Network and Information Security unit2.ppt.ppt

B. Tech.
Information Technology
By
Dr. G.N. Vivekananda
Associate Professor
SCORE, VIT, Vellore
Text Books:
 William Stallings, “Cryptography and Network Security-
Principles and Practice”, 2020, 8th Edition, Pearson Publishers.
 Michael E Whitman and Herbert J Mattord, “Principles of
Information Security”, 2017, 6th Edition, Course Technology Inc.
BITE401L Network and Information Security
Module 2

2
Module II: Public Key Cryptography
●Need and Principles of Public Key
Cryptosystems
●RSA Algorithm
●El Gamal Cryptographic System
●Elliptic Curve Cryptography
●Public Key Distribution and Management
●Diffie-Hellman Key Exchange
Contents

3
Need and Principles of Public Key Cryptosystems
• Public-Key Cryptosystems
• Applications for Public-Key Cryptosystems
• Requirements for Public-Key Cryptography
• Public-Key Cryptanalysis

Symmetric key
Encryption
Plaintext
input
Plaintext
output
Encryption Algorithm
(e.g. AES)
Decryption Algorithm
(reverse of encryption
algorithm)
Secret key shared by
sender and recipient
X
Secret key shared by
sender and recipient
K
Transmitted
cipher text
Y = E(K, X)
K
X

Plaintext
input
Plaintext
output
(e.g. RSA)
X
Transmitted
cipher text
Y = E(PUa, X)
X
Bob’s
Public
key ring
Alice
Ted
Mike
Joy
Alice’s public
key
PUa
Alice’s private
key
PRa
Asymmetric key
Encryption with
Public Key
Bob Alice
 The entire encrypted message
serves as a confidentiality.

Plaintext
input
Plaintext
output
(e.g. RSA)
X
Transmitted
cipher text
Y = E(PRb, X)
X
Alice’s
Public
key ring
Bob
Ted
Mike
Joy
Bob’s public
key
PUb
Bob’s private
key
PRb
Asymmetric key
Encryption with
Private Key
Bob Alice
 The entire encrypted message
serves as a digital signature.

Authentication and
Confidentiality
Message
source
Encryption
Algorithm
Encryption
Algorithm
Decryption
Algorithm
Decryption
Algorithm
Message
Dest.
X Y Y X
Z
Key pair
source
Key pair
source
PRa PUa
PRb
PUb
Source A Source B
Z = E(PUb, E(PRa, X)) X = D(PUa, D(PRB, Z))

8
Principles of public-key cryptosystems
Symmetric encryption has two difficult problems.
 Key distribution problem
 Symmetric encryption requires either
 that two communicants already share a key or
 the use of key distribution center (KDC).
 If the KDC is compromised, …
 Hard to be used for “digital signatures”

9
Public-Key Cryptosystems
Public-key algorithms use two separate key.
 Public key and private key
 It is computationally infeasible to determine the private key given only
knowledge of the cryptographic algorithm and the public key.
 Normally, public key is used for encryption and private key is used for
decryption.
 In some algorithms such as RSA, either of the two keys can be used for
encryption, with the other used for decryption.

10
A public-key encryption scheme has six ingredients.
 Plaintext
 Encryption algorithm
 Ciphertext
 Decryption algorithm
 Public and private key
 One is for encryption and the other is for decryption.

11
The use of public-key encryption
 Each user generate his/her public and private keys.
 Each user places the public key in a public register
and keeps the private key secret.
 If Bob wants to send a message to Alice, Bog
encrypts the message using Alice’s public key.
 Alice decrypts the ciphertext using her private key.

12
The use of public-key encryption. (Bob sends a message to Alice.)

13
A public-key encryption scheme : Secrecy

14
The use of public-key encryption to provide authentication.

15
A public-key encryption scheme : Authentication

16
A public-key encryption scheme : Secrecy and authentication

17
Applications for Public-Key Cryptosystems
The use of public-key cryptosystems
 Encryption/decryption (provide secrecy)
 Digital signatures (provide authentication)
 Key exchange (of session keys)
Some algorithms are suitable for all applications, others can be used
only for one or two.

18
Requirements for Public-Key Cryptography
Diffie and Hellman did lay out the conditions that such algorithms must fulfill
when A sends a message to B.
1. It is easy for B to generate his/her public and private key.
2. It is easy for A to encrypt a message M using B’s public key.
3. It is easy for B to decrypt the ciphertext using B’s private key.
)
(M
E
C b
KU

)]
(
[
)
( M
E
D
C
D
M b
b
b KU
KR
KR 


19
4. It is infeasible for an opponent, knowing the public key, KUb , to determine the
private key, KRb .
5. It is infeasible for an opponent, knowing the public key, KUb , and a ciphertext, C, to
recover the original message, M.
6. (Optional) The encryption and decryption functions can be applied in either order.
)]
(
[
)]
(
[ M
E
D
M
D
E
M b
b
b
b KR
KU
KR
KU 


20
These requirements are hard to achieve so only two algorithms (RSA, elliptic
curve cryptography) have received widespread acceptance.
Why the requirements are so formidable?
 The requirements needs a trap-door one-way function.

21
One-way function
 A one-to-one function such that
 The calculation of the function is easy
 but the calculation of the inverse is infeasible.
Easy
 A problem can be solved in polynomial time.
Infeasible
 It is hard to invert a function for virtually all inputs, not for the worst case or even
average case.
Y = f(X) easy
X = f-1
(Y) infeasible

22
Trap-door one-way function
 Easy to calculate in one direction and infeasible to calculate in the other direction
unless certain additional information is known.
 Thus, the development of a practical public-key scheme depends on discovery of a
suitable trap-door one-way function.
Y = fk(X) easy, if k and X are known
X = fk
-1
(Y) easy, if k and Y are known
X = fk
-1
(Y) infeasible, if Y is known but k is not known

23
Public-Key Cryptanalysis
Brute-force attacks for private keys
 Countermeasure: use large keys
 The key size must be large enough to make brute-force attack
impractical but small enough for practical encryption and
decryption.
Computing the private key given the public key
 No algorithms are proven safe from this attack.

24
Public-Key Cryptanalysis
A probable-message attack
 Suppose that a message were a 56-bit DES key.
 An opponent could encrypt all possible keys using the public
key.
 He could decipher any message by matching the
transmitted ciphertext.
 Countermeasure
 Large key size (?)
 Append some random bits to messages.

Applications for Public-Key Cryptosystems
 Encryption/decryption: The sender encrypts a message with the
recipient’s public key.
 Digital signature: The sender “signs” a message with its private
key. Signing is achieved by a cryptographic algorithm applied to
the message or to a small block of data that is a function of the
message.
 Key exchange: Two sides cooperate to exchange a session key.
Several different approaches are possible, involving the private
key(s) of one or both parties.

26
RSA Algorithm
• Description of the Algorithm
• The Security of RSA

27
The RSA Algorithm
Developed in 1977 by Rivest, Shamir, and Adleman at MIT.
The RSA scheme is a block cipher in which the plaintext / ciphertext are integers
between 0 and n – 1 for some n.
A typical size for n is 1024 bits, or 309 decimal digits.
 n = pq

28
Description of the RSA Algorithm
Plaintext is encrypted in blocks.
 Each block have a binary value less than some number n.
 That is, the block size must be less than or equal to log2(n).
 The block size is k bits, where 2k
< n ≤ 2k+1
.

29
Encryption/Decryption
 M : plaintext block
 C : ciphertext block
 public key: {e, n}
 private key: {d, n}.
n
M
n
M
n
C
M
n
M
C
ed
d
e
d
e
mod
mod
)
(
mod
mod





30
Diffie and Hellman’s requirements
1. It is easy for B to generate his/her public and private key.
 It should be easy for B to find values of e, d, and n.
2. It is easy for A to encrypt a message M using B’s public key.
 It should be easy to calculate Me
.
3. It is easy for B to decrypt the ciphertext using B’s private key.
 It should be easy to calculate Cd
.
4. It is infeasible for an opponent, knowing the public key, KUb , to determine the private key, KRb
.
 It is infeasible to determine d given e and n.
5. It is infeasible for an opponent, knowing the public key, KUb , and a ciphertext, C, to recover
the original message, M.
6. (Optional) The encryption and decryption functions can be applied in either order.

31
First requirement
 It should be easy to find values of e, d, n such that
for all M < n.
n
M
M ed
mod


32
A corollary to Euler’s theorem
 Given two prime numbers, p and q, and two integer, n and m, such that n = pq and 0 <
m < n, and arbitrary integer k,
n
m
m
n
m
m
q
p
k
n
k
mod
mod
1
)
1
)(
1
(
1
)
(







where Φ(n) is the Euler totient function, which is the number of
positive integers less than n and relatively prime to n.

33
If we select e and d such that
they satisfy .
is equivalent to saying
According to the rules of modular arithmetic, this is true
only if e (and therefore d) is relatively prime to Φ(n).
1
)
( 
 n
k
ed 
)
(
mod
)
(
mod
1
1
n
e
d
n
ed





1
)
),
(
gcd( 
e
n

n
M
M ed
mod

1
)
( 
 n
k
ed 

34
RSA’s ingredient.
Public key consist of {e, n} and a private key consist of {d, n}
p, q, two prime numbers (private, chosen)
n = pq (public, calculated)
e, with gcd(Φ(n), e) = 1 1 < e < Φ(n) (public, chosen)
(private, calculated)
)
(
mod
1
n
e
d 



35
RSA’s scheme
 Suppose user B wishes to send the message M to A.
 User A has published its public key, KU={e, n}.
 B calculates C = Me
(mod n) and transmits C.
 Then, user A decrypts by calculating M = Cd
(mod n).
(use KR={d, n})

36
RSA algorithm (example) : the keys generating
 Select two prime number, p = 17 and q = 11.
 Calculate n = pq = 17 X 11 = 187.
 Calculate Φ(n) = (p – 1)(q – 1) = 16 X 10 = 160.
 Select e = 7 (e is relatively prime to Φ(n)).
 Determine d, de = 1 mod 160 (Using extended Euclid’s algorithm).
 d = 23

37
11
187
mod
432
,
894
187
mod
)
132
77
88
(
187
mod
88
132
187
mod
)
77
77
(
187
mod
536
,
969
,
59
187
mod
88
77
187
mod
)
88
88
(
187
mod
7744
187
mod
88
88
187
mod
88
187
mod
)]
187
mod
88
(
)
187
mod
88
(
)
187
mod
88
[(
187
mod
88
7
4
2
1
1
2
4
7

















88
187
mod
243
,
720
,
79
187
mod
)
33
33
55
121
11
(
187
mod
11
33
187
mod
)
55
55
(
881
,
358
,
214
187
mod
11
55
187
mod
)
121
121
(
187
mod
641
,
14
187
mod
11
121
187
mod
)
11
11
(
187
mod
11
11
187
mod
11
187
mod
)]
187
mod
11
(
)
187
mod
11
(
)
187
mod
11
(
)
187
mod
11
(
)
187
mod
11
[(
187
mod
11
23
8
4
2
1
8
8
4
2
1
23
























Encryption
Decryption

38
The Security of RSA
Three possible approaches to attacking the RSA.
 Brute force
 Mathematical attacks
 Timing attacks
Brute force
 trying all possible private keys
 Countermeasures: Use a large key space.

39
The Security of RSA
Mathematical attacks
 Factor n into its two prime factors. This enables calculation of Φ(n) and
determination of d.
 Determine Φ(n) directly, without first determining p and q. This enable determination
of d.
 This is equivalent to factoring n.
 Determine d directly, without first determining Φ(n).
 With presently known algorithms, this appears to be at least as time-consuming as the
factoring problem.

40
The Security of RSA
 Focused on the task of factoring n into its two prime factors.

41
The Security of RSA
To avoid values of n that may be factored more easily, the algorithm’s inventors
suggest constraints on p and q.
 p and q should differ in length by only a few digits.
 Both (p – 1) and (q – 1) should contain a large prime factor.
 gcd (p – 1, q – 1) should be small.
In addition, it has been demonstrated that if e < n and d < n1/4
, then d can be
easily determined.

RSA Algorithm
 RSA is a block cipher in which the Plaintext and Ciphertext are represented as integers
between 0 and n-1 for some n.
 Large messages can be broken up into a number of blocks.
 Each block would then be represented by an integer.
Step-1: Generate Public key and Private key
Step-2: Encrypt message using Public key
Step-3: Decrypt message using Private key

 Select two large prime numbers: p and q
 Calculate modulus : n = p * q
 Calculate Euler’s totient function : φ(n) = (p-1) * (q-1)
 Select e such that e is relatively prime to φ(n) and 1 < e < φ(n)
 Determine d such that d * e ≡ 1 (mod φ(n))
 Publickey : PU = { e, n }
 Privatekey : PR = { d, n }
Two numbers are relatively prime if they have no common factors
other than 1.

 Select two large prime numbers: p = 3 and q = 11
 Calculate modulus : n = p * q, n = 33
 Calculate Euler’s totient function : φ(n) = (p-1) * (q-1)
φ(n) = ( 3 – 1 ) * ( 11 – 1 ) = 20
 Select e such that e is relatively prime to φ(n) and 1 < e < φ(n)
 We have several choices for e : 7, 11, 13, 17, 19 Let’s take e = 7
 Determine d such that d * e ≡ 1 (mod φ(n))
 ? * 7 ≡ 1 (mod 20), 3 * 7 ≡ 1 (mod 20)
 Public key : PU = { e, n } , PU = { 7, 33 }
 Private key : PR = { d, n }, PR = { 3, 33 }
• This is equivalent to
finding d which satisfies
de = 1 + j.φ(n) where j is
any integer.
• We can rewrite this as
d = (1 + j. φ(n)) / e

Step-2 : Encrypt Message
 Encryption Using Public key: C = Me
mod n
Ciphertext Input
Message
Publickey
For message M = 14
C = 147
mod 33
C = [(141
mod 33) X (142
mod 33) X (144
mod 33)] mod 33
C = (14 X 31 X 4) mod 33 = 1736 mod 33
C = 20
PU = { e, n } , PU = { 7, 33 }

Step-3 : Decrypt Message
 Encryption Using Public key: M = Cd
mod n
Plaintext
Message
Cipher
Message
Privatekey
For Ciphertext C = 20
M = 203
mod 33
M = [(201
mod 33) X (202
mod 33)] mod 33
M = (20 X 4) mod 33 = 80 mod 33
M = 14
PR = { d, n } , PR = { 3, 33 }

Example RSA Algorithm
14
7
mod 33 = 20
Plaintext
14
Plaintext
14
20
3
mod 33 = 14
Ciphertext
20
PU = 7, 33 PR = 3, 33
Encryption Decryption

RSA Example
 Find n, φ(n), e, d for p=7 and q= 19 then demonstrate encryption
and decryption for M = 6
n = p * q = 7 * 19 = 133
φ(n) = ( p – 1 ) * ( q – 1) = 108
Finding e relatively prime to 108
e = 2 => GCD( 2, 108 ) = 2 (no)
e = 3 => GCD( 3, 108 ) = 3 (no)
e = 5 => GCD( 5, 108 ) = 1 (Yes)
• Finding d such that (d * e ) mod φ(n) = 1
• We can rewrite this as d = (1 + j . φ(n)) /
e
j = 0 => d = 1 / 5 = 0.2  integer ? (no)
j = 1 => d = 109 / 5 = 21.8  integer ? (no)
j = 2 => d = 217 / 5 = 43.4  integer ? (no)
j = 3 => d = 325 / 5 = 65 integer ? (yes)
Public key :
PU = { e, n } = {5, 133}
Private key :
PR = { d, n } = {65, 133}

RSA Example – cont…
 Encryption:
C = Me
mod n
For message M = 6
C = 65
mod 133
C = 7776 mod 33
C = 62
PU = { e, n } , PU = { 5, 133 }
 Decryption:
M = Cd
mod n
For C = 62
M = 6265
mod 133
M = 2666 mod 33
M = 6
PR = { d, n } , PU = { 65, 133 }

RSA Example
 P and Q are two prime numbers. P=7, and Q=17. Take public key E=5. If plain text value is
10, then what will be cipher text value according to RSA algorithm?
 n = 119
 φ(n) = 96
 e = 5
 d = 77
 PU = { 5, 119 }
 PR = {77, 119}
 C = 105
mod 119 => C = 40

ElGamal Cryptographic System
 public-key cryptosystem related to D-H
 so uses exponentiation in a finite (Galois)
 with security based difficulty of computing
discrete logarithms, as in D-H
 each user (eg. A) generates their key
 chooses a secret key (number): 1 < xA < q-1

compute their public key: yA = a
xA
mod q

ElGamal Message 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 = a
k
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

ElGamal Example
 use field GF(19) q=19 and a=10
 Alice computes her key:

A chooses xA=5 & computes yA=10
5
mod 19 = 3
 Bob send message m=17 as (11,5) by

chosing random k=6

computing K = yA
k
mod q = 3
6
mod 19 = 7

computing C1 = a
k
mod q = 10
6
mod 19 = 11;
C2 = KM mod q = 7.17 mod 19 = 5
 Alice recovers original message by computing:

recover K = C1
xA
mod q = 11
5
mod 19 = 7

compute inverse K-1
= 7-1
= 11

recover M = C2 K-1
mod q = 5.11 mod 19 = 17

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
 newer, but not as well analysed

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
 consider set of points E(a,b) that satisfy
 have addition operation for elliptic curve

geometrically sum of P+Q is reflection of the
intersection R

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

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)

ECC Diffie-Hellman
 can do key exchange analogous to D-H
 users select a suitable curve Eq(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=nAG, PB=nBG
 compute shared key: K=nAPB, K=nBPA

same since K=nAnBG
 attacker would need to find k, hard

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=nAG
 to encrypt Pm : Cm={kG, Pm+kPb}, k random
 decrypt Cm compute:
Pm+kPb–nB(kG) = Pm+k(nBG)–nB(kG) = Pm

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

Comparable Key Sizes for Equivalent
Security
Symmetric
scheme
(key size in bits)
ECC-based
scheme
(size of n in bits)
RSA/DSA
(modulus size in
bits)
56 112 512
80 160 1024
112 224 2048
128 256 3072
192 384 7680
256 512 15360

63
Public Key Distribution and Management
• Key management and distribution
• Symmetric key distribution using symmetric encryption
• Symmetric key distribution asymmetric encryption
• Distribution of public keys
• X.509 certificates
• Public key infrastructure (PKI)

INS is very Interesting Subject
Key Distribution
 Key distribution is the function that delivers a key to two parties
who wish to exchange secure encrypted data.
 Some sort of mechanism or protocol is needed to provide for the
secure distribution of keys.
 Key distribution often involves the use of master keys, which are
infrequently used and are long lasting, and session keys, which
are generated and distributed for temporary use between two
parties.

Key Hierarchy
 Communication between end
systems is encrypted using a
temporary key, often referred
to as a session key.
 Session keys are transmitted
in encrypted form, using a
master key that is shared by
the key distribution center
and an end system or user

Simple Secret Key Distribution
1. A generates a public/private key pair {PUa, PRa} and transmits a
message to B consisting of PUa and an identifier of A, IDA.
2. B generates a secret key, Ks, and transmits it to A, encrypted with
A's public key.
3. A computes D(PRa, E(PUa, Ks)) to recover the secret key. Because
only A can decrypt the message, only A and B will know the
identity of Ks.
4. A discards PUa and PRa and B discards PUa.
Initiator
A
Initiator
B
(2) E(PUa , Ks)
(1) PUa || IDA

Secret Key Distribution with Confidentiality & Authentication
1. A uses B's public key to encrypt a message to B containing an
identifier of A (IA) and a nonce (N1), which is used to identify this
transaction uniquely.
2. B sends a message to A encrypted with PUa and containing A's (N1)
as well as a new nonce generated by B (N2). Because only B could
have decrypted message (1), the presence of N1 in message (2)
assures A that the correspondent is B.
Initiator
A
Initiator
B
(1) E(PUb,[N1||IDA])
(2) E(PUa,[N1|| N2])
(3) E(PUb,N2])
(4) E(Pub,E(PRa,Ks))

Secret Key Distribution with Confidentiality & Authentication
3. A returns N2, encrypted using B's public key, to assure B that its
correspondent is A.
4. A selects a secret key Ks and sends M = E(PUb, E(PRa, Ks)) to B.
Encryption with B's public key ensures that only B can read it;
encryption with A's private key ensures that only A could have
sent it.
5. B computes D(PUa, D(PRb, M)) to recover the secret key.
Initiator
A
Initiator
B
(1) E(PUb,[N1||IDA])
(2) E(PUa,[N1|| N2])
(3) E(PUb,N2])
(4) E(Pub,E(PRa,Ks))

Symmetric key distribution using symmetric encryption
 Two parties A and B, key distribution can be achieved in a number
of ways, as follows:
1. A can select a key and physically deliver key to B.
2. Third party can select the key and physically deliver it to A and
B.
3. If A and B have previously and recently used a key, one party
can transmit the new key to the other, encrypted using the old
key.
4. If A and B each has an encrypted connection to a third party C,

Key Distribution Scenario Key
Distribution
Center (Key)
Initiator
A
Initiator
B
(1) IDA || IDB || N1
(2) E(Ka, [Ks || IDA || IDB || N1]) || E(Kb, [Ks || IDA])
(3) E(Kb, [Ks || IDA])
(4) E(Ks, N2])
(5) E(Ks, f(N2))

Key Distribution Scenario
1. A requests from the KDC a session key to protect a logical connection to B. The message
includes the identity of A and B and a unique nonce N1.
2. The KDC responds with a message encrypted using Ka that includes a one-time session
key Ks to be used for the session, the original request message to enable A to match
response with appropriate request, and info for B
3. A stores the session key for use in the upcoming session and forwards to B the
information from the KDC for B, namely, E(Kb, [Ks || IDA]).
4. At this point, a session key has been securely delivered to A and B, and they may begin
their protected exchange.
5. Using the new session key for encryption B sends a nonce N2 to A.
6. Also using Ks, A responds with f(N2). These steps assure B that the original message it
received (step 3) was not a replay. Note that the actual key distribution involves only steps
1 through 3 but that steps 4 and 5, as well as 3, perform an authentication function.

Distribution of Public Keys
1. Public announcement
2. Publicly available directory
3. Public-key authority
4. Public-key certificates

1. Public Announcement
 Some user could pretend to be user A and send a public key to another
participant or broadcast such a public key.
 Until such time as user A discovers the forgery and alerts other participants, the
forger is able to read all encrypted messages intended for A and can use the
forged keys for authentication

2. Publicly Available Directory
1. The authority maintains a directory with a {name, public key} entry for each participant.
2. Each participant registers a public key with the directory authority.
3. A participant may replace the existing key with a new one at any time.
4. Participants could also access the directory electronically. For this purpose, secure, authenticated
communication from the authority to the participant is mandatory.

3. Public-Key Authority
Initiator
A
Initiator
B
(6) E(PUa , [N1 || N2])
Public-Key
Authority
(1) Request || T1
(2) E(PRauth, [PUb || Request || T1])
(4) Request || T2
(5) E(PRauth, [PUa || Request || T2])
(3) E (PUb, [IDa, N1])
(7) E(PUb, N2)

3. Public-Key Authority – Cont…
1. A sends a timestamped message to the public-key authority containing a request for the current
public key of B.
2. The authority responds with a message that is encrypted using the authority’s private key .
3. Message contains PUb, Original request, Original time stamp T1
A stores B’s public key and also uses it to encrypt a message to B containing an identifier of
A(IDa) and a nonce(N1) , which is used to identify this transaction uniquely.
4, 5. B retrieves A’s public key from the authority in the same manner as A retrieved B’s public key.
6. B sends a message to A encrypted with PUa and containing A’s nonce(N1) as well as a new
nonce generated by B(N2). Because only B could have decrypted message (3), the presence of
N1 in message (6) assures A that the correspondent is B.
7. A returns N2, which is encrypted using B’s public key, to assure B that its correspondent is A.

4. Public-Key Certificates
 Any participant can read a certificate to determine the name and public key of the
certificate’s owner.
 Any participant can verify that the certificate originated from the certificate authority and
is not counterfeit.
 Only the certificate authority can create and update certificates.
 Any participant can verify the currency of the certificate.

4. Public-Key Certificates – Cont…
Initiator
A
Initiator
B
(2) CA
(1) CA
Certificate
Authority
PUa
CA = E(PRauth, [T1 || IDa || PUa])
PUb
CA = E(PRauth, [T2 || IDb || PUb])

4. Public-Key Certificates – Cont…
 Each participant applies to the certificate authority, supplying a public key and requesting
a certificate.
 For participant A, the authority provides a certificate of the form
CA = E (PRauth, [T || IDa || PUa] )
 A may then pass this certificate on to any other participant, who reads and verifies the
certificate as follows:
D(PUauth, CA)
= D(PUauth, E (PRauth, [T || IDa || PUa] ))
= (T || IDa || PUa)

X.509 Certificates
 X.509 defines the format for public-key certificates. used in a variety of applications.
 X.509 defines a framework for the provision of authentication services by the X.500
directory to its users.
 The directory may serve as a repository of public-key certificates.
 Each certificate contains the public key of a user and is signed with the private key of a
trusted certification authority.

X.509
Formats
Version
Certificate
serial number
Algorithm
Parameters
Issuer name
Not before
Not after
Subject name
Algorithms
Parameters
Key
Issuer Unique
identifier
Extensions
Algorithms
Parameters
Encrypted hash
Signature
algorithm
identifier
Proof of
validity
Subject’s
public key
info
Signature
Subject Unique
identifier
Version
1
Version
2
Version
3
All
Version
s

X.509 Format – Cont…
 Version: Differentiates among successive versions of the certificate format; the default is
version 1.
 Serial number: An integer value unique within the issuing CA that is unambiguously
associated with this certificate.
 Signature algorithm identifier: The algorithm used to sign the certificate together with
any associated parameters.
 Issuer name: X.500 name of the CA that created and signed this certificate.
 Period of validity: Consists of two dates: the first and last on which the certificate is valid.
 Subject name: The name of the user to whom this certificate refers.

X.509 Format – Cont…
 Subject’s public-key information: The public key of the subject, plus an identifier of the
algorithm for which this key is to be used, together with any associated parameters.
 Issuer unique identifier: An optional-bit string field used to identify uniquely the issuing
CA in the event the X.500 name has been reused for different entities.
 Subject unique identifier: An optional-bit string field used to identify uniquely the subject
in the event the X.500 name has been reused for different entities.
 Extensions: A set of one or more extension fields.

Public-Key
Certificate Use
H
E D
CA
information
Bob’s Public
key
Bob’s ID
information
H
Generate hash
code of unsigned
certificate
Encrypt hash code
with CA's private key
to form signature
Decrypt signature
with CA's public key
to recover hash code
Recipient can verify
signature by comparing
hash code values
Signed Certificate
Unsigned certificate:
contains user ID,
user's public key

Public key Infrastructure (PKI)
 A public-key infrastructure (PKI) is defined as the set of
hardware, software, people, policies, and procedures needed to
create, manage, store, distribute, and revoke digital certificates
based on asymmetric cryptography.
 The principal objective for developing a PKI is to enable secure,
convenient, and efficient acquisition of public keys.

Public key Infrastructure (PKI)

Public key Infrastructure (PKI) – Cont…
 End entity: A generic term used to denote end users, devices (e.g., servers, routers), or
any other entity that can be identified in the subject field of a public-key certificate.
 Certification authority (CA): The issuer of certificates and (usually) certificate revocation
lists (CRLs).
 Registration authority (RA): An optional component that can assume a number of
administrative functions from the CA.
 CRL issuer: An optional component that a CA can delegate to publish CRLs.
 Repository: A generic term used to denote any method for storing certificates and CRLs so
that they can be retrieved by end entities.

Diffie-Hellman Key Exchange
 The purpose of the Diffie-Hellman algorithm is to enable two users to securely exchange a
key that can be used for subsequent encryption of message.
 This algorithm depends for its effectiveness on the difficulty of computing discrete
logarithms.
Primitive root
 Let be a prime number
 Then is a primitive root for , if the powers of modulo
generates all integers from 1 to – 1 in some
permutation.
 Example: p = 7 then primitive root is 3 because powers
of 3 mod 7 generates all the integers from 1 to 6
𝑎𝑚𝑜𝑑 𝑝,𝑎2
𝑚𝑜𝑑𝑝 ,…,𝑎𝑝−1
𝑚𝑜𝑑𝑝

Discrete Logarithm
 For any integer and a primitive root of prime number , we can find a unique exponent
such that
 The exponent is referred as the discrete logarithm of for the base , mod . It expressed as
below.
 User A and User B agree on two large prime numbers q and α. User A and User B can use
insecure channel to agree on them.
 User A selects a random integer and calculates
𝑏=𝑎𝑖
(𝑚𝑜𝑑 𝑝) h
𝑤 𝑒𝑟𝑒 0 ≤𝑖 ≤(𝑝−1)
𝑏d log𝑎,𝑝 (𝑏)

Diffie-Hellman Key Exchange – Cont…
Global Public Elements
prime number
< and is primitive root of
User A Key Generation
Select private
Calculate public
User B Key Generation
Select private
Calculate public

Select private
Calculate public
Select private
Calculate public
Calculation of Secret Key by User A
Calculation of Secret Key by User b

Private , Public
Private , Public
Secret Key by User A :
Secret Key by User B :

Network and Information Security unit2.ppt.ppt

Diffie-Hellman Key Exchange Example
 Alice and bob agrees on a prime number
 as primitive root of
 Alice selects a private integer
 Alice computes =>
 Bob selects a private integer
 Bob computes =>
 Alice sends to Bob and Bob sends to Alice
 Alice computes key =>
 Bob computes key =>

Diffie-Hellman Key Exchange Illustration

 first public-key type scheme proposed
 by Diffie & Hellman in 1976 along with the exposition of
public key concepts

note: now know that Williamson (UK CESG) secretly proposed
the concept in 1970
 is a practical method for public exchange of a secret key
 used in a number of commercial products

 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

Diffie-Hellman Setup
 all users agree on global parameters:

large prime integer or polynomial q

a being a primitive root mod q
 each user (eg. A) generates their key
 chooses a secret key (number): xA < q

compute their public key: yA = a
xA
mod q
 each user makes public that key yA

 shared session key for users A & B is KAB:
KAB = a
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

Diffie-Hellman Example
 users Alice & Bob who wish to swap keys:
 agree on prime q=353 and a=3
 select random secret keys:

A chooses xA=97, B chooses xB=233
 compute respective public keys:

yA=3
97
mod 353 = 40 (Alice)

yB=3
233
mod 353 = 248 (Bob)
 compute shared session key as:

KAB= yB
xA
mod 353 = 248
97
= 160 (Alice)

KAB= yA
xB
mod 353 = 40
233
= 160 (Bob)

Key Exchange Protocols
 users could create random private/public
D-H keys each time they communicate
 users could create a known private/public
D-H key and publish in a directory, then
consulted and used to securely
communicate with them
 both of these are vulnerable to a meet-in-
the-Middle Attack
 authentication of the keys is needed

Dr. G.N. Vivekananda
Associate Professor
School of Computer Science Engineering & Information Systems
Vellore Institute of Technology, Vellore
BITE401L Network and Information Security

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