cs457 – introduction to information systems security cryptography 1b

CS457 – Introduction to Information Systems Security

Cryptography 1b

Elias [email protected]

Elias Athanasopoulos 2

Cryptography Elements

Symmetric Encryption- Block Ciphers- Stream Ciphers

Asymmetric EncryptionCryptographic Hash FunctionsApplications

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The need for randomness

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The need for randomness

Key distributionReplay attacks (nonces)Session key generation Generation of keys for the RSA public-key

encryption algorithmStream ciphers

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Randomness

Uniform distribution - 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 - No one subsequence in the sequence can be

inferred from the others. Security requirement

- 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

- Occurrence of a zero or one is equally likely. The expected number of zeros (or ones) is n/2, where n = the sequence length

Scalability - 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- The behavior of a generator must be consistent across

starting values (seeds)

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Tests Frequency test

- Determine whether the number of ones and zeros in a sequence is approximately the same as would be expected for a truly random sequence

Runs test - 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

- 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- 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 - 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

- Designed specifically and solely for the purpose of generating pseudorandom bit streams.

Algorithms based on existing cryptographic algorithms - 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- Stream ciphers

Asymmetric ciphers- RSA, compute primes

Hash functions and message authentication codes

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Example

Xn+1 = (aXn + c) mod mSelection of a, c, and m, is very critical:a=7, c=0, m=32

- {7, 17, 23, 1, 7, etc.} a=5

- {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

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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]);

/* 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];


Asymmetric Encryption

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What is a prime number?

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An integer p > 1 is a prime number if and only if its only divisors are: 1, +p, and -p.

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More maths

Any integer a > 1 can be factored in a unique way as:

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Determining the prime factors of a large number is no easy task!

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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)

Easy algorithm, but hard to reverse- Y = f(X), easy- X = f-1(X), computationally hard - Computationally hard means solvable in non-

polynomial time

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RSA

Plaintext = M, cipher = C

C = Me mod nM = 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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n=pq, p, q are prime numbersφ(n) = φ(pq) =φ(p) φ(q) =(p-1)(q-1)

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Just believe me that this holds! (i.e., φ(pq) =φ(p) φ(q))


RSA Steps

p, q, two prime numbers- Private

n = pq- 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)

- Public- Recall φ(n) = (p-1)(q-1)

d from e, and φ(n)- Private

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RSA example

1. Select p = 17 and q = 11 2. Then, n = pq = 17×11 = 187. 3. φ(n) = (p-1)(q-1) = 16×10 = 160. 4. Select e relatively prime to φ(n) = 160 and less

than φ(n); e = 7. 5. 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 operationSide channels

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Cryptographic Hash Functions

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How it works?

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Integrity and Message Authentication

Integrity- (e.g., download a file)- Message digest

Message Authentication Code (MAC)- 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

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Passwords- Never stored in plain- Server stores only the hash value- Salt (same plain goes to different hash)

Cracking- GPUs- Dictionary attacks

cs457 – introduction to information systems security cryptography 1b

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