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UNIT III
PUBLIC KEY CRYPTOSYSTEM / ASYMMETRIC
CRYPTOSYSTEM
Prepared by
Dr. R. Arthy, AP/IT
Kamaraj College of Engineering and Technology (Autonomous), Madurai
CS8792 - CRYPTOGRAPHY AND
NETWORK SECURITY
Outline
 MATHEMATICS OF
ASYMMETRIC KEY
CRYPTOGRAPHY:
 Primes
 Primality Testing
 Factorization
 Euler„s totient function
 Fermat„s Theorem
 Euler„s Theorem
 Chinese Remainder Theorem
 Exponentiation and logarithm
 ASYMMETRIC KEY
CIPHERS:
 RSA cryptosystem
 Key distribution
 Key management
 Diffie Hellman key exchange
 Elgamal cryptosystem
 Elliptic curve arithmetic
 Elliptic curve cryptography
Prepared by R.Arthy, AP/IT
MATHEMATICS OF ASYMMETRIC
KEY CRYPTOGRAPHY
Prime Number
 A whole number greater than 1 that can not be made
by multiplying other whole numbers.
 eg. 2,3,5,7 are prime
 4,6,8,9,10 are not prime numbers are central to number
theory
 list of prime number less than 200 is:
 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83
89 97 101 103 107 109 113 127 131 137 139 149 151 157 163
167 173 179 181 191 193 197 199
Prepared by R.Arthy, AP/IT
Primality Testing
 To find large prime numbers
 Traditionally sieve using trial division
 ie. divide by all numbers (primes) in turn less than the
square root of the number
 only works for small numbers
 Alternatively can use statistical primality tests based
on properties of primes
 for which all primes numbers satisfy property
 but some composite numbers, called pseudo‐primes, also
satisfy the property
 can use a slower deterministic primality test
Prepared by R.Arthy, AP/IT
Primality Testing - Miller Rabin
Algorithm
 A test based on prime properties that result from
Fermat‟s Theorem
TEST (n) is:
1. Find integers k, q, k > 0, q odd, so that (n–1)=2kq
2. Select a random integer a, 1<a<n-1
3. if aq mod n = 1 then
4. return (“inconclusive");
5. for j = 0 to k – 1 do
6. if (a2jq mod n = n-1) then
7. return(“inconclusive")
8. return (“composite")
Prepared by R.Arthy, AP/IT
Prepared by R.Arthy, AP/IT
Prepared by R.Arthy, AP/IT
[contd…] Probabilistic Consideration
 if Miller‐Rabin returns “composite” the number is
definitely not prime
 otherwise is a prime or a pseudo‐prime
 chance it detects a pseudo‐prime is < 1/4 chance it
detects a pseudo prime is < /4
 hence if repeat test with different random a then
chance n is prime after t tests is:
 Pr(n prime after t tests) = 1‐4‐t
 eg. for t=10 this probability is > 0.99999
 could then use the deterministic AKS test
Prepared by R.Arthy, AP/IT
Factorization
 To factor a number n is to write it as a product of other
numbers: n=a x b x c
 note that factoring a number is relatively hard compared
to multiplying the factors together to generate the number
 The prime factorisation of a number n is when its
written as a product of primes
 eg. 91=7x13 ; 3600=24x32x52
Prepared by R.Arthy, AP/IT
Euler‟s Totient Function
 Number of elements in reduced set of residues
is called the Euler Totient Function ø(n)
 To compute ø(n) need to count number of
residues to be excluded
 In general need prime factorization
 for p (p prime)
 ø(p) = p-1
 for p.q (p,q prime)
 ø(p.q)=(p-1)x(q-1)
 eg. ø(37) = 36
ø(21) = (3–1)x(7–1) = 2x6 = 12
Prepared by R.Arthy, AP/IT
Euler‟s Theorem
 if n is a positive integer and a, n are coprime, then
aφ(n) ≡ 1 mod n
where φ(n) is the Euler's totient function.
Prepared by R.Arthy, AP/IT
Prepared by R.Arthy, AP/IT
Fermat‟s Theorem
 ap-1 = 1 (mod p)
where p is prime and gcd(a,p)=1
also known as Fermat‟s Little Theorem
also have: ap = a (mod p) • useful in public
key and primality testing
Prepared by R.Arthy, AP/IT
Prepared by R.Arthy, AP/IT
Chinese Remainder Theorem
 Let us consider the set of equations
a = x1 mod m1
a = x2 mod m2
…
a = xk mod mk
 Then there exist a unique solution if the m1, m2,.. are
pairwise coprime (i.e.) gcd(mi, mj) = 1
Prepared by R.Arthy, AP/IT
[contd…]
 Steps:
 Find M = m1 * m2 * m3 *… *mk
 Calculate
M1 = M/m1, M2 = M/m2, …, Mk = M/mk
 Find the modulo inverse of Mi, i.e. M-1
1, M-1
2, …, M-1
k
 Unique solution is
a = ((x1 * M1 * M-1
1) + (x2 * M2 * M-1
2) + … + (xk * Mk *
M-1
k)) mod M
Prepared by R.Arthy, AP/IT
Prepared by R.Arthy, AP/IT
Prepared by R.Arthy, AP/IT
Primitive Roots
 From Euler‟s theorem have aø(n)mod n=1
 Consider am=1 (mod n), GCD(a,n)=1
 must exist for m = ø(n) but may be smaller
 once powers reach m, cycle will repeat
 If smallest is m = ø(n) then a is called a primitive root
 if p is prime, then successive powers of a "generate"
the group mod p
 These are useful but relatively hard to find
Prepared by R.Arthy, AP/IT
Exponentiation and Logarithm
 The inverse problem to exponentiation is to find the
discrete logarithm of a number modulo p
 That is to find i such that b = ai (mod p) • this is
written as i = dloga b (mod p)
 If a is a primitive root then it always exists, otherwise
it may not
 eg. x = log3 4 mod 13 has no answer x = log2 3 mod 13 =
4 by trying successive powers
 whilst exponentiation is relatively easy, finding
discrete logarithms is generally a hard problem
Prepared by R.Arthy, AP/IT
ASYMMETRIC KEY CIPHERS
Private-Key Cryptography
 Traditional private/secret/single key cryptography
uses one key
 Shared by both sender and receiver
 If this key is disclosed communications are
compromised
 also is symmetric, parties are equal
 Hence does not protect sender from receiver forging a
message & claiming is sent by sender
Prepared by R.Arthy, AP/IT
Public-Key Cryptography
 Probably most significant advance in the 3000 year
history of cryptography
 Uses two keys – a public & a private key
 Asymmetric since parties are not equal
 Uses clever application of number theoretic concepts
to function
 Complements rather than replaces private key crypto
Prepared by R.Arthy, AP/IT
Public-Key Cryptography
 Public-key/two-key/asymmetric cryptography
involves the use of two keys:
 A public-key, which may be known by anybody, and can
be used to encrypt messages, and verify signatures
 A private-key, known only to the recipient, used to
decrypt messages, and sign (create) signatures
 Is asymmetric because
 Those who encrypt messages or verify signatures cannot
decrypt messages or create signatures
Prepared by R.Arthy, AP/IT
Public-Key Cryptography
Prepared by R.Arthy, AP/IT
Why Public-Key Cryptography?
 Developed to address two key issues:
 Key distribution – how to have secure communications
in general without having to trust a kdc with your key
 Digital signatures – how to verify a message comes
intact from the claimed sender
 Public invention due to whitfield diffie & martin
hellman at stanford uni in 1976
 Known earlier in classified community
Prepared by R.Arthy, AP/IT
Public-Key Characteristics
 Public-key algorithms rely on two keys with the
characteristics that it is:
 Computationally infeasible to find decryption key
knowing only algorithm & encryption key
 Computationally easy to en/decrypt messages when the
relevant (en/decrypt) key is known
 Either of the two related keys can be used for encryption,
with the other used for decryption (in some schemes)
Prepared by R.Arthy, AP/IT
Public-Key Cryptosystems
Prepared by R.Arthy, AP/IT
Public-Key Applications
 Can classify uses into 3 categories:
 Encryption/decryption (provide secrecy)
 Digital signatures (provide authentication)
 Key exchange (of session keys)
 Some algorithms are suitable for all uses, others are
specific to one
Prepared by R.Arthy, AP/IT
Security of Public Key Schemes
 Like private key schemes brute force exhaustive
search attack is always theoretically possible
 But keys used are too large (>512bits)
 Security relies on a large enough difference in
difficulty between easy (en/decrypt) and hard
(cryptanalyse) problems
 More generally the hard problem is known, its just
made too hard to do in practise
 Requires the use of very large numbers
 Hence is slow compared to private key schemes
Prepared by R.Arthy, AP/IT
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
 Nb. Exponentiation takes o((log n)3) operations (easy)
 Uses large integers (eg. 1024 bits)
 Security due to cost of factoring large numbers
 Nb. Factorization takes o(e log n log log n) operations (hard)
Prepared by R.Arthy, AP/IT
RSA Key Setup
 Each user generates a public/private key pair by:
 Selecting two large primes at random - p, q
 Computing their system modulus n=p.Q
 Note ø(n)=(p-1)(q-1)
 Selecting at random the encryption key e
 Where 1<e<ø(n), gcd(e,ø(n))=1
 Solve following equation to find decryption key d
 E.D=1 mod ø(n) and 0≤d≤n
 Publish their public encryption key: ku={e,n}
 Keep secret private decryption key: kr={d,p,q}
Prepared by R.Arthy, AP/IT
RSA Use
 To encrypt a message M the sender:
 Obtains public key of recipient ku={e,n}
 Computes: c=me mod N, where 0≤M<N
 To decrypt the ciphertext C the owner:
 Uses their private key kr={d,p,q}
 Computes: m=cd mod N
 Note that the message M must be smaller than the
modulus N (block if needed)
Prepared by R.Arthy, AP/IT
Prepared by R.Arthy, AP/IT
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
Prepared by R.Arthy, AP/IT
Prepared by R.Arthy, AP/IT
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
 Eg. 75 = 74.71 = 3.7 = 10 mod 11
 Eg. 3129 = 3128.31 = 5.3 = 4 mod 11
Prepared by R.Arthy, AP/IT
Exponentiation
Prepared by R.Arthy, AP/IT
RSA Key Generation
 Users of RSA must:
 Determine two primes at random - p, q
 Select either e or d and compute the other
 Primes p,q must not be easily derived from modulus
n=p.Q
 Means must be sufficiently large
 Typically guess and use probabilistic test
 Exponents e, d are inverses, so use inverse algorithm
to compute the other
Prepared by R.Arthy, AP/IT
RSA Security
 Three approaches to attacking RSA:
 Brute force key search (infeasible given size of numbers)
 Mathematical attacks (based on difficulty of computing
ø(n), by factoring modulus N)
 Timing attacks (on running of decryption)
Prepared by R.Arthy, AP/IT
Factoring Problem
 Mathematical approach takes 3 forms:
 Factor n=p.Q, hence find ø(n) and then d
 Determine ø(n) directly and find d
 Find d directly
 Currently believe all equivalent to factoring
 Have seen slow improvements over the years
 As of aug-99 best is 130 decimal digits (512) bit with GNFS
 Biggest improvement comes from improved algorithm
 Cf “quadratic sieve” to “generalized number field sieve”
 Barring dramatic breakthrough 1024+ bit RSA secure
 Ensure p, q of similar size and matching other constraints
Prepared by R.Arthy, AP/IT
Timing Attacks
 Developed in mid-1990‟s
 Exploit timing variations in operations
 Eg. Multiplying by small vs large number
 Or if's varying which instructions executed
 Infer operand size based on time taken
 RSA exploits time taken in exponentiation
 Countermeasures
 Use constant exponentiation time
 Add random delays
 Blind values used in calculations
Prepared by R.Arthy, AP/IT
KEY DISTRIBUTION AND KEY
MANAGEMENT
Symmetric Key Distribution Using
Symmetric Encryption
 Symmetric schemes require both parties to share a
common secret key
 Issue is how to securely distribute this key
 Frequent key changes can be desirable
 Often secure system failure due to a break in the key
distribution scheme
Prepared by R.Arthy, AP/IT
[contd…]
 Given parties A and B have various key distribution
alternatives:
 A can select key and physically deliver to B
 Third party can select & deliver key to A & B
 If A & B have communicated previously can use
previous key to encrypt a new key
 If A & B have secure communications with a third party
C, C can relay key between A & B
Prepared by R.Arthy, AP/IT
[contd…]
 Key distribution using Key Distribution Center (KDC)
Prepared by R.Arthy, AP/IT
[contd…]
 Hierarchies of KDC‟s required for large networks, but
must trust each other
 Session key lifetimes should be limited for greater
security
 Use of automatic key distribution on behalf of users,
but must trust system
 Use of decentralized key distribution
 Controlling key usage
Prepared by R.Arthy, AP/IT
Symmetric Key Distribution Using
Public-key Encryption
 Public key cryptosystems are inefficient
 So almost never use for direct data encryption
 Rather use to encrypt secret keys for distribution
 Merkle proposed this very simple scheme
 allows secure communications
 no keys before/after exist
Prepared by R.Arthy, AP/IT
[contd…]
 Secret Key Distribution with Confidentiality and
Authentication
Prepared by R.Arthy, AP/IT
Distribution of Public Keys
 Public announcement
 Publicly available directory
 Public-key authority
 Public-key certificates
Prepared by R.Arthy, AP/IT
Public Announcement
 Users distribute public keys to recipients or broadcast
to community at large
 Eg. Append PGP keys to email messages or post to news
groups or email list
 Major weakness is forgery
 Anyone can create a key claiming to be someone else and
broadcast it
 Until forgery is discovered can masquerade as claimed
user
Prepared by R.Arthy, AP/IT
Publicly Available Directory
 Can obtain greater security by registering keys with a
public directory
 Directory must be trusted with properties:
 Contains {name,public-key} entries
 Participants register securely with directory
 Participants can replace key at any time
 Directory is periodically published
 Directory can be accessed electronically
 Still vulnerable to tampering or forgery
Prepared by R.Arthy, AP/IT
Public-Key Authority
 Improve security by tightening control over
distribution of keys from directory
 Has properties of directory
 And requires users to know public key for the
directory
 Then users interact with directory to obtain any desired
public key securely
 Does require real-time access to directory when keys are
needed
Prepared by R.Arthy, AP/IT
Public-Key Authority
Prepared by R.Arthy, AP/IT
Public-Key Certificates
 Certificates allow key exchange without real-time
access to public-key authority
 A certificate binds identity to public key
 Usually with other info such as period of validity, rights
of use etc
 With all contents signed by a trusted public-key or
certificate authority (CA)
 Can be verified by anyone who knows the public-key
authorities public-key
Prepared by R.Arthy, AP/IT
Public-Key Certificates
Prepared by R.Arthy, AP/IT
DIFFIE HELLMAN KEY
EXCHANGE ALGORITHM
Diffie-Hellman Key Exchange
 The Diffie–Hellman (DH) key exchange technique
was first defined in their seminal paper in 1976.
 DH key exchange is a method of exchanging public
(i.e. non-secret) information to obtain a shared secret.
 DH is not an encryption algorithm.
Prepared by R.Arthy, AP/IT
Diffie-Hellman Key Exchange
 DH key exchange has the following important
properties:
 The resulting shared secret cannot be computed by either
of the parties without the cooperation of the other.
 A third party observing all the messages transmitted
during DH key exchange cannot deduce the resulting
shared secret at the end of the protocol.
Prepared by R.Arthy, AP/IT
[contd…]
 A public-key distribution scheme
 Cannot be used to exchange an arbitrary message
 Rather it can establish a common key
 Known only to the two participants
 Value of key depends on the participants (and their
private and public key information)
 Based on exponentiation in a finite (galois) field
(modulo a prime or a polynomial) - easy
 Security relies on the difficulty of computing discrete
logarithms (similar to factoring) – hard
Prepared by R.Arthy, AP/IT
Diffie-Hellman Setup
 All users agree on global parameters:
 Large prime integer or polynomial q
 Α a primitive root mod q
 Each user (eg. A) generates their key
 Chooses a secret key (number): xa < q
 Compute their public key: ya = α
xa
mod q
 Each user makes public that key ya
Prepared by R.Arthy, AP/IT
Diffie-Hellman Key Exchange
 shared session key for users A & B is KAB:
KAB = α
xA.xB
mod q
= yA
xB
mod q (which B can compute)
= yB
xA
mod q (which A can compute)
 KAB is used as session key in private-key
encryption scheme between Alice and Bob
 if Alice and Bob subsequently communicate, they
will have the same key as before, unless they
choose new public-keys
 attacker needs an x, must solve discrete log
Prepared by R.Arthy, AP/IT
[contd…]
Prepared by R.Arthy, AP/IT
Prepared by R.Arthy, AP/IT
Prepared by R.Arthy, AP/IT
Man-in-the-middle Attack
1.Darth prepares by creating two private / public keys
2.Alice transmits her public key to Bob
3.Darth intercepts this and transmits his first public key to
Bob. Darth also calculates a shared key with Alice
4.Bob receives the public key and calculates the shared key
(with Darth instead of Alice)
5.Bob transmits his public key to Alice
6.Darth intercepts this and transmits his second public key
to Alice. Darth calculates a shared key with Bob
7.Alice receives the key and calculates the shared key (with
Darth instead of Bob)
 Darth can then intercept, decrypt, re-encrypt, forward all
messages between Alice & Bob Prepared by R.Arthy, AP/IT
ELGAMAL CRYPTOGRAPHY
Objective
 To show that RSA is not the only public key system
 To exhibit a public key system based on a different one
way function
 ElGamal is the basis for several well-known
cryptographic primitives
Prepared by R.Arthy, AP/IT
Elgamal Key Exchange
 Public-key cryptosystem related to D-H
 Uses exponentiation in a finite field
 With security based difficulty of computing discrete
logarithms, as in D-H
 Each user (e.g. A) generates their key
 Chooses a secret key (number): 1 < XA < q-1
 Computes their public key: YA = a XA mod q
Prepared by R.Arthy, AP/IT
Elgamal Key Exchange
 Bob encrypt a message to send to A computing
 represent message M in range 0 <= M <= q-1
 longer messages must be sent as blocks
 chose random integer k with 1 <= k <= q-1
 compute one-time key K = YA
k mod q
 encrypt M as a pair of integers (C1,C2) where
 C1 = ak mod q ; C2 = KM mod q
 A then recovers message by
 recovering key K as K = C1
xA mod q
 computing M as M = C2 K-1 mod q
 a unique k must be used each time
 otherwise result is insecure Prepared by R.Arthy, AP/IT
Prepared by R.Arthy, AP/IT
Prepared by R.Arthy, AP/IT
ELLIPTIC CURVE
CRYPTOGRAPHY
Elliptic Curve Cryptography
 Majority of public-key crypto (RSA, D-H) use either
integer or polynomial arithmetic with very large
numbers/polynomials
 Imposes a significant load in storing and processing
keys and messages
 An alternative is to use elliptic curves
 Offers same security with smaller bit sizes
Prepared by R.Arthy, AP/IT
Real Elliptic Curves
 an elliptic curve is defined by an equation in two
variables x & y, with coefficients
 consider a cubic elliptic curve of form
 y2 = x3 + ax + b
 where x,y,a,b are all real numbers
 also define zero point O
 have addition operation for elliptic curve
 geometrically sum of Q+R is reflection of intersection R
Prepared by R.Arthy, AP/IT
Real Elliptic Curve Example
Prepared by R.Arthy, AP/IT
Finite Elliptic Curves
 Elliptic curve cryptography uses curves whose
variables & coefficients are finite
 Have two families commonly used:
 Prime curves ep(a,b) defined over zp
 Use integers modulo a prime
 Best in software
 Binary curves e2m(a,b) defined over gf(2n)
 Use polynomials with binary coefficients
 Best in hardware
Prepared by R.Arthy, AP/IT
Elliptic Curve Cryptography
 ECC addition is analog of modulo multiply
 ECC repeated addition is analog of modulo
exponentiation
 Need “hard” problem equiv to discrete log
 Q=kp, where Q,P belong to a prime curve
 Is “easy” to compute Q given k,p
 But “hard” to find k given Q,P
 Known as the elliptic curve logarithm problem
 Certicom example: E23(9,17)
Prepared by R.Arthy, AP/IT
ECC Diffie-Hellman
 can do key exchange analogous to D-H
 users select a suitable curve Ep(a,b)
 select base point G=(x1,y1) with large order n s.t. nG=O
 A & B select private keys nA<n, nB<n
 compute public keys: PA=nA×G, PB=nB×G
 compute shared key: K=nA×PB, K=nB×PA
 same since K=nA×nB×G
Prepared by R.Arthy, AP/IT
ECC Encryption/Decryption
 several alternatives, will consider simplest
 must first encode any message M as a point on the
elliptic curve Pm
 select suitable curve & point G as in D-H
 each user chooses private key nA<n
 and computes public key PA=nA×G
 to encrypt Pm : Cm={kG, Pm+k Pb}, k random
 decrypt Cm compute:
Pm+kPb–nB(kG) = Pm+k(nBG)–nB(kG) = Pm
Prepared by R.Arthy, AP/IT
ECC Security
 relies on elliptic curve logarithm problem
 fastest method is “Pollard rho method”
 compared to factoring, can use much smaller key sizes
than with RSA etc
 for equivalent key lengths computations are roughly
equivalent
 hence for similar security ECC offers significant
computational advantages
Prepared by Dr. R.Arthy, AP/IT

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Cs8792 cns - Public key cryptosystem (Unit III)

  • 1. UNIT III PUBLIC KEY CRYPTOSYSTEM / ASYMMETRIC CRYPTOSYSTEM Prepared by Dr. R. Arthy, AP/IT Kamaraj College of Engineering and Technology (Autonomous), Madurai CS8792 - CRYPTOGRAPHY AND NETWORK SECURITY
  • 2. Outline  MATHEMATICS OF ASYMMETRIC KEY CRYPTOGRAPHY:  Primes  Primality Testing  Factorization  Euler„s totient function  Fermat„s Theorem  Euler„s Theorem  Chinese Remainder Theorem  Exponentiation and logarithm  ASYMMETRIC KEY CIPHERS:  RSA cryptosystem  Key distribution  Key management  Diffie Hellman key exchange  Elgamal cryptosystem  Elliptic curve arithmetic  Elliptic curve cryptography Prepared by R.Arthy, AP/IT
  • 4. Prime Number  A whole number greater than 1 that can not be made by multiplying other whole numbers.  eg. 2,3,5,7 are prime  4,6,8,9,10 are not prime numbers are central to number theory  list of prime number less than 200 is:  2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 Prepared by R.Arthy, AP/IT
  • 5. Primality Testing  To find large prime numbers  Traditionally sieve using trial division  ie. divide by all numbers (primes) in turn less than the square root of the number  only works for small numbers  Alternatively can use statistical primality tests based on properties of primes  for which all primes numbers satisfy property  but some composite numbers, called pseudo‐primes, also satisfy the property  can use a slower deterministic primality test Prepared by R.Arthy, AP/IT
  • 6. Primality Testing - Miller Rabin Algorithm  A test based on prime properties that result from Fermat‟s Theorem TEST (n) is: 1. Find integers k, q, k > 0, q odd, so that (n–1)=2kq 2. Select a random integer a, 1<a<n-1 3. if aq mod n = 1 then 4. return (“inconclusive"); 5. for j = 0 to k – 1 do 6. if (a2jq mod n = n-1) then 7. return(“inconclusive") 8. return (“composite") Prepared by R.Arthy, AP/IT
  • 9. [contd…] Probabilistic Consideration  if Miller‐Rabin returns “composite” the number is definitely not prime  otherwise is a prime or a pseudo‐prime  chance it detects a pseudo‐prime is < 1/4 chance it detects a pseudo prime is < /4  hence if repeat test with different random a then chance n is prime after t tests is:  Pr(n prime after t tests) = 1‐4‐t  eg. for t=10 this probability is > 0.99999  could then use the deterministic AKS test Prepared by R.Arthy, AP/IT
  • 10. Factorization  To factor a number n is to write it as a product of other numbers: n=a x b x c  note that factoring a number is relatively hard compared to multiplying the factors together to generate the number  The prime factorisation of a number n is when its written as a product of primes  eg. 91=7x13 ; 3600=24x32x52 Prepared by R.Arthy, AP/IT
  • 11. Euler‟s Totient Function  Number of elements in reduced set of residues is called the Euler Totient Function ø(n)  To compute ø(n) need to count number of residues to be excluded  In general need prime factorization  for p (p prime)  ø(p) = p-1  for p.q (p,q prime)  ø(p.q)=(p-1)x(q-1)  eg. ø(37) = 36 ø(21) = (3–1)x(7–1) = 2x6 = 12 Prepared by R.Arthy, AP/IT
  • 12. Euler‟s Theorem  if n is a positive integer and a, n are coprime, then aφ(n) ≡ 1 mod n where φ(n) is the Euler's totient function. Prepared by R.Arthy, AP/IT
  • 14. Fermat‟s Theorem  ap-1 = 1 (mod p) where p is prime and gcd(a,p)=1 also known as Fermat‟s Little Theorem also have: ap = a (mod p) • useful in public key and primality testing Prepared by R.Arthy, AP/IT
  • 16. Chinese Remainder Theorem  Let us consider the set of equations a = x1 mod m1 a = x2 mod m2 … a = xk mod mk  Then there exist a unique solution if the m1, m2,.. are pairwise coprime (i.e.) gcd(mi, mj) = 1 Prepared by R.Arthy, AP/IT
  • 17. [contd…]  Steps:  Find M = m1 * m2 * m3 *… *mk  Calculate M1 = M/m1, M2 = M/m2, …, Mk = M/mk  Find the modulo inverse of Mi, i.e. M-1 1, M-1 2, …, M-1 k  Unique solution is a = ((x1 * M1 * M-1 1) + (x2 * M2 * M-1 2) + … + (xk * Mk * M-1 k)) mod M Prepared by R.Arthy, AP/IT
  • 20. Primitive Roots  From Euler‟s theorem have aø(n)mod n=1  Consider am=1 (mod n), GCD(a,n)=1  must exist for m = ø(n) but may be smaller  once powers reach m, cycle will repeat  If smallest is m = ø(n) then a is called a primitive root  if p is prime, then successive powers of a "generate" the group mod p  These are useful but relatively hard to find Prepared by R.Arthy, AP/IT
  • 21. Exponentiation and Logarithm  The inverse problem to exponentiation is to find the discrete logarithm of a number modulo p  That is to find i such that b = ai (mod p) • this is written as i = dloga b (mod p)  If a is a primitive root then it always exists, otherwise it may not  eg. x = log3 4 mod 13 has no answer x = log2 3 mod 13 = 4 by trying successive powers  whilst exponentiation is relatively easy, finding discrete logarithms is generally a hard problem Prepared by R.Arthy, AP/IT
  • 23. Private-Key Cryptography  Traditional private/secret/single key cryptography uses one key  Shared by both sender and receiver  If this key is disclosed communications are compromised  also is symmetric, parties are equal  Hence does not protect sender from receiver forging a message & claiming is sent by sender Prepared by R.Arthy, AP/IT
  • 24. Public-Key Cryptography  Probably most significant advance in the 3000 year history of cryptography  Uses two keys – a public & a private key  Asymmetric since parties are not equal  Uses clever application of number theoretic concepts to function  Complements rather than replaces private key crypto Prepared by R.Arthy, AP/IT
  • 25. Public-Key Cryptography  Public-key/two-key/asymmetric cryptography involves the use of two keys:  A public-key, which may be known by anybody, and can be used to encrypt messages, and verify signatures  A private-key, known only to the recipient, used to decrypt messages, and sign (create) signatures  Is asymmetric because  Those who encrypt messages or verify signatures cannot decrypt messages or create signatures Prepared by R.Arthy, AP/IT
  • 27. Why Public-Key Cryptography?  Developed to address two key issues:  Key distribution – how to have secure communications in general without having to trust a kdc with your key  Digital signatures – how to verify a message comes intact from the claimed sender  Public invention due to whitfield diffie & martin hellman at stanford uni in 1976  Known earlier in classified community Prepared by R.Arthy, AP/IT
  • 28. Public-Key Characteristics  Public-key algorithms rely on two keys with the characteristics that it is:  Computationally infeasible to find decryption key knowing only algorithm & encryption key  Computationally easy to en/decrypt messages when the relevant (en/decrypt) key is known  Either of the two related keys can be used for encryption, with the other used for decryption (in some schemes) Prepared by R.Arthy, AP/IT
  • 30. Public-Key Applications  Can classify uses into 3 categories:  Encryption/decryption (provide secrecy)  Digital signatures (provide authentication)  Key exchange (of session keys)  Some algorithms are suitable for all uses, others are specific to one Prepared by R.Arthy, AP/IT
  • 31. Security of Public Key Schemes  Like private key schemes brute force exhaustive search attack is always theoretically possible  But keys used are too large (>512bits)  Security relies on a large enough difference in difficulty between easy (en/decrypt) and hard (cryptanalyse) problems  More generally the hard problem is known, its just made too hard to do in practise  Requires the use of very large numbers  Hence is slow compared to private key schemes Prepared by R.Arthy, AP/IT
  • 32. 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  Nb. Exponentiation takes o((log n)3) operations (easy)  Uses large integers (eg. 1024 bits)  Security due to cost of factoring large numbers  Nb. Factorization takes o(e log n log log n) operations (hard) Prepared by R.Arthy, AP/IT
  • 33. RSA Key Setup  Each user generates a public/private key pair by:  Selecting two large primes at random - p, q  Computing their system modulus n=p.Q  Note ø(n)=(p-1)(q-1)  Selecting at random the encryption key e  Where 1<e<ø(n), gcd(e,ø(n))=1  Solve following equation to find decryption key d  E.D=1 mod ø(n) and 0≤d≤n  Publish their public encryption key: ku={e,n}  Keep secret private decryption key: kr={d,p,q} Prepared by R.Arthy, AP/IT
  • 34. RSA Use  To encrypt a message M the sender:  Obtains public key of recipient ku={e,n}  Computes: c=me mod N, where 0≤M<N  To decrypt the ciphertext C the owner:  Uses their private key kr={d,p,q}  Computes: m=cd mod N  Note that the message M must be smaller than the modulus N (block if needed) Prepared by R.Arthy, AP/IT
  • 36. 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 Prepared by R.Arthy, AP/IT
  • 38. 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  Eg. 75 = 74.71 = 3.7 = 10 mod 11  Eg. 3129 = 3128.31 = 5.3 = 4 mod 11 Prepared by R.Arthy, AP/IT
  • 40. RSA Key Generation  Users of RSA must:  Determine two primes at random - p, q  Select either e or d and compute the other  Primes p,q must not be easily derived from modulus n=p.Q  Means must be sufficiently large  Typically guess and use probabilistic test  Exponents e, d are inverses, so use inverse algorithm to compute the other Prepared by R.Arthy, AP/IT
  • 41. RSA Security  Three approaches to attacking RSA:  Brute force key search (infeasible given size of numbers)  Mathematical attacks (based on difficulty of computing ø(n), by factoring modulus N)  Timing attacks (on running of decryption) Prepared by R.Arthy, AP/IT
  • 42. Factoring Problem  Mathematical approach takes 3 forms:  Factor n=p.Q, hence find ø(n) and then d  Determine ø(n) directly and find d  Find d directly  Currently believe all equivalent to factoring  Have seen slow improvements over the years  As of aug-99 best is 130 decimal digits (512) bit with GNFS  Biggest improvement comes from improved algorithm  Cf “quadratic sieve” to “generalized number field sieve”  Barring dramatic breakthrough 1024+ bit RSA secure  Ensure p, q of similar size and matching other constraints Prepared by R.Arthy, AP/IT
  • 43. Timing Attacks  Developed in mid-1990‟s  Exploit timing variations in operations  Eg. Multiplying by small vs large number  Or if's varying which instructions executed  Infer operand size based on time taken  RSA exploits time taken in exponentiation  Countermeasures  Use constant exponentiation time  Add random delays  Blind values used in calculations Prepared by R.Arthy, AP/IT
  • 44. KEY DISTRIBUTION AND KEY MANAGEMENT
  • 45. Symmetric Key Distribution Using Symmetric Encryption  Symmetric schemes require both parties to share a common secret key  Issue is how to securely distribute this key  Frequent key changes can be desirable  Often secure system failure due to a break in the key distribution scheme Prepared by R.Arthy, AP/IT
  • 46. [contd…]  Given parties A and B have various key distribution alternatives:  A can select key and physically deliver to B  Third party can select & deliver key to A & B  If A & B have communicated previously can use previous key to encrypt a new key  If A & B have secure communications with a third party C, C can relay key between A & B Prepared by R.Arthy, AP/IT
  • 47. [contd…]  Key distribution using Key Distribution Center (KDC) Prepared by R.Arthy, AP/IT
  • 48. [contd…]  Hierarchies of KDC‟s required for large networks, but must trust each other  Session key lifetimes should be limited for greater security  Use of automatic key distribution on behalf of users, but must trust system  Use of decentralized key distribution  Controlling key usage Prepared by R.Arthy, AP/IT
  • 49. Symmetric Key Distribution Using Public-key Encryption  Public key cryptosystems are inefficient  So almost never use for direct data encryption  Rather use to encrypt secret keys for distribution  Merkle proposed this very simple scheme  allows secure communications  no keys before/after exist Prepared by R.Arthy, AP/IT
  • 50. [contd…]  Secret Key Distribution with Confidentiality and Authentication Prepared by R.Arthy, AP/IT
  • 51. Distribution of Public Keys  Public announcement  Publicly available directory  Public-key authority  Public-key certificates Prepared by R.Arthy, AP/IT
  • 52. Public Announcement  Users distribute public keys to recipients or broadcast to community at large  Eg. Append PGP keys to email messages or post to news groups or email list  Major weakness is forgery  Anyone can create a key claiming to be someone else and broadcast it  Until forgery is discovered can masquerade as claimed user Prepared by R.Arthy, AP/IT
  • 53. Publicly Available Directory  Can obtain greater security by registering keys with a public directory  Directory must be trusted with properties:  Contains {name,public-key} entries  Participants register securely with directory  Participants can replace key at any time  Directory is periodically published  Directory can be accessed electronically  Still vulnerable to tampering or forgery Prepared by R.Arthy, AP/IT
  • 54. Public-Key Authority  Improve security by tightening control over distribution of keys from directory  Has properties of directory  And requires users to know public key for the directory  Then users interact with directory to obtain any desired public key securely  Does require real-time access to directory when keys are needed Prepared by R.Arthy, AP/IT
  • 56. Public-Key Certificates  Certificates allow key exchange without real-time access to public-key authority  A certificate binds identity to public key  Usually with other info such as period of validity, rights of use etc  With all contents signed by a trusted public-key or certificate authority (CA)  Can be verified by anyone who knows the public-key authorities public-key Prepared by R.Arthy, AP/IT
  • 59. Diffie-Hellman Key Exchange  The Diffie–Hellman (DH) key exchange technique was first defined in their seminal paper in 1976.  DH key exchange is a method of exchanging public (i.e. non-secret) information to obtain a shared secret.  DH is not an encryption algorithm. Prepared by R.Arthy, AP/IT
  • 60. Diffie-Hellman Key Exchange  DH key exchange has the following important properties:  The resulting shared secret cannot be computed by either of the parties without the cooperation of the other.  A third party observing all the messages transmitted during DH key exchange cannot deduce the resulting shared secret at the end of the protocol. Prepared by R.Arthy, AP/IT
  • 61. [contd…]  A public-key distribution scheme  Cannot be used to exchange an arbitrary message  Rather it can establish a common key  Known only to the two participants  Value of key depends on the participants (and their private and public key information)  Based on exponentiation in a finite (galois) field (modulo a prime or a polynomial) - easy  Security relies on the difficulty of computing discrete logarithms (similar to factoring) – hard Prepared by R.Arthy, AP/IT
  • 62. Diffie-Hellman Setup  All users agree on global parameters:  Large prime integer or polynomial q  Α a primitive root mod q  Each user (eg. A) generates their key  Chooses a secret key (number): xa < q  Compute their public key: ya = α xa mod q  Each user makes public that key ya Prepared by R.Arthy, AP/IT
  • 63. Diffie-Hellman Key Exchange  shared session key for users A & B is KAB: KAB = α xA.xB mod q = yA xB mod q (which B can compute) = yB xA mod q (which A can compute)  KAB is used as session key in private-key encryption scheme between Alice and Bob  if Alice and Bob subsequently communicate, they will have the same key as before, unless they choose new public-keys  attacker needs an x, must solve discrete log Prepared by R.Arthy, AP/IT
  • 67. Man-in-the-middle Attack 1.Darth prepares by creating two private / public keys 2.Alice transmits her public key to Bob 3.Darth intercepts this and transmits his first public key to Bob. Darth also calculates a shared key with Alice 4.Bob receives the public key and calculates the shared key (with Darth instead of Alice) 5.Bob transmits his public key to Alice 6.Darth intercepts this and transmits his second public key to Alice. Darth calculates a shared key with Bob 7.Alice receives the key and calculates the shared key (with Darth instead of Bob)  Darth can then intercept, decrypt, re-encrypt, forward all messages between Alice & Bob Prepared by R.Arthy, AP/IT
  • 69. Objective  To show that RSA is not the only public key system  To exhibit a public key system based on a different one way function  ElGamal is the basis for several well-known cryptographic primitives Prepared by R.Arthy, AP/IT
  • 70. Elgamal Key Exchange  Public-key cryptosystem related to D-H  Uses exponentiation in a finite field  With security based difficulty of computing discrete logarithms, as in D-H  Each user (e.g. A) generates their key  Chooses a secret key (number): 1 < XA < q-1  Computes their public key: YA = a XA mod q Prepared by R.Arthy, AP/IT
  • 71. Elgamal Key Exchange  Bob encrypt a message to send to A computing  represent message M in range 0 <= M <= q-1  longer messages must be sent as blocks  chose random integer k with 1 <= k <= q-1  compute one-time key K = YA k mod q  encrypt M as a pair of integers (C1,C2) where  C1 = ak mod q ; C2 = KM mod q  A then recovers message by  recovering key K as K = C1 xA mod q  computing M as M = C2 K-1 mod q  a unique k must be used each time  otherwise result is insecure Prepared by R.Arthy, AP/IT
  • 75. Elliptic Curve Cryptography  Majority of public-key crypto (RSA, D-H) use either integer or polynomial arithmetic with very large numbers/polynomials  Imposes a significant load in storing and processing keys and messages  An alternative is to use elliptic curves  Offers same security with smaller bit sizes Prepared by R.Arthy, AP/IT
  • 76. Real Elliptic Curves  an elliptic curve is defined by an equation in two variables x & y, with coefficients  consider a cubic elliptic curve of form  y2 = x3 + ax + b  where x,y,a,b are all real numbers  also define zero point O  have addition operation for elliptic curve  geometrically sum of Q+R is reflection of intersection R Prepared by R.Arthy, AP/IT
  • 77. Real Elliptic Curve Example Prepared by R.Arthy, AP/IT
  • 78. Finite Elliptic Curves  Elliptic curve cryptography uses curves whose variables & coefficients are finite  Have two families commonly used:  Prime curves ep(a,b) defined over zp  Use integers modulo a prime  Best in software  Binary curves e2m(a,b) defined over gf(2n)  Use polynomials with binary coefficients  Best in hardware Prepared by R.Arthy, AP/IT
  • 79. Elliptic Curve Cryptography  ECC addition is analog of modulo multiply  ECC repeated addition is analog of modulo exponentiation  Need “hard” problem equiv to discrete log  Q=kp, where Q,P belong to a prime curve  Is “easy” to compute Q given k,p  But “hard” to find k given Q,P  Known as the elliptic curve logarithm problem  Certicom example: E23(9,17) Prepared by R.Arthy, AP/IT
  • 80. ECC Diffie-Hellman  can do key exchange analogous to D-H  users select a suitable curve Ep(a,b)  select base point G=(x1,y1) with large order n s.t. nG=O  A & B select private keys nA<n, nB<n  compute public keys: PA=nA×G, PB=nB×G  compute shared key: K=nA×PB, K=nB×PA  same since K=nA×nB×G Prepared by R.Arthy, AP/IT
  • 81. ECC Encryption/Decryption  several alternatives, will consider simplest  must first encode any message M as a point on the elliptic curve Pm  select suitable curve & point G as in D-H  each user chooses private key nA<n  and computes public key PA=nA×G  to encrypt Pm : Cm={kG, Pm+k Pb}, k random  decrypt Cm compute: Pm+kPb–nB(kG) = Pm+k(nBG)–nB(kG) = Pm Prepared by R.Arthy, AP/IT
  • 82. ECC Security  relies on elliptic curve logarithm problem  fastest method is “Pollard rho method”  compared to factoring, can use much smaller key sizes than with RSA etc  for equivalent key lengths computations are roughly equivalent  hence for similar security ECC offers significant computational advantages Prepared by Dr. R.Arthy, AP/IT