COEN 350: Network Security
Overview of Cryptography
Overview of Cryptography
Table of contents
 Introduction
 Cryptographic Security
 One Way Functions
 Secret Key Cryptography
 Public Key Cryptography
 Message Authentication Codes
 Zero Knowledge Proofs
 Diffie Hellman Key Exchange
Cryptography

Traditional use of cryptography

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Secret Key (Symmetric) Cryptography


Encrypt a plain text into cypher
Only people with the right knowledge can recover
plain text.
Encryption and decryption use secret key c.
Public Key (Asymmetric) Cryptography

Encryption and decryption use two different keys.
Cryptography

Other uses of cryptography




Secure data while stored.
Authenticate entities.
Ensure integrity of data.
Sign statements so that signature cannot
be repudiated.
Cryptography

Other uses of cryptography

Fast file destruction:



Encrypt files with a secret key.
Destroy secret key to securely delete the file.
E-cash
Hash Functions




Given an object, create a hash (short
bit-string) of the object.
Hashs differ  Objects differ
Objects differ with overwhelming prob.
Hashes differ
Cryptographically secure hash:

Given a hash, cannot find object with that
hash.
Hash Functions

Tripwire



Protect OS against trojans.
Maintain hashes of all system libraries in a
secure area.
Check hash against known hash
periodically.
Overview of Cryptography
Table of contents
 Introduction
 Cryptographic Security
 One Way Functions
 Secret Key Cryptography
 Public Key Cryptography
 Message Authentication Codes
 Zero Knowledge Proofs
 Diffie Hellman Key Exchange
Cryptographic Security


Leverage in cryptography comes from
functions that are hard to compute
without special knowledge.
“Hard to compute” difficult to
substantiate
Cryptographic Security

“Hard to compute” = NP complete

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Problem is P: can be solved deterministically in polynomial
time.
Problem is NP: solution can be verified in polynomial time.
Central Conjecture: NP  P.
NP-complete: If this problem can be solved in polynomial
time then all NP problems can be solved in polynomial time.
NP-complete problems: Intrinsically difficult problems to
solve on a computer.
But: NP completeness is tendency.
Instances of NP-complete problems can be easy to
solve.

Knapsack problem.
Cryptographic Security


“Computationally hard” = “Takes n
years to solve on best machine.”
Breaking codes is usually parallelizable.
Use distributed attack.


SETI@home
Moore’s law: Computers double in
speed every 16 months.
Cryptographic Security

UNIX password cracking

UNIX passwords are 8 characters long.


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
Assume 102 printable characters in a password.
1016 possible passwords.
10000 password attempts a second takes
1012/2 seconds to find random password.
16,000 years to find password
Dictionary attacks take much less.
Cryptographic Security


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DES Data encryption standard
Published in 1977 by National Bureau of Standards.
Uses 56 bit key
Brute-Force attack succeeds after ~1016 tries.
1977: Diffie Hellman:


Spend $20,000,000.- to build parallel machine that can find
key in 12 hours.
1998: Electronic Frontier Association


Build DES cracker for $250,000.- that could break a key in 4
days.
$150,000.- for second cracker
Cryptographic Security

Security of Algorithms

Fundamental Security Paradigm
"If a lot of smart people have tried to
crack a paradigm for a long time, then it
is impossible to crack the paradigm."

Cryptographic Security
Models for evaluating security
 Unconditional Security



Adversary has unlimited computational resources, but there
is not enough information available to defeat the system.
Example: One Time Pad
Complexity Theoretic Security

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Defines an appropriate model of computation
Adversaries can mount attacks that use space and time
polynomial resources.
These attacks might be in practice impossible.
True attacks might be non-polynomial.
Cryptographic Security
Models for evaluating security
 Provable Security


Difficulty of defeating a protocol is at least
as hard as another (supposedly difficult)
problem.
Computational Security

Measures the amount of effort (using the
best methods available now) required to
defeat a system.
Overview of Cryptography
Table of contents
 Introduction
 Cryptographic Security
 One Way Functions
 Secret Key Cryptography
 Public Key Cryptography
 Message Authentication Codes
 Zero Knowledge Proofs
 Diffie Hellman Key Exchange
One-Way Functions

One way function


Easy to compute
Hard to invert.

“Hard” means computationally infeasible.
One-Way Functions



Example
X = {1, 2, ... , 16}
Define f: X → X, x → x3 mod 17.



This function is reasonably easy to compute.
Surprisingly hard to calculate logarithms in a finite
field.
Use the following table.
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
1
8
10 13
6
12
3
2
15 14
5
11
4
7
9
16l
One-Way Functions

Pre-image resistance:


Given a possible image y, it is
computationally impossible to find any
preimage x such that f (x) = y.
Second pre-image resistance:

Given a pre-image x, it is computationally
infeasible to find another preimage z, z  x,
such that f (x) = f (y).
One-Way Functions

Collision resistant:

It is computationally infeasible to find any
two distincts inputs x, x', x'  x such that
f(x) = f(x').
One-Way Functions
Definition: A function f is a strong oneway hash function (also known as a
collision resistant (one-way) hash
function) if and only if

f is easily computable, that is, given x, it is easy to
calculate f(x).

f is pre-image resistant.

f is second pre-image resistant.

f is collision resistant.
One-Way Functions

One-Way function with trapdoors


Much in cryptography is based on being
able to do a difficult thing when possessing
a secret.
There are one-way functions that are easy
to invert if one knows a secret.
One-Way Functions

Choose

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p = 48611 (a prime)
q = 53993 (a prime)
n = p·q.
Define f


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f (x) = x 3 mod n.
f is one way, if we only know n.
If we know the secret that n = pq, then there is
an algorithm that solves x 3 = y mod n for given y
and unknown x.
One-Way Functions

One-way function with trapdoor



Family of functions fi where i  I, an index
set.
Each fi is one-way.
There exists functions hi and a secret s
such that


hi (s, .) is easy to compute
fi (hi (s, y)) = y.

That is, hi (s, .) is the inverse function of fi
Overview of Cryptography
Table of contents
 Introduction
 Cryptographic Security
 One Way Functions
 Secret Key Cryptography
 Public Key Cryptography
 Message Authentication Codes
 Zero Knowledge Proofs
 Diffie Hellman Key Exchange
Secret Key Cryptography

Conventional encryption uses a secret
to convert plaintext to cipher and the
same secret to convert cipher to
plaintext.

A Greek general tattoos the message into the crown of the head of
a slave who then lets his hair grow again. When the slave reaches
the destination, the recipient reads the message after the slave has
shaven his head again.

One-time pad

Caesar’s cypher
Secret Key Cryptography


Encryption uses an algorithm publicly
known.
Sender and receiver use a key.
Secret Key Cryptography

Generic recipe:

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Take the plain text.
Apply a transformation (based on secret,
reversible with secret).
Repeat until result is sufficiently disguised
Product cipher

Use first one transformation, then another
one.
Secret Key Cryptography

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Substitution Permutation Network
Each state involves substitutions and
permutations.
Substitutions:


Take an input, replace it by an output.
Often implemented as a table.

Input needs to be small.
Secret Key Cryptography

Permutations

Take the bits and reorder them.
Secret Key Cryptography

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Substitution
Permutation
Network
Encode from top to
bottom
Decode from bottom
to top
Secret Key Cryptography

Iterated block cipher

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Made up of rounds.
In each round, apply an transformation with a
separate key (the round key).
Feistel Cipher
Secret Key Cryptography

Feistel Cipher



Iterated Block cipher
Block size is 2t.
Each round:
 Breaks input into left half L(n) and right half
R(n)
 L(n+1) = R(n).
 R(n+1) = Mangler(R(n), Kn)  L(n)
 Kn is round key.
Secret Key Cryptography
Feistel round for encryption (left) and decryption (right)
Secret Key Cryptography

DES (1977)

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uses a 64b key with a parity check, so that effective key
size is 56b.
Derives 16 round keys of 48b each.
Works on input of size 64.
Uses 16 round Feistel algorithm
IDEA (1991)


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Uses a 128b key
Uses 8 computationally identical rounds based on
generalized Feistel algorithm
Additional beginning and ending transformation.
Secret Key Cryptography

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Typical block code takes 64b plaintext and
changes it to 64b cipher text.
Electronic Code Book:



Break plain text into 64b-blocks.
Encrypt all blocks.
Vulnerable to attacks

Two identical text blocks are encrypted the same way.

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Allows guessing contents.
Reordering of plain text = Reordering of cipher text.

Change meaning of cipher text.
Secret Key Cryptography

Example:


Database contains employee and salary
information.
Encrypted:
Secret Key Cryptography

Switch portion of cipher text

Resulting plaintext
Secret Key Cryptography
Cipher Block Chaining
Encryption and Decryption
Secret Key Cryptography

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Cipher Block Chaining
If we do not mind to mangle some
preceding data, we can switch bits.
How? Your turn.
Secret Key Cryptography


Output Feedback modes
Same idea, but prevents these types of
attacks.
Output Feed Back
Cipher Feed Back
Secret Key Cryptography

One-Time Pad

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Only proven secure cryptographic method
But the pad needs to be transmitted between
sender and receiver.
XORing with a short string is not secure.

See projects
Secret Key Cryptography

Message Authentication Code


Can be calculated with cipher block chaining or
similar method.
c6 is the MAC
Overview of Cryptography
Table of contents
 Introduction
 Cryptographic Security
 One Way Functions
 Secret Key Cryptography
 Public Key Cryptography
 Message Authentication Codes
 Zero Knowledge Proofs
 Diffie Hellman Key Exchange
Public Key Cryptography

Asymmetric Key Cryptograpy.
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Use one key for encryption, another for
decryption.
E(e,.) encryption with key e
D(d,.) is decryption with key d
D(d,E(e,m)) = E(e,D(d,m)) = m for all
messages m.
Public Key Cryptography


Keep one key public, the other one private.
Use public key to encrypt, give Bob secret key to
decrypt.
Public Key Cryptography

Signing Messages.


Alice creates a public key pair (e,d) and gives or
publishes e.
Alice uses private key to calculate s = D(d,m)



Pad with zeroes if necessary.
Bob uses Alice's public key to decrypt E(e,s) =
E(e,D(d,m')) = m.
If m is in the format that a signed message has,
then Bob accepts the message as truly Alice's.
Public Key Cryptography
RSA

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RSA: Rivest Shamir Adleman
Choose n = pq, p, q large primes
Select e that is coprime to
(n)=(p-1)(q-1)
Find d such that e d = 1 mod (n).

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Only computationally feasible if n = pq is known.
Public key: (e,n)
Private key: (d,n)
Public Key Cryptography
RSA

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Encryption with private key.
Divide messages into chunks < n
Encrypt chunk c as c1 = ce mod n.
Decryption


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Calculate c = c1d mod n.
c1d = (ce)d = ced = cx(n)+1 = c
Using Euler’s theorem

a(n) = 1 mod n.
Public Key Cryptography

RSA is safe if used with caution.
Overview of Cryptography
Table of contents
 Introduction
 Cryptographic Security
 One Way Functions
 Secret Key Cryptography
 Public Key Cryptography
 Message Authentication Codes
 Zero Knowledge Proofs
 Diffie Hellman Key Exchange
Message Authentication Code

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

Also known as MIC (Message Integrity
Code).
Append MAC to message.
Nobody can change message without
changing MAC.
Easy to check whether MAC belongs to
the message.
Message Authentication Code

Symmetric key MACs using a hash
function


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Message
Calculate hash value
Protect hash value by encrypting it with a
secret key.
Sender and receiver share the secret
key.
Message Authentication Code

ISO 8732-2: (Banking - Approved Algorithm for Message
Authentication)
MAC(Message M) {
for(i=0; i <= LengthOfMessage; i++)
{
v = v << 1
e = v XOR w
x = [ ((e + y mod 2**32) or A and C) * (x XOR M(i) ] mod 2**32-1
y = [ ((e + x mod 2**32) or B and D) * (y XOR M(i) ] mod 2**32-2
}
return x XOR y;
}
where A, B, C, and D are constants, and v and w are determined by the key.
Message Authentication Code

Cipher Block Chaining (CBC) derived MAC
Message Authentication Code

Public Key Message Authentication

If message is small


Alice encrypts with private key.
Bob decrypts with public key.


If message looks right, it comes from someone who knows
Alice’s key.
If message is large



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Calculate a digest (hash) of the message
Alice encrypts digest with private key.
Bob decrypts digest with public key.
If digest matches, it comes from Alice.
Message Authentication Code

MD5



MD5 was developed by Rivest in 1994.
Its 128 bit (16 byte) message digest
SHA1




Secure Hash Algorithm
developed by NIST
published in 1994
produces a 160-bit (20 byte) message digest
Overview of Cryptography
Table of contents
 Introduction
 Cryptographic Security
 One Way Functions
 Secret Key Cryptography
 Public Key Cryptography
 Message Authentication Codes
 Zero Knowledge Proofs
 Diffie Hellman Key Exchange
Zero Knowledge Proof


A potentially new way for identification.
Challenge-Response

Claimant proves identity to verifier by
demonstrating knowledge of a secret.


E.g. Verifier gives Claimant random number.
 Claimant can encode it, thus proving
knowledge of a secret key.
Password: Forces claimant to give out
password.
Zero-Knowledge Proofs


Interactive proof system that claimant
knows secret.
But secret is not revealed to verifier or
an observer.
Zero-Knowledge Proof


Alice wants to convince Bob that she
knows the secret word to the door in
this cave.
But Alice doesn’t want to show Bob how
she does it.
Zero-Knowledge Proof




Bob and Alice walk
to A.
Alice walks to either
C or D.
Bob goes to B and
cries out “left” or
“right”
Alice can satisfy the
request.
Zero-Knowledge Proof



Repeat n times.
Alice is lucky with
probability ½ and
does not have to
open door.
Alice is always lucky
with probability 1/2n
Zero-Knowledge Proof

Hostile observer
cannot distinguish
between Alice and
Bob playing a
charade or Alice
knowing how to get
through the door.
Zero Knowledge Proof.

Fiat Shamir:



A trusted center publishes a modulus n = p·q that
is the product of two primes.
Alice selects a secret s coprime to n and publishes
v = s 2 mod n as its public key.
Repeat n times:




Alice chooses a random number r and sends r2 to Bob.
Bob randomly selects e = 0 or e = 1.
Alice sends y = r · se mod n to Bob
Bob accepts this proof if y 2 = r 2 v e mod n.
Zero Knowledge Proof


Assume Alice is an impostor.
If Alice guesses that Bob will send e =
0.




Alice picks s and sends v = s2 to Bob.
Bob asks for e = 0.
Alice sends rs
Bob checks out that (rs)2 = r 2 v
Zero Knowledge Proof


Assume Alice is an impostor.
If Alice guesses that Bob will send e =
1.




Alice picks a and sends v = a2/v to Bob.
Bob asks for e = 1.
Alice sends a
Bob checks out that a 2 = a 2/v · v
Overview of Cryptography
Table of contents
 Introduction
 Cryptographic Security
 One Way Functions
 Secret Key Cryptography
 Public Key Cryptography
 Message Authentication Codes
 Zero Knowledge Proofs
 Diffie Hellman Key Exchange
Diffie Hellmann



First public key system.
Two partners share secret number.
No eavesdropper can deduce secret
number.
Diffie Hellman


p large prime
g<p

Best choice is a generator modulo p:

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
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
 n  i : n = gi.
(p,g) are public.
Alice picks secret r.
Bob picks secret s.
Alice sends gr mod p to Bob.
Bob sends gs mod p to Alice.
Common key is t = (gr)s = (gs)r mod t.
Diffie Hellman


Snooper needs to derive t from gr and
gs.
This is computationally equivalent to
calculate r from gr mod p.
Diffie Hellman

Man-in-the-middle-attack:








Mallory intercepts communications between Alice
and Bob.
Alice sends gr to Mallory.
Mallory sends gr’ to Bob.
Bob sends gs to Mallory.
Mallory sends gs’ to Alice.
Alice establishes common key grs’ with Mallory.
Bob establishes common key gr’s with Mallory.
Alice and Bob communicate via Mallory, who reads
the traffic (or changes it).
Diffie Hellman

Social defense against this attack:

Alice publicly distributes gr mod p.