3 public key cryptography

UNIT-3 PUBLIC KEY
CRYPTOGRAPHY
Information Security

Principles of Public key
Cryptography
 Asymmetric encryption is a form of
cryptosystem in which encryption and
decryption are performed using the different
keys- one public key and one private key
 Also known as public-key encryption
 It uses mathematical functions rather than
substitution and permutation
 More secure from cryptanalysis than the
symmetric encryption

 Asymmetric keys
 Two related keys, a public key and a private key,
that are used to perform complementary
operations, such as encryption and decryption or
signature generation and signature verification
 Public key certificate
 A digital document issued and digitally signed by
the private key of a Certification authority that
binds the name of a subscriber to a public key.
The certificate indicates that the subscriber
identified in the certificate has sole control and
access to the corresponding private key

 Public key cryptographic algorithm
 A cryptographic algorithm that uses two related
keys, a public key and a private key
 Public key infrastructure
 A set of policies, processes, server platform,
software and workstations used for the
purpose of administrating certificates and
public-private key pairs, including the ability to
issue, maintain, and revoke public certificate

Public-key cryptosystem
 Asymmetric algorithms rely on one key for
encryption and a different but related key for
decryption
 These algorithms have the following important
characteristics
 It is computationally infeasible to determine
the decryption key given only knowledge of the
cryptographic algorithm and the encryption key

 Six ingredients:
 Plaintext - This is a readable message or data
that is fed into the algorithm as the input
 Encryption algorithm - The encryption algorithm
performs various transformations on the plaintext
 Public and private keys - this is a pair of keys
that have been selected so that if one is used for
encryption, the other is used for decryption. The
exact transformations performed by the algorithm
depend on the public key and the private key that
is provided as input

 Cipher text - this is the scrambled message
produced as output
 Decryption algorithm - The algorithm that
accepts the cipher text and matching key and
produces the original plain text

 The essential steps are
1. Each user generates a pair of keys to be used for the
encryption and decryption of messages
2. Each user places one of the two keys in public register or
other accessible file. This is public key. The other key is
kept private. Each user maintains a collection of public keys
obtained from others
3. If Bob wishes to send a confidential message to Alice, Bob
encrypts the message using Alice’s public key
4. When Alice receives the message, she decrypts it using her
private key
5. No other recipient can decrypt the message because only
Alice knows her private key

 Here, all participants have access to public
keys, and private keys are generated locally
by each participant and therefore need never
be distributed
 As long as a user’s private key remains
protected and secret, incoming communication
is secure
 At any time, a system can change its private
key and publish the related public key to
replace its old public key

Difference between conventional
encryption and Public-key
encryption

Public-key Cryptosystem:
Secrecy

Authentication

Authentication and Secrecy

Applications of Public-key
Cryptosystem
 A applications are divided in three broad
categories:
 Encryption/decryption – The sender encrypts
the message with the recipient’s public key
 Digital Signature – The sender “signs” a
message with its private key
 Key exchange – two sides cooperate to
exchange a session key

RSA Algorithm
 Asymmetric key cryptographic algorithm
 Rivest-Shamir-Adleman (RSA) name is given by taking the
firstname of its inventors
 It uses prime numbers
 This algorithm is based on the fact that it is easy to find and
multiply large prime numbers together, but it is extremely
difficult to factor their product
 The private and public keys in RSA are based on very large
prime numbers
 The real challenge in RSA is the selection and generation of
the public key and private key
 Lets know how private key and public key are generated and,
using them, how can we perform encryption and decryption

Algorithm
1. Choose two prime numbers P and Q
2. Calculate N = P * Q
3. Select the public key E (i.e. Encryption key) such that it is not a
factor of (P–1) and (Q–1)
4. Select the private key D (i.e. Decryption key) such that the
following equation is true
(D * E) mod (P–1) * (Q–1) = 1
5. For encryption, calculate the cipher text CT from the plain text PT
as follows
CT = PTE mod N
6. Send CT as the cipher text to the receiver
7. For decryption, calculate the plain text PT from the cipher text CT
as follows
PT = CTD mod N

Example of RSA
1. Choose two large prime numbers P and Q
 Let P= 7, Q=17
2. Calculate N = P * Q
 N= 7 * 17= 119
3. Select the public key E such that it is not a
factor of (P-1) * (Q-1)
 Lets find (7-1) * (17-1)= 6*16= 96
 The factors of 96 are 2,2,2,2,2 and 3 ( because 96 =
2*2*2*2*2*3)
 Thus we have to choose E such that none of the
factors of E is 2 and 3
 Lets choose E as 5

4. Select the private key D such that the
following equation is true
(D * E) mod (P–1) * (Q–1) = 1
 Lets substitute the values of E, P and Q in the
equation
 We have (D * 5) mod (7-1) * (17-1) = 1
 i.e. (D * 5) mod (6) * (16) = 1
 i.e. (D * 5) mod (96) = 1
 After some calculations, let us take D=77
 So that (77 * 5) mod (96) = 385 mod 96 = 1

5. For encryption, calculate the cipher text CT
from the plain text PT as follows
CT = PTE mod N
 Lets assume that plaintext PT = 10
 Then, CT = 105 mod 119 = 100000 mod 119 = 40
6. Send CT as the cipher text to the receiver
 Send 40 as the cipher text to the receiver
7. For decryption, calculate the plain text PT
from the cipher text CT as follows
PT = CTD mod N
 PT = 4077 mod 119 = 10

The Security of RSA
 Four possible approaches to attacking the RSA
algorithm are as follows:
 Brute force: This involves trying all possible
private keys.
 Mathematical attacks: There are several
approaches, all equivalent in effort to factoring the
product of two primes.
 Timing attacks: These depend on the running
time of the decryption algorithm.
 Chosen cipher text attacks: This type of attack
exploits properties of the RSA algorithm.

Key management
 One of the major roles of public-key encryption
has been to address the problem of key
distribution
 There are actually two distinct aspects to the
use of public-key cryptography in this regard:
 The distribution of public keys
 The use of public-key encryption to distribute
secret keys

Distribution of Public Keys
 Schemes for Key distribution:
 Public announcement
 Publicly available directory
 Public-key authority
 Public-key certificates

Public announcement of Public
Keys
 The point of public-key encryption is that the public key is
public
 Thus, if there is some broadly accepted public-key algorithm,
such as RSA, any participant can send his or her public key
to any other participant or broadcast the key to the
community at large
 Although this approach is convenient, it has a major
weakness
 Anyone can forge such a public announcement
 That is, 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

Publicly Available Directory
 A greater degree of security can be achieved
by maintaining a publicly available dynamic
directory of public keys
 Maintenance and distribution of the public
directory would have to be the responsibility of
some trusted entity or organization
 Such a scheme would include the following
elements:

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. Registration would have to be in
person or by some form of secure authenticated
communication.
3. A participant may replace the existing key with a new
one at any time, either because of the desire to
replace a public key that has already been used for a
large amount of data, or because the corresponding
private key has been compromised in some way.
4. Participants could also access the directory
electronically. For this purpose, secure,
authenticated communication from the authority to
the participant is mandatory.

 This scheme is clearly more secure than
individual public announcements but still has
vulnerabilities
 If an adversary succeeds in obtaining or
computing the private key of the directory
authority, the adversary could authoritatively
pass out counterfeit public keys and
subsequently impersonate any participant and
eavesdrop on messages sent to any
participant

Public-Key Authority
 Stronger security for public-key distribution
can be achieved by providing tighter control
over the distribution of public keys from the
directory
 As before, the scenario assumes that a central
authority maintains a dynamic directory of
public keys of all participants
 Each participant reliably knows a public key for
the authority, with only the authority knowing
the corresponding private key

 The following steps occur:
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, PRauth Thus, A is able to
decrypt the message using the authority's public key.
Therefore, A is assured that the message originated with the
authority. The message includes the following:
1. B's public key, PUb which A can use to encrypt messages
destined for B
2. The original request, to enable A to match this response with
the corresponding earlier request and to verify that the original
request was not altered before reception by the authority
3. The original timestamp, so A can determine that this is not an
old message from the authority containing a key other than B's
current public key

3. 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, encrypted using B's public key, to
assure B that its correspondent is A.

Public-Key Certificates
 The public-key authority could be somewhat of a bottleneck
in the system, for a user must appeal to the authority for a
public key for every other user that it wishes to contact
 The directory of names and public keys maintained by the
authority is vulnerable to tampering.
 An alternative approach is to use certificates that can be
used by participants to exchange keys without contacting a
public-key authority
 A certificate consists of a public key plus an identifier of the
key owner, with the whole block signed by a trusted third
party
 The third party is a certificate authority, such as a government
agency or a financial institution, that is trusted by the user
community

 A user can present his or her public key to the
authority in a secure manner, and obtain a
certificate And then can publish the certificate
 Anyone needed this user's public key can
obtain the certificate and verify that it is valid
by way of the attached trusted signature
 A participant can also convey its key
information to another by transmitting its
certificate. Other participants can verify that
the certificate was created by the authority

 We can place the following requirements on
this scheme:
1. Any participant can read a certificate to
determine the name and public key of the
certificate's owner.
2. Any participant can verify that the certificate
originated from the certificate authority and is
not counterfeit.
3. Only the certificate authority can create and
update certificates.

Distribution of Secret Keys Using
Public-Key Cryptography
 Simple Secret Key Distribution
 Secret Key Distribution with Confidentiality and
Authentication
 A Hybrid Scheme

Simple Secret Key Distribution
 If A wishes to communicate with B, the following
procedure is employed:
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.

 A and B can now securely communicate using
conventional encryption and the session key
Ks
 At the completion of the exchange, both A and
B discard Ks.
 Despite its simplicity, this is an attractive
protocol. No keys exist before the start of the
communication and none exist after the
completion of communication.
 The risk of compromise of the keys is minimal

 The protocol depicted in Figure is insecure
against an adversary who can intercept
messages and then either relay the
intercepted message or substitute another
message
 Such an attack is known as a man-in-the-
middle attack
 In this case, If an adversary, E, has control of
the intervening communication channel, then
E can compromise the communication in the
following fashion without being detected:

1. A generates a public/private key pair {PUa, PRa}
and transmits a message intended for B
consisting ofP Ua and an identifier of A, IDA.
2. E intercepts the message, creates its own
public/private key pair {PUe, PRe} and transmits
PUe||IDA to B.
3. B generates a secret key, Ks, and transmits
E(PUe, Ks).
4. E intercepts the message, and learns Ks by
computing D(PRe, E(PUe, Ks)).
5. E transmits E(PUa, Ks) to A.

Secret Key Distribution with
Confidentiality and Authentication
 It provides protection against both active and passive attacks
 It is assumed that A and B have exchanged public keys by one of the
schemes
1. A uses B's public key to encrypt a message to B containing an identifier
of A (IDA) 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 nonce
(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.
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 of this message 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.

A Hybrid Scheme
 This scheme retains the use of a key
distribution center (KDC) that shares a secret
master key with each user and distributes
secret session keys encrypted with the master
key
 A public key scheme is used to distribute the
master keys
 The following rationale is provided for using
this three-level approach:

 Performance:
 There are many applications, especially transaction-
oriented applications, in which the session keys
change frequently.
 Distribution of session keys by public-key encryption
could degrade overall system performance because of
the relatively high computational load of public-key
encryption and decryption. With a three-level
hierarchy, public-key encryption is used only
occasionally to update the master key between a user
and the KDC.
 Backward compatibility:
 The hybrid scheme is easily overlaid on an existing
KDC scheme, with minimal disruption or software
changes.

 The addition of a public-key layer provides a
secure, efficient means of distributing master
keys
 This is an advantage in a configuration in
which a single KDC serves a widely distributed
set of users

Diffie-Hellman key exchange
Algorithm
 The two parties, who want to communicate
securely, can agree on a symmetric key using
this technique
 The can then can be used for encryption and
decryption
 This algorithm can be used only for key
agreement, but not for encryption and
decryption
 Once both parties agree on the key to be
used, they need to use other symmetric
encryption algorithms

Description of algorithm
 Lets assume that Alice and Bob want to agree
upon a key to be used for encrypting/decrypting
messages that would be exchanged between
them
 Then the Diffie-Hellman algorithm works as
follows:
1. Firstly, Alice and Bob agree on two large prime
numbers, n and g. these two integers need not
be kept secret. Alice and Bob can use insecure
channel to agree on them
2. Alice chooses another large random number x,
and calculates A such that
A = gx mod n

3. Alice sends the number A to Bob
4. Bob independently chooses another large
random integer y and the n calculates B such
that
B = gy mod n
5. Bob sends the number B to Alice
6. A now computes the secret key K1
K1 = Bx mod n
7. B now computes the secret key K2
K2 = Ay mod n

Example
1. Let n=11, g=7
2. Let x=3. then, we have A=73 mod 11=343
mod 11 =2
3. Alice sends 2 to Bob
4. Let y=6. then we have, B=76 mod 11=117649
mod 11 =4
5. Bob sends the 4 to Alice
6. We have, K1=43 mod 11= 64 mod 11 = 9
7. We have K2 = 26 mod 11=64 mod 11 = 9

Elliptic Curve
Cryptography(ECC)
 An elliptic curve (EC) is a smooth, projective algebraic curve
on which there is a specified point O
 The main difference between RSA and ECC is that unlike
RSA, ECC offers the same level of security for smaller key
sizes
 ECC requires much smaller keys than those used in
conventional public key cryptosystem, while maintaining the
same level of security
 The use of elliptic cures therefore allows faster encryption
and decryption
 ECC is highly mathematical in nature
 An Elliptic Curve is similar to a normal curve draw as a graph
on x-axes and y-axes and has points
 Each point can be designated by an (x , y) coordinate

 This is how elliptic curve public key
cryptography works
 For Alice and Bob to communicate securely
over an unsecure network they can exchange
a private key over this network in the following
way:
 A particular rationale base point P is published
in a public domain for use with a particular
elliptic curve E also published in a public
domain
 Alice and Bob choose random integers Ka and
Kb respectively, which they use as private key

 Alice computes Ka * P, Bob computes Kb * P and
they exchange these values over the network
 Using the information they received from each
other and their private keys, both Alice and Bob
compute (Ka * Kb)*P = Ka * (Kb * P) =Kb * (Ka *
P)
 This value is then the shared secret that only Alice
and Bob possess.
 The private keys Ka and Kb and the shared
secrety (Ka * Kb)*P are difficult to compute given
Ka * P and Kb * P
 Thus Alice and Bob do not compromise their
private keys or their shared secret in exchange

3 public key cryptography

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to 3 public key cryptography (20)

Recently uploaded (20)

3 public key cryptography