CPS 214 Computer Networks and
Distributed Systems
Cryptography Basics
RSA
SSL
SSH
Kerberos
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Basic Definitions
Plaintext
Key1
Encryption Ekey1(M) = C
Cyphertext
Key2
Decryption
Dkey2(C) = M
Original Plaintext
Private Key or Symmetric: Key1 = Key2
Public Key or Asymmetric: Key1  Key2
Key1 or Key2 is public depending on the protocol
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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 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.
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One-way functions:
possible definition
1. F(x) is polynomial time
2. 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 = 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 privatekey protocols?
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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.
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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 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)
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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
hard
Used 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 source
2. Do not allow repudiation, i.e., that’s not my
signature.
3. Do not allow tampering with the message
without invalidating the signature
Item 2 turns out to be really hard to do
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Using Public Keys
Alice
Dk1(m)
Bob
K1 = Alice’s private key
Bob decrypts it with her public key
More Efficiently
Alice
Dk1(h(m)) + m
Bob
h(m) is a one-way hash of m
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Key Exchange
Private Key method
Eka(k)
Trent
Generates k
Alice
Public Key method
Alice
Ekb(k)
Bob
Ek1(k)
Generates k
Bob
k1 = Bob’s public key
Or we can use a direct protocol, such as DiffieHellman (discussed later)
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Private Key Algorithms
Plaintext
Key1
Encryption Ek(M) = C
Cyphertext
Key1
Decryption
Dk(C) = M
Original Plaintext
What granularity of the message does Ek encrypt?
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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, …
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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
m
key
R
R
.
.
.
s1
R = the “round” function
si = state after round i
ki = the ith round key
k2
s2
R
Consists of n rounds
k1
.
.
.
kn
c
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Iterated Block Ciphers: Decryption
m
R-1
s1
R-1
.
.
.
Run the rounds in
reverse.
Requires that R has an
inverse.
key
k1
k2
s2
R-1
.
.
.
kn
c
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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
F
R-1
ki
F
XOR
ki
XOR
Forwards
Backwards
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 Cipher
Avalanche 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.
Plaintext
K1
Encryption Ek(M) = C
Cyphertext
K2
Public Key systems
K1 = public key
K2 = private key
Digital signatures
Decryption Dk(C) = M
K1 = private key
K2 = public key
Original Plaintext
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
handA->B: server finish: ([amazon,prev-messages,masterkey])key1 shake
B->A: client finish: ([bob,prev-messages,masterkey])key2
A->B: server message: (message1,[message1])key1
data
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 from masterkey and session ID
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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
ga
Alice
gc
Mallory
gd
Bob
gb
Key1 = gad
Key1 = gcb
Mallory gets to listen to everything.
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RSA
Invented by Rivest, Shamir and Adleman in 1978
Based on difficulty of factoring.
Used to hide the size of a group Zn* since:
*

n   (n)  n  (1  1/ p)
.
p| n
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.
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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 p1 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 Performance
Performance: (600Mhz PIII) (from: ssh toolkit):
Algorithm
Bits/key
Mbits/sec
1024
.35sec/key
2048
2.83sec/key
1024
1786/sec
3.5
2048
672/sec
1.2
1024
74/sec
.074
2048
12/sec
.024
ElGamal Enc.
1024
31/sec
.031
ElGamal Dec.
1024
61/sec
.061
RSA Keygen
RSA Encrypt
RSA Decrypt
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):
T ( n)  e
(1 o ( n ))(ln n )1 / 2 (ln(ln n ))1 / 2
– Used in 1994 to factor a 129 digit (428-bit)
number. 1600 Machines, 8 months.
Number field Sieve (NFS):
T ( n)  e
(1.923 o (1))(ln n )1 / 3 (ln(ln n ))2 / 3
– 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)
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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 publicprivate 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)
Kerberos
Authentication
Server
2
1
Client
1.
2.
3.
4.
5.
3
Ticket Granting Server
(TGS)
4
5
Service Server
Request ticket-granting-ticket (TGT)
<TGT>
Request server-ticket (ST)
<ST>
Request service
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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
1.
2.
3.
4.
5.
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
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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.
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