Lecture 3: Cryptography Support Services: Key Management

Anish Arora

CSE5473

Introduction to Network Security

Outline

A. Distribution via symmetric keys

B. Distribution via public keysI. of public keys

II. of session keys

C. Group Key Management

A. Key distribution assuming symmetric keys

• how to securely distribute this key is an issue

• often security failure is due to a break in key distribution scheme

• given parties A and B have various key distribution alternatives:

1. A can select key and physically deliver to B

2. third party can select & deliver key to A & B

3. if A & B have communicated previously can use previous key to

encrypt a new key

4. if A & B have secure communications with a third party C, C can

relay key between A & B

A key distribution protocol

Another protocol (for connection-oriented networks)

A decentralized key distribution protocol

Assume a master key is known to principals j and k :

j k : request, n

k j : Smaster ‹ S , request , k , n+1 , m ›

j k : S ‹m+1›

Merkle’s puzzles

• Each puzzle requires O(n) work

• Alice sends O(n) puzzles to Bob, puzzle=EP(“message”)

• Bob chooses one, and spends O(n) effort to break it and get key

• Bob communicates choice index (which was encrypted by Alice) to Alice

• Eve has to perform O(n2) work to guess the key

More on Merkle’s Puzzle

Alice: for i=1, …, 232 choose random Pi ∈{0,1}32 ; xi,ki∈{0,1}128

set puzzlei ⟵ E096 ll Pi (“Puzzle # xi” ll ki)

Send puzzle1 , … , puzzle232 to Bob

Bob: choose a random puzzlej and solve it. Obtain (xj, kj ) .

Send xj to Alice

Alice: lookup puzzle with number xj, use kj as shared secret

[Dan Boneh]

B. Public key management

• public-key encryption helps address key distribution problems

• two aspects:

I. distribution of public keys

II. use of public-key encryption to distribute secret keys

I. Distribution of public keys

• via one of:

public announcement

publicly available directory

public-key authority

public-key certificates

Public announcement

• users distribute public keys to recipients or broadcast to

community at large

e.g. append PGP keys to email messages or post to news groups

or email list

• major weakness is forgery: anyone can

create a key claiming to be someone else and broadcast it

masquerade as claimed user until forgery is discovered

Publicly available directory

• users obtain greater security by registering keys with a public directory

• directory must be trusted, and with these properties: contains {name, public-key} entries

participants register securely with directory

participants can replace key at any time

directory is periodically published

directory can be accessed electronically

• still vulnerable to tampering or forgery, if channel or access to directory is vulnerable

Public-key authority

• improves security by tightening control over distribution of keys from directory

• has same properties as directory + requires users to know public key for the directory

• users interact with directory to obtain any desired public key securely requires real-time access to directory when keys are needed

Deriving a protocol for authority based distribution

Consider the basic protocol:

j k : B.j

k j : B.k

j k : B.k ‹m›

k j : B.j ‹m’›

Subject to man-in-the-middle attack

Man-in-the-middle attack

Recall the attack

j k : B.j intercepted by Mal

Mal k : B.Mal

k j : B.k intercepted by Mal

Mal j : B.Mal

j k : B.Mal ‹m›

“Mallory”-in-the-middle can now passively receive the messages sent by j to k and vice versa

To foil attack: get Trent to sign & send public keys of one to other

Foiling the attack: use signatures

One solution: get Trent to sign and send public keys of the

one to the other

T k : R.T ‹ B.j ›

T j : R.T ‹ B.k ›

But freshness of exchange remains an issue:

how to tolerate replay attacks

Public-key authority

Public-key certificates

• certificates allow key exchange without real-time access to public-key authority

• users contact authority only on behalf of self as opposed to others

• a certificate binds identity to public key usually with other info such as period of validity

with all contents signed by a trusted Public-Key or Certificate Authority (CA)

• certificates can be verified by anyone who knows the public-key authorities public-key

Public-key certificates

Light-weight public key certificates

CA structures

• One universally trusted authority issues: monopoly pricing, risk of all eggs in one basket, cost of getting

certificate in first place could have local registration authorities (RAs) to simplify getting

certificate initially could replace one with many (monopoly -> oligarchy; as in trusted

roots of IE) but less secure, since one “weak” CA compromises all

• Top-down hierarchy, starting from universally trusted authority

certificate chains, a CA certifies a public key to below to subordinate CA need to verify multiple certificates at user end but don’t have to go to original CA to get certificate in first place

Organizing CAs

alternatively, assume name subordination: each CA only responsible for its name subspace more secure in practice

bottom-up version (as opposed to building trust from the top-down): extend to traverse up and down intranet namespace hierarchy & across

extranet namespaces security within organization (intranet) is controlled by organization easy configuration: start with own public key

• Many independent CAs: configure which ones to trust issue: anarchy doesn’t scale either

• X.509 is an IEEE standard for certificate syntax, PKIX is an extension to this standard, SPKI is a competing IETF standard

Revoking certificates

• If certificate compromised, notify CA and ask for a new certificate

• How to revoke certificate: Supplement certificate lifetimes with certificate revocation lists (CRLs) or a black list server (OLRS)

• These can be maintained on-line

II. Public-key distribution of secret keys

• use previous methods to obtain public-key

• then use public-key for secrecy or authentication is slow

• so use private-key encryption to protect message contents

• hence need a session key

• have several alternatives for negotiating a suitable session

Simple secret key distribution

• proposed by Merkle in 1979

j generates a new temporary public key pair

j sends k the public key and its identity

k generates a session key S & sends it to j encrypted using the

supplied public key

j decrypts the session key and both use the key

j k : B.j

k j : B.j ‹ S ›

Man-in-the-middle attack

Here’s one attack:

j k : B.j intercepted by Mal

Mal j : B.j ‹S›

Mal k : B.Mal

k j : B.Mal ‹S’› intercepted by Mal

j k : S‹m› intercepted by Mal

Mal k : S’‹m›

“Mallory”-in-the-middle can now actively receive the messages sent by j to k and vice versa

Foiling the attack: use signatures

One solution: get Trent to sign and send public keys of the one to

the other

T k : R.T ‹ B.j ›

T j : R.T ‹ B.k ›

j k : R.j ‹ S.jk ‹m›, B.k ‹S.jk› ›

But freshness of exchange remains an issue !

Foiling replay attacks: use nonce exchange

• To deal with freshness, assuming securely exchanged public-keys:

Diffie-Hellman key exchange

• first public-key scheme proposed

• by Diffie & Hellman in 1976, along with exposition of public key concepts

• is a practical method for public exchange of a secret key as opposed to secure communication of messages

• used in a number of commercial products

Diffie-Hellman key exchange

• shared session key for users A & B is KAB:

KAB = αxA.xB mod q

= yA

xB mod q (which B can compute)

= yB

xA mod q (which A can compute)

• KAB is used as session key in private-key encryption scheme between Alice and Bob

• if Alice and Bob subsequently communicate, they will have the same key as before, unless they choose new public-keys

• attacker needs an x, must solve discrete log

Diffie-Hellman key exchange

• value of key depends on the participants (and their private and public key information)

• based on exponentiation in a finite (Galois) field (modulo a prime or a polynomial) – easy

• security relies on the difficulty of computing discrete logarithms (similar to factoring) – hard i.e., given α, q, y = α

x mod q computing x is hard

discrete log computation takes more time than factoring a

composite of magnitude of q

Diffie-Hellman setup

• all users agree on global parameters: large prime q α a primitive root mod q

powers of α generate all numbers 1..q-1

• each user (e.g. A) generates their key chooses a secret key (number): xA < q

Computes its public key: yA = αxA mod q

• each user makes public that key yA

Diffie-Hellman example

Users Alice & Bob who wish to swap keys:

• agree on prime q=353 and α=3

• select random secret keys: A chooses xA=97, B chooses xB=233

• compute public keys: yA=3

97 mod 353 = 40 (Alice)

yB=3233 mod 353 = 248 (Bob)

• compute shared session key as:

KAB= yB

xA mod 353 = 24897 = 160 (Alice)

KAB= yA

xB mod 353 = 40233 = 160 (Bob)

Man-in-the-Middle attack for D-H

• Mallory intercepts exchange with Alice and sets up key with her, likewise sets up key with Bob

traps all exchanges of data and faithfully forwards after decrypting with one key and then re-encrpyting with other key

can now actively enable communications between Alice and Bob

j k : αxj mod q intercepted by Mal

Mal j : αxMal mod q

Mal k : αxMal mod q

k j : αxk mod q intercepted by Mal

j k : α xj

xMal mod q ‹m› intercepted by Mal

Mal k : α xk

xMal mod q ‹m›

Dealing with Man-in-the-Middle attack for D-H

• Avoided by sending messages not in the clear, but encrypted: with private keys with public keys and signed (in reverse order) by only one side

• But if private keys already exist, then why have D-H to begin with? “Forward secrecy”: if former private key compromised, latter

keys not deducible

Denial-of-Service protection for D-H

• Mallory may send too many request for key exchanges to Bob

• To avoid this, add a preliminary message Bob first sends a cookie Alice’s response includes her public key and the cookie Bob verifies cookie before sending his public key in response

Key distribution systems issues

• hierarchies of KDC’s required for large networks, but must trust each other

• session key lifetimes should be limited for greater security

• use of automatic key distribution on behalf of users, but must trust system

• controlling purposes keys are used for

C. Group Key Management

A. Distribution via symmetric keys

B. Distribution via public keysI. of public keys

II. of session keys

C. Distribution via “group” keyI. The key-tree approach

II. The grid approach (for sensor networks)

The Key Tree Approach (Wong, Gouda, Lam)

• Keys represented as nodes Group key is the root Auxiliary keys are internal nodes Individual keys are leaves

• Member u holds all keys in ancestor nodes Example: u1 holds keys k1 and kG

kG

k1

u2u1 u3

k2

u5u4 u6

k3

u8u7 u9

Scalability of Key Trees

• Reduces DELETE(u) communication costs from O(n) to O(log n)

• Example: DELETE(u9)

Must change 2 shared keys: kG and k3

Keys are changed bottom up in the tree

Change k3 with 2 messages: E(k’3,u7), E(k’3,u8)kG

k1

u2u1 u3

k2

u5u4 u6

k3

u8u7 u9

Scalability of Key Trees

• Change kG with 3 messages:

E(k’G,k1), E(k’G,k2), E(k’G,k’3)

kG

k1

u2u1 u3

k2

u5u4 u6

k’3

u8u7

u9

User-Oriented Rekeying

• Encryption Cost Join: 1 + 2 + … + h-1 + h-1 Leave: (d-1)(1+2+…+h-1)

• Rekey Messages Join: h Leave: (d-1)(h-1)

k9

u9

• Join rekey messages • Leave rekey messages

Key-Oriented Rekeying

• Encryption Cost Join: 2(h-1) Leave: d(h-1)

• Rekey Messages Join: 2(h-1) Leave: (d-1)(h-1)

k9

u9

• Join rekey messages • Leave rekey messages

Group-Oriented Rekeying

k9

u9

Two rekey messages for join:

Encryption cost : 2(h-1)

Leave Operation:

Encryption cost: d(h-1)

Rekey messages: 1

Grid Protocol (Kulkarni, Gouda, Arora)

• Arrange the secrets in a grid

• Each user is also assigned to some location in the grid

• Each user gets secrets in its row and in its column

= user + secret

Grid Protocol (Continued)

= user + secret= communicating users= secrets used

When two users in different rows and different columns communicate

• Consider the rectangle formed by those two users

• Choose secrets at the other two corners of the rectangle

When users is same row (or column) communicate

• Maintain a secret that is shared between only those two users