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CPS 214 Computer Networks and Distributed Systems

Cryptography BasicsRSASSLSSHKerberos

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Basic Definitions

Private Key or Symmetric: Key1 = Key2

Public Key or Asymmetric: Key1 Key2

Key1 or Key2 is public depending on the protocol

Encryption

Decryption

Key1

Key2

Cyphertext

Ekey1(M) = C

Dkey2(C) = M

Original Plaintext

Plaintext

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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.

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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 protocolsAnd, the security of most protocols rely on their

existence.Unfortunately, not known to exist. This is true

even if we assume P NP.

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One-way functions: possible definition

1. F(x) is polynomial time2. F-1(x) is NP-hard

What is wrong with this definition?

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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.

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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)

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One-way functions in private-key protocols

y = ciphertext m = plaintext k = keyIs 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?

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One-way functions in private-key protocols

y = ciphertext m = plaintext k = keyHow 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.

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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?

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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 nWhere n = pq (p, q are prime)p or q or d (where ed = 1 mod (p-1)(q-1)) can

be used as trapdoorsIn public-key algorithms

f(x) = public key (e.g., e and n in RSA)Trapdoor = private key (e.g., d in RSA)

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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 hardUsed in digital signatures and other protocols.

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Protocols: Digital Signatures

Goals:1. Convince recipient that message was

actually sent by a trusted source2. Do not allow repudiation, i.e., that’s not my

signature.3. Do not allow tampering with the message

without invalidating the signatureItem 2 turns out to be really hard to do

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Using Public Keys

More Efficiently

Alice BobDk1(m)

Alice BobDk1(h(m)) + m

K1 = Alice’s private keyBob decrypts it with her public key

h(m) is a one-way hash of m

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Key ExchangePrivate Key method

Public Key method

Trent

Alice Bob

Eka(k) Ekb(k)Generates k

Alice Bob

Generates k

Ek1(k)

k1 = Bob’s public key

Or we can use a direct protocol, such as Diffie-Hellman (discussed later)

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Private Key Algorithms

Encryption

Decryption

Key1

Key1

Cyphertext

Ek(M) = C

Dk(C) = M

Original Plaintext

Plaintext

What granularity of the message does Ek encrypt?

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Private Key AlgorithmsBlock 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, …

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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

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Iterated Block Ciphers

Consists of n rounds

R = the “round” function

si = state after round i

ki = the ith round key

R

R

R

s1

.

.

.

m

c

.

.

.

key

k1

k2

kn

s2

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Iterated Block Ciphers: Decryption

Run the rounds in reverse.

Requires that R has an inverse.

R-1

R-1

R-1

s1

.

.

.

m

c

.

.

.

key

k2

kn

s2

k1

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Feistel Networks

If function is not invertible rounds can still be made invertible. Requires 2 rounds to mix all bits.

Fki

XOR

Fki

XOR

high bits low bits

Forwards Backwards

R R-1

Used by DES (the Data Encryption Standard)

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Product Ciphers

Each round has two components:– Substitution on smaller blocks

Decorrelate input and output: “confusion”– Permutation across the smaller blocks

Mix the bits: “diffusion”Substitution-Permutation Product CipherAvalanche Effect: 1 bit of input should affect all

output bits, ideally evenly, and for all settings of other in bits

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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)

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Public Key Cryptosystems

Introduced by Diffie and Hellman in 1976.

Encryption

Decryption

K1

K2

Cyphertext

Ek(M) = C

Dk(C) = M

Original Plaintext

Plaintext Public Key systemsK1 = public key

K2 = private keyDigital

signaturesK1 = private key

K2 = public key

Typically used as part of a more complicated protocol.

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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)

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Example of SSL (3.0)SSL (Secure Socket Layer) is the standard for the web (https).Protocol (somewhat simplified): Bob -> amazon.com B->A: client hello: protocol version, acceptable ciphers A->B: server hello: 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 encryptionkey1 and key2 are derived from masterkey and session ID

hand-shake

data

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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?

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Person-in-the-middle attack

Alice BobMallory

ga

gbgd

gc

Key1 = gad Key1 = gcb

Mallory gets to listen to everything.

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RSA

Invented by Rivest, Shamir and Adleman in 1978Based 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.

)/11()(|

* pnnnp

n

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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

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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

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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.

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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”

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RSA PerformancePerformance: (600Mhz PIII) (from: ssh toolkit):

AlgorithmBits/key

Mbits/sec

RSA Keygen1024 .35sec/key

2048 2.83sec/key

RSA Encrypt1024 1786/sec 3.5

2048 672/sec 1.2

RSA Decrypt1024 74/sec .074

2048 12/sec .024

ElGamal Enc. 1024 31/sec .031

ElGamal Dec. 1024 61/sec .061

DES-cbc 56 95

twofish-cbc 128 140

Rijndael 128 180

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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).

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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)

2/12/1 ))(ln(ln)))(ln(1()( nnnoenT

3/23/1 ))(ln(ln)))(ln1(923.1()( nnoenT

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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.

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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.

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Kerberos (kinit)

KerberosAuthentication

Server

Client Service Server

Ticket Granting Server(TGS)

12 34

5

1. Request ticket-granting-ticket (TGT)2. <TGT>3. Request server-ticket (ST)4. <ST>5. Request service

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C = client S = server K = key or session keyT = timestamp V = time rangeTGS = 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

1. Client to Kerberos: C,TGS2. Kerberos to Client: {KC,TGS}KC, TC,TGS

3. Client to TGS: {TC,TGS , S}, AC,TGS

4. TGS to Client: {KC,S}KC,TGS, TC,S

5. Client to Server: AC,S, TC,S

Kerberos V Message Formats

Possibly repeat

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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.