Lecture 1: Cryptography for Network Security

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Lecture 1 Cryptography for Network Security

• Anish Arora
• CSE5473
• Introduction to Network Security

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Symmetric encryption
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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

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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
• substitution (S-box) provides confusion
• permutation (P-box) provides diffusion
• stream ciphers process messages a bit or byte at
  a time
• typically have a (pseudo) random stream key

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Block versus stream ciphers contd

• key should satisfy
• statistical uniformity of distribution of
  numbers in sequence
• 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

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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

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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

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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)

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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

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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

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DEA
Decryption runs backward
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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

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Breaking DES
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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

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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

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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

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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

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Cipher Block Chaining
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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

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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

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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

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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

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Asymmetric encryption Public key cryptography
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Security of public key schemes

• brute force attacks infeasible since keys used
  are too large (gt 512bits)
• security relies on a large computation difference
  in difficulty between easy (en/decrypt) and hard
  (cryptanalyse) problems
• the hard problem is generally known, its just
  made too hard to do in practice
• requires the use of very large numbers
• hence is slow compared to private key schemes

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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)

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Authentication using public keys
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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)

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RSA

• To encrypt a message M the sender
• obtain public key of recipient KUe,N
• computes CMe mod N, where 0MltN
• To decrypt the ciphertext C the receiver
• uses its private key KRd,p,q
• computes MCd mod N
• Message M is smaller than modulus N (so block if
  needed)

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RSA key generation

• 1 determine two primes at random - p, q
• primes p,q must not be easily derived from mod
  Np.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
• its okay for e to be predictable (e.g. small),
  so encr. fast
• but e should not be too small, e.g. 3

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RSA Example

• Select primes p17 q11
• Compute n pq 1711187
• Compute ø(n)(p1)(q-1)1610160
• Select e gcd(e,160)1 choose e7
• Determine d de1 mod 160 and d lt 160
• i.e. d23 since 237161 101601
• Publish public key KU7,187
• Keep secret private key KR23,17,11

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RSA Example (contd.)

• sample RSA encryption/decryption is
• given message M 88 (note 88lt187)
• encryption
• C 887 mod 187 11
• decryption
• M 1123 mod 187 88

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RSA Key Setup

• Each user generates a public/private key pair by
  
• selecting two large primes at random - p, q
• computing their system modulus Np.q
• note ø(N)(p-1)(q-1)
• selecting at random the encryption key e
• where 1lteltø(N), gcd(e,ø(N))1
• solve following equation to find decryption key d
  
• e.d1 mod ø(N) and 0dN
• publish their public encryption key KUe,N
• keep secret private decryption key KRd,p,q

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Fermats 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
• gt
• a x 2a x x (n-1)a (n-1)! mod n
• an-1 1 mod n

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Why RSA Works

• because of Euler's Theorem
• aø(n)mod N 1 where gcd(a,N)1
• in RSA have
• Np.q
• ø(N)(p-1)(q-1)
• carefully chosen e d to be inverses mod ø(N)
• hence e.d1k.ø(N) for some k
• hence Cd (Me)d M1k.ø(N) M1.(Mø(N))q
  M1.(1)q M1 M mod N

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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

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Small M

• M0 and M1 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 (s1 ...) to guess M

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Avoiding Message Guessing

• Add random padding to make M large and
  unpredictable
• Public Key Cryptographic Standard replace M with
  M
• M0 padding 10 gt64 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

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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

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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

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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

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Requirements for hash functions

• can be applied to any sized message M
• produces fixed-length output h
• is easy to compute hH(M) for any message M
• given h is infeasible to find x s.t. H(x)h
• one-way property
• given x is infeasible to find y s.t. H(y)H(x)
• weak collision resistance
• is infeasible to find any x,y s.t. H(y)H(x)
• strong collision resistance

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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

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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

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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-k1) / 365k
• 365!/((365-k)! x 365k)

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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 gt 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 !

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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

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Block ciphers as hash functions

• can use block ciphers as hash functions
• using H00 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

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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

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MD5 overview

• pad message so its length is 448 mod 512
• append a 64-bit length value to message
• initialise 4-word (128-bit) MD buffer (A,B,C,D)
• 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
• output hash value is the final buffer value

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MD5 overview
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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)

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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 cant 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

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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

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SHA overview

• pad message so its length is 448 mod 512
• append a 64-bit length value to message
• initialise 5-word (160-bit) buffer (A,B,C,D,E) to
  
• (67452301,efcdab89,98badcfe,10325476,c3d2e1f0)
• 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
• output hash value is the final buffer value

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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 CPUs)

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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

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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