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CS457 – Introduction to Information Systems
Security
Cryptography 1b
Elias Athanasopoulos
[email protected]
Cryptography Elements
 Symmetric Encryption
Block Ciphers
- Stream Ciphers
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 Asymmetric Encryption
 Cryptographic Hash Functions
 Applications
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The need for randomness
 Key distribution
 Replay attacks (nonces)
 Session key generation
 Generation of keys for the RSA public-key
encryption algorithm
 Stream ciphers
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Randomness
 Uniform distribution
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The distribution of bits in the sequence should be
uniform; that is, the frequency of occurrence of
ones and zeros should be approximately equal.
 Independence
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No one subsequence in the sequence can be
inferred from the others.
 Security requirement
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Unpredictability
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Random Generator Types
 True Random Number Generators (TRNGs)
 Pseudo-random Number Generators (PRNGs)
 Pseudo-random Functions (PRFs)
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TRNGs
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PRNGs
r = f(seed);
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Requirements

Uniformity
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
Scalability
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
Occurrence of a zero or one is equally likely. The expected
number of zeros (or ones) is n/2, where n = the sequence
length
Any test applicable to a sequence can also be applied to
subsequences extracted at random. If a sequence is
random, then any such extracted subsequence should also
be random
Consistency
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The behavior of a generator must be consistent across
starting values (seeds)
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Tests

Frequency test
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
Runs test
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
Determine whether the number of ones and zeros in a
sequence is approximately the same as would be expected
for a truly random sequence
Determine whether the number of runs of ones and zeros
of various lengths is as expected for a random sequence
Maurer’s universal statistical test
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Detect whether or not the sequence can be significantly
compressed without loss of information. A significantly
compressible sequence is considered to be non-random
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Unpredictability
 Forward unpredictability
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If the seed is unknown, the next output bit in the
sequence should be unpredictable in spite of any
knowledge of previous bits in the sequence
 Backward unpredictability
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It should also not be feasible to determine the seed
from knowledge of any generated values. No
correlation between a seed and any value generated
from that seed should be evident; each element of the
sequence should appear to be the outcome of an
independent random event whose probability is 1/2
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Seed
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Cryptographic PRNGs

Purpose-built algorithms
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
Algorithms based on existing cryptographic algorithms
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
Stream ciphers
Asymmetric ciphers
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
Cryptographic algorithms have the effect of randomizing input.
Indeed, this is a requirement of such algorithms. Three broad
categories of cryptographic algorithms are commonly used to
create PRNGs:
Symmetric block ciphers
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
Designed specifically and solely for the purpose of generating
pseudorandom bit streams.
RSA, compute primes
Hash functions and message authentication codes
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Example
Xn+1 = (aXn + c) mod m
Selection of a, c, and m, is very critical:
 a=7, c=0, m=32
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
{7, 17, 23, 1, 7, etc.}
a=5
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{5, 25, 29, 17, 21, 9, 13, 1, 5, etc.}
 In theory m should be very large (2^31)
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Stream ciphers
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RC4
/* Stream Generation */
i, j = 0;
while (true)
i = (i + 1) mod 256;
j = (j + S[i]) mod 256;
Swap (S[i], S[j]);
t = (S[i] + S[j]) mod 256;
k = S[t];
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/* Initialization */
for i = 0 to 255 do
S[i] = i;
T[i] = K[i mod keylen];
/* Initial Permutation of S */
j = 0;
for i = 0 to 255 do
j = (j + S[i] + T[i]) mod 256;
Swap (S[i], S[j]);
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More maths
 Any
integer a > 1 can be factored in a
unique way as:
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Public-Key Cryptography
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Properties
 2 keys
Public Key (no secrecy)
- Private Key (if stolen everything is lost)
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 Easy algorithm, but hard to reverse
Y = f(X), easy
- X = f-1(X), computationally hard
- Computationally hard means solvable in nonpolynomial time
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RSA
Plaintext = M, cipher = C
C = Me mod n
M = Cd mod n = (Me mod n)d = Med mod n
Public Key = {e, n}
Private Key = {d, n}
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Euler’s totient function
 Written φ(n), and defined as the number of
positive integers less than n and relatively
prime to n. By convention, φ(1) = 1.
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Just believe me that this holds!
(i.e., φ(pq) =φ(p) φ(q))
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RSA Steps
 p, q, two prime numbers
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Private
 n = pq
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n can be public, but recall that it is hard to infer p and
q by just knowing n
 e is relative prime to φ(n)
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Public
Recall φ(n) = (p-1)(q-1)
 d from e, and φ(n)
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Private
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RSA example
1.
2.
3.
4.
5.
Select p = 17 and q = 11
Then, n = pq = 17×11 = 187.
φ(n) = (p-1)(q-1) = 16×10 = 160.
Select e relatively prime to φ(n) = 160 and less
than φ(n); e = 7.
Determine d
- de = 1 (mod 160) and d < 160,
- The correct value is d = 23, because 23 × 7 = 161 = (1 ×
160) + 1;
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Computational Aspects
 RSA builds on exponents
 Intensive operation
 Side channels
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How it works?
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Integrity and Message Authentication
 Integrity
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(e.g., download a file)
Message digest
 Message Authentication Code (MAC)
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-
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Used between two parties that share a secret key to
authenticate information exchanged between those
parties
Input is a secret key and a data block and the product
is their hash value, referred to as the MAC
An attacker who alters the message will be unable to
alter the MAC value without knowledge of the secret
key
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Digital Signatures
 The hash value of a message is encrypted with
a user’s private key. Anyone who knows the
user’s public key can verify the integrity of the
message that is associated with the digital
signature.
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Simple Hash Functions
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Essentially based on compression
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Requirements
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Applications for Hash Functions
 Passwords
Never stored in plain
- Server stores only the hash value
- Salt (same plain goes to different hash)
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 Cracking
GPUs
- Dictionary attacks
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