must not be able to reuse authenticators • Servers should store all previous and valid tickets • Help prevent replays • Client keys are typically a one-way hash of the password. Clients do not keep these keys. • Kerberos 5 uses CBC mode for encryption Kerberos 4 was insecure because it used a nonstandard mode. 15-853

More Related