Public key algorithms like RSA and ElGamal allow for secure encryption without a shared private key. RSA uses a public and private key pair generated from large prime numbers such that a message encrypted with the public key can only be decrypted by the corresponding private key. It is widely used due to its security being based on the difficulty of factoring large numbers, though it is less efficient than symmetric algorithms due to involving modular exponentiation. ElGamal also uses a public/private key approach and its security relies on the discrete logarithm problem.

1. Public Key
Algorithm
By
Pankaj Jatav (110101168)
Rahul Kumar (120101813)
Rahul Aggarwal
(110101189)
Prateek Pandey
(110101180)

2. Cryptography
 The art of the Secret (crypto-) writing (-graphy). It enables you to
store information or transmit
it across insecure networks, so that it cannot be read by anyone
except the intended recipient.

4. TERMINOLOGY
 Plain Text:- The message or data that is to be transmitted over the
network.
 Cipher :- A mapping algorithm which is used to encrypt or decrypt
the message.
 Key : A key is a number (or a set of numbers) that the cipher
implements to encrypt or decrypt a message.
To encrypt a message we need to convert the plaintext to ciphertext
using an encryption algorithm and encryption key whereas to
decrypt the message we require a decryption algorithm and a
decryption key to reveal the plaintext

5. Symmetric Key Cryptography
 Symmetric-key cryptography refers to encryption methods in which
both the sender and receiver share the same key (or, less
commonly, in which their keys are different, but related in an easily
computable way). This was the only kind of encryption publicly
known until June 1976.
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6. Symmetric Key Cryptography
System

7. Asymmetric Key Cryptography
Asymmetric cryptography or public-key cryptography is cryptography
in which a pair of keys is used to encrypt and decrypt a message so
that it arrives securely. Initially, a network user receives a public and
private key pair from a certificate authority. Any other user who wants
to send an encrypted message can get the intended recipient's public
key from a public directory. They use this key to encrypt the message,
and they send it to the recipient. When the recipient gets the message,
they decrypt it with their private key, which no one else should have
access to.
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8. ASYMMETRIC KEY
Cryptography System

9. Why use Public Key
Algorithms??

10. ASYMMETRIC KEY
Cryptography System

11. Public Key Algorithms
 Proposed by Diffie and Hellman in 1976.
 The encryption algorithm must meet 3 requirements:
1. D(E(P))=P
2. It is Exceedingly difficult to deduce D from E.
3. E cannot be broken by a chosen plaintext attack.

13. Public key blueprint
 The keys used to encrypt and decrypt are
different.
 Anyone who wants to be a receiver needs to
“publish” an encryption key, which is known as
the public key.
 Anyone who wants to be a receiver needs a
unique decryption key, which is known as the
private key.
 It should not be possible to deduce the plaintext
from knowledge of the ciphertext and the public
key.
 Some guarantee needs to be offered of the
authenticity of a public key.
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14. Public Key Algorithms
Algorithms Satisfying all three requirements
are:
 RSA
 ElGamal

15. RSA
Given by Rivest, Shamir & Adleman of MIT in
1977
Best known & widely used public-key scheme
Based on exponentiation in a finite field over
integers modulo a prime
Uses large integers (eg. 1024 bits)
Security due to cost of factoring large numbers

16. RSA
 Choose two large prime numbers p, q. (e.g.,
1024 bits each)
 Compute n = p*q and z = (p-1)(q-1).
 Choose d (with d<n) that has no common
factors with z. (e, z are “relatively prime”).
 Choose e such that ed-1 is exactly divisible by
z. (in other words: ed mod z = 1 ).

17. RSA: Encryption, decryption
 Given (n,e) and (n,d) as computed above
 Dividing Plain Text into Blocks: 0<=P<n
 To encrypt message P (<n), compute
e
C = M (mod
n)
 To decrypt received bit pattern, C, compute
d
M = C (mod
n)
 Public key is (n,e).
 Private key is (n,d).

18. RSA example:
Let us choose p=3 and q=11.
Then n=33 and z=20.
d=7 (so d, z relatively prime).
e=3 (so ed-1 exactly divisible by z).
Encrypting messages:
M M
e
C = Me mod n
19 6859 28
d
C C
M = Cd mod n
28 13492928512 19
Encrypt:
Decrypt:

19. RSA Issues
 RSA is computationally intense.
 Commonly used key lengths are 1024
bits
 The plain text should be smaller than
the key length
 The encrypted text is of the same size
as the key length
 Generally used to encrypt secret keys.
 Basis: Factoring a big number is hard

20. Summary
 Public key systems replace the problem of distributing
symmetric keys with one of authenticating public keys
 Public key encryption algorithms need to be trapdoor one-way
functions
 RSA is a public key encryption algorithm whose security is
believed to be based on the problem of factoring large
numbers
 ElGamal is a public key encryption algorithm whose
security is believed to be based on the discrete logarithm
problem
 RSA is generally favoured over ElGamal for practical rather
than security reasons
 RSA and ElGamal are less efficient and fast to operate
than most symmetric encryption algorithms because they
involve modular exponentiation
 DH key exchange is an important protocol on which many
real key exchange protocols are based

21. Thank You

22. Attacks on RSA
 Smooth Number Attack:
 If you sign m1 and m2
 S1 = md
1 mod n
 S2 = m2d mod n
 Attacker can sign m1m2, m1/m2, m12, m1jm2k
 Easy to do if mi’s are small (smooth) numbers.
 Cube Root Problem of RSA
 If public exponent e=3:
 hde mod n = h
 hd mod n = h1/3
 Simply compute h1/3 mod n