CS363Week 4 - Friday

Last time

What did we talk about last time? Public key cryptography A little number theory

Questions?

Project 1

Kiefer Weis Presents

More Number Theory!

Fermat’s Little Theorem

If p is prime and a is a positive integer not divisible by p, then:

ap –1 1 (mod p)

Proof of Fermat's Theorem Assume a is positive and less than p Consider the sequence a, 2a, 3a, …, (p – 1)a If these are taken mod p, we will get:

1, 2, 3, …, p – 1 This bit is the least obvious part of the proof However (because p is prime) if you add any non-zero

element repeatedly, you will eventually get back to the starting point, covering all values (except 0) once

Multiplying this sequence together gives: a ∙ 2a ∙ 3a ∙ … ∙ (p – 1)a 1 ∙ 2 ∙ 3 ∙ … ∙ (p – 1) (mod

p) ap – 1(p – 1)! (p – 1)! (mod p) ap – 1 1 (mod p)

Euler's in the mix too

Euler’s totient function (n)(n) = the number of positive

integers less than n and relatively prime to n (including 1)

If p is prime, then (p) = p – 1 If we have two primes p and q

(which are different), then:(pq) = (p)∙(q) = (p – 1)(q – 1)

Take that, Fermat

Euler’s Theorem:For every a and n that are relatively prime,

a(n) 1 (mod n)

This generalizes Fermat’s Theorem because (p) = p – 1 if p is prime

Proof is messier

RSA

RSA Algorithm

Named for Rivest, Shamir, and Adleman

Take a plaintext M converted to an integer

Create an ciphertext C as follows:C = Me mod n

Decrypt C back into M as follows:M = Cd mod n = (Me)d mod n = Med mod n

The piecesTerm Details Source

M Message to be encrypted Sender

C Encrypted message Computed by sender

n Modulus, n = pq Known by everyonep Prime number Known by receiverq Prime number Known by receivere Encryption exponent Known by everyoned Decryption exponent Computed by

receiver(n) Totient of n Known by receiver

How it Works

To encrypt:C = Me mod n

e could be 3 and is often 65537, but is always publically known

To decrypt:M = Cd mod n = Med mod n

We get d by finding the multiplicative inverse of e mod (n)

So, ed 1 (mod (n))

Why it Works

We know that ed 1 (mod (n)) This means that ed = k(n) + 1 for

some nonnegative integer kMed = Mk(n) + 1 M∙(M(n))k (mod n) By Euler’s Theorem

M(n) 1 (mod n) So, M∙(M(n))k M (mod n)

An example

M = 26p = 17, q = 11, n = 187, e = 3C = M3 mod 187 = 185(n) = (p – 1)(q – 1) = 160d = e-1 mod 160 = 107Cd = 185107 mod 187 = 26 If you can trust my modular

arithmetic

Why it’s safe

You can’t compute the multiplicative inverse of e mod (n) unless you know what (n) is

If you know p and q, finding (n) is easy

Finding (n) is equivalent to finding p and q by factoring n

No one knows an efficient way to factor a large composite number

Future risks

Public key cryptography would come crashing down if Advances in number theory could make

RSA easy to break Quantum computers could make it easy

to factor large composites

Practical considerations Choose your primes carefully

p < q < 2p But, the primes can't be too close together either Some standards insist that p and q are strong

primes, meaning that p – 1 = 2m and p + 1 = 2n where m and n have large prime factors

There are ways to factor poorly chosen pairs of primes Pad your data carefully Take the example of a credit card number

If you know a credit card number is encrypted using RSA using a public n and an e of 3, how do you discover the credit card number?

Key Management

Key management Once you have great cryptographic

primitives, managing keys is still a problem How do you distribute new keys?

When you have a new user When old keys have been cracked or need to be

replaced How do you store keys? As with the One Time Pad, if you could

easily send secret keys confidentially, why not send messages the same way?

Notation for sending We will refer to several schemes for sending

data Let X and Y be parties and Z be a message { Z } k means message Z encrypted with key k Thus, our standard notation will be:

X Y: { Z } k Which means, X sends message Z, encrypted with

key k, to Y X and Y will be participants like Alice and Bob

and k will be a clearly labeled key A || B means concatenate message A with B

Kinds of keys

Typical to key exchanges is the idea of interchange keys and session keys

An interchange key is a key associated with a particular user over a (long) period of time

A session key is a key used for a particular set of communication events

Why have both kinds of keys?

Possible attacks using single keys If only a single key (instead of interchange and

session keys) were used, participants are more vulnerable to: Known plaintext attacks (and potentially chosen

plaintext attacks) Attacks requiring many copies of encrypted material

for comparison Replay attacks in which old encrypted data is sent

again from a malicious party Forward search attacks in which a user computes

many likely messages using a public key and thereby learns the contents of such a message when it is sent

Key exchange criteria

To be secure, a key exchange whose goal is to allow secret communication from Alice to Bob must meet this criteria:1. Alice and Bob cannot transmit their key

unencrypted2. Alice and Bob may decide to trust a

third party (Cathy or Trent)3. Cryptosystems and protocols must be

public, only the keys are secret

Classical exchange: Attempt 0 If Bob and Alice have no prior arrangements,

classical cryptosystems require a trusted third party Trent

Trent and Alice share a secret key kAlice and Trent and Bob share a secret key kBob

Here is the protocol:1. Alice Trent: {request session key to Bob} kAlice

2. Trent Alice: { ksession } kAlice || { ksession } kBob

3. Alice Bob: { ksession } kBob

What's the problem? Unfortunately, this protocol is vulnerable to a

replay attack (Evil user) Eve records { ksession } kBob sent in step

3 and also some message enciphered with ksession (such as "Deposit $500 in Dan's bank account")

Eve can send the session key to Bob and then send the replayed message

Maybe Eve is in cahoots with Dan to get him paid twice

Eve may or may not know the contents of the message she is sending

The real problem is no authentication

Needham-Schroeder: Attempt 1 We modify the protocol to add

random numbers (called nonces) and user names for authentication1. Alice Trent: { Alice || Bob || rand1 }

kAlice

2. Trent Alice: { Alice || Bob || rand1 || ksession || {Alice || ksession }kBob } kAlice

3. Alice Bob: { Alice || ksession }kBob

4. Bob Alice: { rand2 } ksession 5. Alice Bob: { rand2 – 1 } ksession

Problems with Needham-Schroeder Needham-Schroeder assumes that all keys are

secure Session keys may be less secure since they are

generated with some kind of (possibly predictable) pseudorandom generator

If Eve can recover a session key (maybe after a great deal of computational work), she can trick Bob into thinking she's Alice as follows:1. Eve Bob: { Alice || ksession }kBob

2. Bob Alice: { rand3 } ksession [intercepted by Eve]3. Eve Bob: { rand3 – 1 } ksession

Denning and Sacco: Attempt 2 Denning and Sacco use timestamps (T) to let

Bob detect the replay1. Alice Trent: { Alice || Bob || rand1 } kAlice

2. Trent Alice: { Alice || Bob || rand1 || ksession || {Alice || T || ksession }kBob } kAlice

3. Alice Bob: { Alice || T || ksession }kBob

4. Bob Alice: { rand2 } ksession 5. Alice Bob: { rand2 – 1 } ksession

Unfortunately, this system requires synchronized clocks and a useful definition of when timestamp T is "too old"

Otway-Rees: Attempt 3 The Otway-Rees protocol fixes these problem

by using a unique integer num to label each session1. Alice Bob: num || Alice || Bob || { rand1 || num ||

Alice || Bob } kAlice

2. Bob Trent: num || Alice || Bob || {rand1 || num || Alice || Bob } kAlice || { rand2 || num || Alice || Bob } kBob

3. Trent Bob: num || { rand1 || ksession } kAlice || { rand2 || ksession } kBob

4. Bob Alice: num || { rand1 || ksession } kAlice

Kerberos Strange as it seems, these key exchange protocols

are actually used Kerberos was created at MIT as a modified

Needham-Schroeder protocol (with timestamps) Originally used to control access to network services for

MIT students and staff Current versions of Windows use a modified version of

Kerberos for authentication Many Linux and Unix implementations have an

implementation of Kerberos Kerberos uses a central server that issues tickets to

users which give them the authority to access a service on some other server

Public Key Exchange

Public key exchange

Suddenly, the sun comes out! Public key exchanges should be

really easy The basic outline is:

1. Alice Bob: { ksession } eBob

eBob is Bob's public key Only Bob can read it, everything's

perfect! Except… There is still no authentication

Easily fixable

Alice only needs to encrypt the session key with her private key

That way, Bob will be able to decrypt it with her public key when it arrives

New protocol:1. Alice Bob: {{ ksession } dAlice }eBob

Any problems now?

(Wo)man in the middle A vulnerability arises if Alice needs to fetch

Bob's public key from a public server Peter Then, Eve can cause problems Attack:

1. Alice Peter: Send me Bob's key [intercepted by Eve]

2. Eve Peter: Send me Bob's key3. Peter Eve: eBob

4. Eve Alice: eEve

5. Alice Bob: { ksession } eEve [intercepted by Eve]6. Eve Bob { ksession } eBob

Key Infrastructure and Storage

Key problems The previous man in the middle attack

shows a significant problem How do we know whose public key is whose? We could sign a public key with a private

key, but then… We would still be dependent on knowing the

public key matching the private key used for signing

It's a massive chicken and egg or bootstrapping problem

Certificate signature chains A typical approach is to create a long chain of

individuals you trust Then, you can get the public key from someone

you trust who trusts someone else who… etc. This can be arranged in a tree layout, with a

central root certificate everyone knows and trusts This system is used by X.509

Alternatively, it can be arranged haphazardly, with an arbitrary web of trust This system is used by PGP, which incorporates

different levels of trust

Upcoming

Next time…

Finish key management Hash functions David Gallop presents

Reminders

Read section 12.4 Finish Project 1

Due tonight by midnight!