Lecture 1: Cryptography for Network Security

Anish Arora

CSE5473

Introduction to Network Security

Symmetric encryption

Symmetric encryption requirements

• two requirements for secure use of symmetric encryption:

a strong encryption algorithm

a secret key known only to sender / receiver

y= S ‹x› in notation of book : Y = EK(X)

x= S ‹y› X = DK(Y)

• assume encryption algorithm is known

• implies a secure channel to distribute key

Block versus stream ciphers

• block ciphers divide messages into blocks, each is then en/decrypted

like a substitution on very big characters 64-bits or more

would need table of 264 entries for a 64-bit block

instead create from smaller building blocks using idea of a product cipher

o substitution (S-box) provides confusiono permutation (P-box) provides diffusion

• stream ciphers process messages a bit or byte at a time typically have a (pseudo) random stream key

Block versus stream ciphers … contd

key should satisfy o statistical uniformity: of distribution of numbers in sequenceo unpredictability: of successive members of sequence

randomness of key destroys statistical properties in message

but must not reuse stream key e.g. RC4 used in SSL and WEP

• many current ciphers are block ciphers

• many symmetric block ciphers use Feistel Cipher Structure

Pseudo random functions (PRFs)

• A pseudo random function is an efficiently computable function that emulates a random oracle and there is no efficient algorithm for distinguishing a PRF function,

chosen randomly from its PRF family, from a random oracle

• Random oracle is a function that responds to every query with a random response from its range note that response is deterministic note that it is impossible to implement a random oracle

• Main property of PRF: all its outputs appear to be random assuming function is chosen in random

Pseudo random generators (PRGs)

• PRFs should not be confused with PRGs

• Main property of PRG a single output of a PRG appears to be random

• Cryptographically secure (CS) PRGs are used for generating a pool of randomness

for selecting keys, seeds, nonces, one-time pads

sources of physical randomness may suffice geiger counters, thermal noise measurement observing random activity on keyboard or screen

CSPRGs can be constructed using crypto primitives, or number theoretically

• PRFs can be generated from PRGs

Fesitel schema for symmetric encryption

• Overall processing at each iteration: use two 32-bit halves L and R

• Li = Ri-1

• Ri = Li-1 F(Ri-1, Ki)

Data Encryption Standard (DES)

• A widely used symmetric encryption scheme

• Algorithm is referred to as Data Encryption Algorithm (DEA)

• DES is a block cipher

• Plaintext is processed in 64-bit blocks

• Key is usually 56-bits in length

• n rounds, in each

every block first undergoes key-based substitution

then all blocks are collated and undergo key-based permutation

• Easy in hardware, slow in software

selection of block size, key size, #rounds, round function, subkey

generation scheme trades off security vs speed

The function F in DES

• takes 32-bit R half & 48-bit subkey and:• expands R to 48-bits using perm E• adds to subkey• passes through 8 S-boxes to get 32-bit result• finally permutes this using 32-bit perm P

DEA

Decryption runs backward

DES History

• IBM developed Lucifer cipher by team led by Feistel used 64-bit data blocks with 128-bit key

• Then redeveloped as a commercial cipher with input from NSA & others

• In 1973, NBS issued request for proposals for a national cipher standard

• IBM submitted their revised Lucifer which was eventually accepted as the DES

• DES standard is public

• But there has been considerable controversy over design in choice of 56-bit key (vs original Lucifer 128-bit) and because design criteria were classified

Breaking DES

Concerns about DEA

• Key length of only 56-bits is insufficient

• Even with larger keys, breaking is feasible if you have

known plaintext or have repeated encryptions generally these are statistical attacks

access to timing or power consumption information

use knowledge of implementation to derive subkey bits

exploit fact that calculations take varying times based on input value

particularly problematic on smartcards

Weaknesses in DES

• DES has Weak and Semi-Weak Keys: The round key construction is such that

Any key comprising All 0s, All 1s, Alternating 0s and 1a, or Alternating 1s and 0s is its own inverse (these are the 4 weak keys)

The set of keys composed of two halves each of the above sorts is such that each key has an inverse in this set (these are 12 semi-weak keys)

• Complement key property means that brute force search for DES is of complexity 255, not 256

DES Electronic Code Book

• In encryption via ECB, repeated 64-bit blocks are identically encrypted

• ECB attackers who know the data structure (e.g. fields such as salary) can reorder blocks while preserving structure

Cipher Block Chaining

• To overcome ECB weakness, add (i.e. XOR) a random number to each 64-bit block being encrypted, and additionally communicate the random number in the clear

• This is inefficient

• Approximation: only communicate the initial random number, and derive the successive random numbers from the previously encrypted message Initial random number is called the initialization vector Default IVs, such as All Zeroes, can be used, but is insecure

for repeated transmissions of the same message sequence

Cipher Block Chaining

A CBC threat

• If message structure is known, intruder can systematically ensure

that a modified message is delivered, by changing the previous

ciphertext

but then the previous plaintext is deciphered in a way not controlled by

intruder

• An alternative to CBC is the Counter Mode (CTR):

precompute encryptions of a counter value and XOR with

successive messages (this method enjoys parallelism)

to avoid reusing the same sequence of precomputed encryptions, prefix a

nonce (which is made public) to the counter value sequence

note: message blocks are not encrypted, just XOR-ed, unlike CBC mode

Multiple DES, 3DES

• Two successive encryptions with different keys are better than

one 56 bit key

E2.E1 to encrypt and D2.D1 to decrypt

Combinatorially, two keys yields more permutations than those

possible with one key

However, meet-in-the-middle cryptanalysis reduces complexity of

attack to 256, so net improvement is not large

• 3DES uses two keys: E1.D2.E1 to encrypt and D1.E2.D1 to decrypt

or three keys: E3.D2.E1 to encrypt and D3.E2.D1 to decrypt

Other symmetric block ciphers

• Blowfish Easy to implement High execution speed Run in less than 5K of memory Uses a 32 to 448 bit key

• RC5 Suitable for hardware and software Fast, simple, but proprietary Adaptable to processors of different word lengths Variable number of rounds Variable-length key Low memory requirement High security Data-dependent rotations

AES

• AES, Elliptic Curve, IDEA, Public Key cryptography concern numbers & finite fields

• US NIST issued call for ciphers in 1997 15 candidates accepted in Jun 98 5 were shortlisted in Aug 99 Rijndael was selected as the AES in Oct 2000 issued as a standard in Nov 2001

• Symmetric block cipher, 128-bit data, 128/192/256-bit keys provide full specification & design details both C & Java implementations NIST have released all submissions & unclassified analyses iterative (vs feistel) cipher, operates on entire block per round

Asymmetric encryption: Public key cryptography

Security of public key schemes

• brute force attacks infeasible since keys used are too large (> 512bits)

• security relies on a large computation difference in difficulty between easy (en/decrypt) and hard (cryptanalyse) problems

• the hard problem is generally known, it’s just made too hard to do in practice

• requires the use of very large numbers

• hence is slow compared to private key schemes

Background

• Asymmetric cryptography invented by Diffie and Helman

’76

• 3 categories of uses:

encryption/decryption (provide secrecy)

digital signatures (provide authentication)

key exchange (of session keys)

Authentication using public keys

RSA

• by Rivest, Shamir & Adleman of MIT in 1977

• best known & widely used public-key scheme

• based on exponentiation in a finite (Galois) field over integers modulo a prime

exponentiation takes O((log n)3) operations (easy)

• uses large integers (e.g. 1024 bits)

• security due to cost of factoring large numbers

factorization takes O(e log n log log n) operations (hard)

RSA

• To encrypt a message M the sender: obtain public key of recipient KU={e,N} computes: C=Me mod N, where 0≤M<N

• To decrypt the ciphertext C the receiver: uses its private key KR={d,p,q} computes: M=Cd mod N

• Message M is smaller than modulus N (so block if needed)

RSA key generation

1: determine two primes at random - p, q

• primes p,q must not be easily derived from mod N=p.q means must be sufficiently large typically guess and use probabilistic test

2: select either e or d and compute the other

• exponents e, d are inverses in mod (p-1).(q-1)

• the goal is that selection of d should be random and unpredictable e is computed using extended Euclidean algorithm

• it’s okay for e to be predictable (e.g. small), so encr. fast but e should not be too small, e.g. 3

RSA Example

1. Select primes: p=17 & q=11

2. Compute n = pq =17×11=187

3. Compute ø(n)=(p–1)(q-1)=16×10=160

4. Select e : gcd(e,160)=1; choose e=7

5. Determine d: de=1 mod 160 and d < 160

i.e. d=23 since 23×7=161= 10×160+1

1. Publish public key KU={7,187}

2. Keep secret private key KR={23,17,11}

RSA Example (contd.)

• sample RSA encryption/decryption is:

• given message M = 88 (note 88<187)

• encryption:

C = 887 mod 187 = 11

• decryption:

M = 1123 mod 187 = 88

Fermat’s Little Theorem

Theorem: If n is prime,

an-1 ≡ 1 mod n

Proof: {a mod n, 2a mod n, … (n-1)a mod n}

= since n is prime , a is not divisible by n

{1, 2, … n}

=>

a x 2a x … x (n-1)a ≡ (n-1)! mod n

=

an-1 ≡ 1 mod n

Why RSA Works

• because of Euler's Theorem:

aø(n)mod N = 1 where gcd(a,N)=1

• in RSA have:

N=p.q ø(N)=(p-1)(q-1) carefully chosen e & d to be inverses mod ø(N) hence e.d=1+k.ø(N) for some k

• hence :Cd = (Me)d = M1+k.ø(N) = M1.(Mø(N))q = M1.(1)q = M1 = M mod N

Security of RSA

• Factoring numbers is hard

• Breaking (p-1)(q-1) or d is not easier than factoring n

i.e. there is an easy way to factor n once (p-1)(q-1) is broken likewise if d is broken

Small M

• M=0 and M=1 are obviously not secure on encryption, even if e is very large

• Likewise, for other small M, Me would be smaller than N and Me mod N can be precomputed and checked by the passive eavesdropper

• Even if Alice adds a large salt s to M, the attacker can compute the encryption of s as well as its successors (s+1 ...) to guess M

Avoiding Message Guessing*

• Add random padding to make M large and unpredictable

• Public Key Cryptographic Standard: replace M with M’

M’=0 padding || 10 || >64 random bits || 00000000 || M

• OAEP Scheme: Optimal asymmetric encryption padding

sender XORs message with output of the cryptographic hash function with random input; recipient obtains random and XORs out cryptographic hash

* [David Evans]

Exponentiation

• can use the Square and Multiply Algorithm

• a fast, efficient algorithm for exponentiation

• concept is based on repeatedly squaring base

• and multiplying in the ones that are needed to compute the result

• look at binary representation of exponent

• only takes O(log2 n) multiples for number n

e.g. 75 = 74.71 = 3.7 = 10 mod 11 e.g. 3129 = 3128.31 = 5.3 = 4 mod 11

RSA (contd.)

• Due to Rivest, Shamir & Adleman of MIT in 1977

• Best known & widely used public-key scheme

• Based on exponentiation in a finite (Galois) field over integers modulo a prime exponentiation takes O((log n)3) operations (easy)

• Uses large integers (e.g. 1024 bits)

• Security due to cost of factoring large numbers factorization takes O(e log n log log n) operations (hard) barring dramatic breakthrough 1024+ bit RSA secure

• Timing attacks possible exploit time taken in exponentiation to infer operands countermeasures

use constant exponentiation time, add random delays

Hash functions

• a hash function produces a fingerprint of some file/message/data

h = H(M)

condenses a variable-length message M

to a fixed-sized fingerprint

• usually assume that the hash function is public, not keyed cf. MAC which is keyed

• hash used to detect changes to message, e.g. by creating a digital signature or fingerprint of a message

• cryptographically secure hashes also used to generate PRFs, e.g. to derive keys

Requirements for hash functions

1. can be applied to any sized message M

2. produces fixed-length output h

3. is easy to compute h=H(M) for any message M

4. given h is infeasible to find x s.t. H(x)=h• one-way property

5. given x is infeasible to find y s.t. H(y)=H(x)• weak collision resistance

6. is infeasible to find any x,y s.t. H(y)=H(x)• strong collision resistance

Simple hash functions

• there are several proposals for simple functions, based on XOR of message blocks:

e.g. longitudinal redundancy check: xor of columns of n-bit block arranged in rows

e.g. above+ circular left shift of hash after each row effect of rotated XOR (RXOR) is to randomize the input

• but these lack weak collision resistance simply “add a block” to obtain desired hash

• need a stronger cryptographic function, which tolerates strong collision resistance as well

More on weak collision resistance

• How big should our hash function be?

• If attacker can perform 263 hashes, hash function should have 64 bits is output so that probability that the attacker can find another message with the same hash is less than 0.5

• Assuming hash function distribution is uniform Prob (one guess gives same hash)= 2-64

Prob (the guess does not give same hash)= 1- 2-64

Prob (2L hash guesses dont give the same hash) = (1- 2-64)263

Birthday attacks imply need longer hash values

• You might think a 64-bit hash is secure, but by Birthday Paradox is not

• Given k people, what is the probability that there are two people with the same birthday

If birthdays uniformly distributed over 365 days, probability of no duplicates = 365x364x … x (365-k+1) / 365k

= 365!/((365-k)! x 365k)

Birthday attacks imply need longer hash values

• birthday attack on strong-collision resistance works thus:

opponent generates 2m/2 variations of a valid message all with

essentially the same meaning

opponent also generates 2m/2 variations of a desired fraudulent

message

two sets of messages are compared to find pair with same hash (probability > 0.5 by birthday paradox)

have user sign the valid message, then substitute the forgery which will have a valid signature

• conclusion is that need to use longer hash values

• also, you might wish to change every message you sign !

Hash algorithms

• similarities in evolution of hash functions & block ciphers increasing power of brute-force attacks led to evolution in

algorithms from DES to AES in block ciphers

from MD4 & MD5 to SHA-1 in hash algorithms

• likewise tend to use common iterative structure as do block ciphers iteration of collision-resistant round compression function

preserves collision resistance

• good round functions should have an avalanche effect small changes in input should have large changes in output

Block ciphers as hash functions

• can use block ciphers as hash functions

using H0=0 and zero-pad of final block

compute: Hi = EMi [Hi-1]

and use final block as the hash value

similar to cipher block chaining but without a key

• but resulting hash should not be too small (64-bit)

• like block ciphers have brute-force attacks, and a number of analytic attacks on iterated hash functions

MD5

• designed by Ronald Rivest (the R in RSA)

• latest in a series of MD2, MD4

• produces a 128-bit hash value

• until recently was the most widely used hash algorithm

in recent times had both brute-force & cryptanalytic concerns

• specified as Internet standard RFC1321

MD5 overview

1. pad message so its length is 448 mod 512

2. append a 64-bit length value to message

3. initialise 4-word (128-bit) MD buffer (A,B,C,D)

4. process message in 16-word (512-bit) blocks:

using 4 rounds of 16-bit operations on message block & buffer

add output to buffer input to form new buffer value

5. output hash value is the final buffer value

MD5 overview

MD4

• precursor to MD5

• also produces a 128-bit hash of message

• has 3 rounds of 16 steps vs 4 in MD5

• design goals:

collision resistant (hard to find collisions)

direct security (no dependence on "hard" problems)

fast, simple, compact

favours little-endian systems (e.g., PCs)

Strength of MD5

• MD5 hash is dependent on all message bits

• Rivest claimed security is as strong as can be with 128 bit code

• known attacks are:

Berson 92 attacked any 1 round using differential cryptanalysis (but can’t

extend)

Boer & Bosselaers 93 found a pseudo collision (again unable to extend)

Dobbertin 96 created collisions on MD compression function (but initial

constants prevent exploit)

conclusion was that MD5 should be vulnerable soon

• In 2004, an attack was found

Secure Hash Algorithm (SHA)

• SHA was designed by NIST & NSA in 1993, revised 1995 as SHA-1

• US standard for use with DSA signature scheme

standard is FIPS 180-1 1995, also Internet RFC3174

nb. the algorithm is SHA, the standard is SHS

• produces 160-bit hash values

• now the generally preferred hash algorithm SHA-1 is now regarded as broken (with a theoretical attack of 251) This year SHA-3 is being finalized

• based on design of MD4 with key differences

SHA overview

1. pad message so its length is 448 mod 512

2. append a 64-bit length value to message

3. initialise 5-word (160-bit) buffer (A,B,C,D,E) to

(67452301,efcdab89,98badcfe,10325476,c3d2e1f0)

4. process message in 16-word (512-bit) chunks: expand 16 words into 80 words by mixing & shifting use 4 rounds of 20 bit operations on message block & buffer add output to input to form new buffer value

5. output hash value is the final buffer value

SHA-1 verses MD5

• brute force attack is harder (160 vs 128 bits for MD5)

• was regarded as less vulnerable to attacks (compared to MD4/5), but this is no longer true

• a little slower than MD5 (80 vs 64 steps)

• both designed as simple and compact

• optimised for big endian CPU's (vs MD5 which is optimised for little endian CPU’s)

Revised secure hash standard

• NIST has issued a revision FIPS 180-2

• adds 3 additional hash algorithms

• SHA-256, SHA-384, SHA-512

• designed for compatibility with increased security provided by the AES cipher

• structure & detail is similar to SHA-1

• hence analysis should be similar

Reading on Crypto

Comparable to the extent covered in class, read

• Chapter 3: 3.1-3.4, 3.6

• Chapter 5: 5.1

• Chapter 6: 6.2-6.5

• Chapter 7: 7.4

• Chapter 9: 9.1-9.2

• Chapter 11: 11.4-11.5

• Chapter 12: 12.1-12.2