The document presents an overview of public key cryptography and the RSA algorithm, detailing the principles of encryption and decryption, the distinction between symmetric and asymmetric key cryptography, and the mathematical foundations of RSA. It describes the RSA algorithm's steps for key generation, encryption, and decryption, and discusses its practical applications in securing web traffic, emails, and remote connections. Despite RSA's popularity and advantages in security, it faces challenges such as slow key generation and decryption compared to symmetric key systems like DES.

1. Presented by-
Pallabi Patowary(Roll no-CSI15018)
Indrajit Kalita(Roll no-CSI15004)
Public Key Cryptography
& RSA Algorithm

2. CRYPTOGRAPHYCRYPTOGRAPHY
– study of techniques for secure communication in the presence of third
parties
– The science of using mathematics to encrypt and decrypt data.
– Cryptography enables us to store sensitive information or transmit it across
insecure networks so that it cannot be read by anyone except the intended
recipient
 Plaintext: The original intelligible message
 Cipher text: The transformed message
– Key: A key is a complex value that is determined using an algorithm .
 Encryption: The process of converting plaintext to cipher text
 Decryption: The process of converting cipher text back into plaintext.
Types of Cryptography-
1. Private Key Cryptography
2. Public Key Cryptography

3. PUBLIC-KEY CRYPTOGRAPHYPUBLIC-KEY CRYPTOGRAPHY
• Also known as Asymmetric-key cryptography
• Encryption and decryption are
carried out using two different
keys
• Symmetric key cryptography
uses same secret key for
encryption and decryption
• In 1976 Diffie and Hellman
Introduced public-key cryptography
• Based on mathematical
calculation

4. Public-Key CharacteristicsPublic-Key Characteristics
– computationally infeasible to find private key
knowing only algorithm & public key
– computationally easy to en/decrypt messages
when the relevant (private/public) key is known
– either of the two related keys can be used for
encryption and other is used for decryption

5. Pictorial scenario of PKCPictorial scenario of PKC

6. CONFIDENTIALITY
• protecting the information from disclosure to
unauthorized parties.
• ensures that only the right people (people
who knows the key) can read the information.

7. How PKC works when ConfidentialityHow PKC works when Confidentiality
is neededis needed
1. Each system generates a pair of keys.
2. Each system publishes its public key keeping its
companion key private.
3. If A wishes to send a message to B then it
encrypts the message using B’s public key.
4. When B receives the message, it decrypts the
message using its private key. No one else can
decrypt the message because only B knows its
private key.

8. A’s public and private keys – PUA and PRA.
B’s public and private keys -- PUB and PRB.
C = E(PUB ; M)
M= D(PRB ; C )
A’s public and private keys – PUA and PRA.
B’s public and private keys -- PUB and PRB.
C = E(PUB ; M)
M= D(PRB ; C )

9. AUTHENTICATION
• The sender and receiver can confirm each
other’s identity and origin/destination of the
information

10. A’s public and private keys – PUA and PRA.
B’s public and private keys -- PUB and PRB.
C = E(PRA ; M)
M= D(PUA ; C )
A’s public and private keys – PUA and PRA.
B’s public and private keys -- PUB and PRB.
C = E(PRA ; M)
M= D(PUA ; C )

11. A’s public and private keys – PUA and PRA.
B’s public and private keys - PUB and PRB.
C = E[PUB ; E(PRA ; M)]
M = D[PUA ; D(PRB ; C)]
A’s public and private keys – PUA and PRA.
B’s public and private keys - PUB and PRB.
C = E[PUB ; E(PRA ; M)]
M = D[PUA ; D(PRB ; C)]

12. DIFFERENCE BETWEEN SYMMETRIC AND ASYMMETRIC KEYDIFFERENCE BETWEEN SYMMETRIC AND ASYMMETRIC KEY
CRYPTOGRAPHYCRYPTOGRAPHY
Characteristic Symmetric key cryptography
Asymmetric key
cryptography
Key used for
encryption/decryption
Same key is used
One key is used for encryption
and another ;different key is
used for decryption
Speed of encryption/decryption Very fast Slower
Size of resulting encrypted text
Usually same as or less than the
original plain text size.
More than the original plain text
size
Known keys
Both parties should know the key
in symmetric key encryption
Only, either one of the keys is
known by the two parties in
public key encryption.
Usage Confidentiality
Confidentiality, digital signature
etc.

13. Public-Key ApplicationsPublic-Key Applications
– encryption/decryption(provide Confidentiality)
– digital signatures (provide authentication or proof
of origin)

14. RSA Algorithm - Introduction
 by Rivest, Shamir & Adleman of MIT in 1977
 best known & widely used public-key scheme

15. • RSA is the first practical public-key cryptosystems and is
widely used for secure data transmission
• Every number has a prime factorization and it is hard to
find if the number is very large. The time complexity is
also very high. But on the other hand the multiplication
is easy and time complexity is also not very high.
– For example the multiplication of 15 and 30 is 450 but the prime
factorization of 450 is not easy and also it takes more time as
compare to multiplication
• The RSA algorithm involves three steps: key generation,
encryption and decryption.

16. Mathematics behind RSAMathematics behind RSA
• Theorem 1 : Let a, p be two integers that are co-prime to each
other. Then, there is only a unique integer x Zp satisfying∈
Regardless of the value of b.
• Corollary(1) : If a and p are co-prime to each other, then 0, a, 2a, ...,
(p - 1)a are all distinct after modulo p.
• Theorem 2(Fermat’s Little Theorem) : If p is a prime number, for
any non-zero a Zp, it holds that∈
• Theorem 3(Chinese Remainder Theorem) : Let p and q be two co-
prime integers. Then
modax p b=
( 1)
mod 1p
a p−
=
If mod 1 and mod then
mod
x p x q a
x pq a
= =
=

17. Mathematics behind RSAMathematics behind RSA
• Eulers Totient Function(ø) : When doing arithmetic
modulo n then
The complete set of residues is: 0...(n-1)
The reduced set of residues is those numbers (residues)
which are relatively prime to n
Number of elements in reduced set of residues is called
the Euler Totient Function ø(n)
If the the number ‘n’ is a prime number then the value of
ø(n)=n-1

18. RSA ALGORITHMRSA ALGORITHM
• Choose two large prime numbers p and q randomly
• Let n = pq.
• Let (n)ɸ = (p - 1)(q - 1).
• Choose a large number e ϵ [2, -1]ɸ that is co-prime to (n)ɸ .
• Compute d [2, -1]ɸ such that
ed mod (n) = 1 or ed = k× (n)+1ɸ ɸ
There is a unique such d. Furthermore, d must be co-prime to (n)ɸ
• Announce to the whole word the pair (e; n), which is his public key.
• Keep the pair (d; n) secret to himself, which is his private key
• Sender will now use the public key to encrypt the message M as follows
C = Me
mod n. Here, C is the encrypted message.
• Receiver will use its own private key to decrypt the message by using the equation
M=Cd
mod n

19. PROOF OF RSAPROOF OF RSA
• The private key d is unique and d must be co-prime to (n)ɸ
Proof : From the above corollary (1) , we found that e and (n)ɸ are co-prime.
Therefore, there exists only one unique integer d, such that
ed mod (n) = 1ɸ .
Hence d is unique
Again, Let d is not co-prime so t is the greatest common divisor of d and (n)ɸ
Suppose
d = c1t and (n) = cɸ 2t.
Hence:
ec1t = c3c2t + 1
t(ec1- c3c2)=1
From ed mod (n) = 1.ɸ We know ed = c3 (n) + 1ɸ for some integer c3.
Therefore t=1.
Hence it is proved that the private key d is unique and co-prime with (n)ɸ .

20. PROOF OF RSAPROOF OF RSA
• How to Prove
Proof : Since n=pq, Therefore we need to prove the following two condition
mod moded
M n M n=
mod mod and
mod mod
ed
ed
M p M p
M q M q
=
=
1 ( ) 1
Now,
mod mod (Since 1 ( ) 1, 1 some constant)ed k n
M p M p ed k n kφ
φ× +
= = × + =
1 ( )
modk n
M M pφ×
=
1 ( 1)( 1)
mod (Since ( ) ( 1)( 1))k p q
M M p n p qφ× − −
= = − −
2 ( 1)
mod (Let 2 1 ( 1))k p
M M p k k q× −
= = × −
( 1) 2
( mod ) modp k
M p M p−
=
Now let us assume that M and p are co-prime then according to Fermat’s theorem
( 1)
mod 1p
M p−
=
Then
2
mod 1 mod moded k
M p M p M p= =

21. PROOFPROOF OF RSAOF RSA
Again if M is a multiple of prime p then
mod 0M p =
( 1) 2
Therefore,
mod ( mod ) 0 moded p k
M p M p M p−
= × =
Hence it is proved that,
mod moded
M p M p=
Similarly we can prove that mod moded
M q M q=
Now we can conclude that mod moded
M n M n=

22. PROOFPROOF OF RSAOF RSA
Proof:
Proved that modd
M C n=
We know that therefore,M n<
modM M n=
mod (Since mod mod )ed ed
M n M n M n= =
( mod ) mode d
M n n=
mod (Since modn)d e
C n C M= =
Hence proved

23. RSA ExampleRSA Example
• Select primes: p=17 & q=11
• Compute n = pq =17×11=187
• Compute ø(n)=(p–1)(q-1)=16×10=160
• Select e : gcd(e,160)=1; choose e=7
• Determine d: de=1 mod 160 and d < 160 Value is d=23 since
23×7=161= 10×160+1
• Publish public key KU={7,187}
• Keep secret private key KR={23,187}
• sample RSA encryption/decryption is:
• given message M = 88 (note that 88<187)
• encryption:
C = 887
mod 187 = 11
• decryption:
M = 1123
mod 187 = 88

24. PRACTICAL APPLICATIONS OF THE RSA ALGORITHMPRACTICAL APPLICATIONS OF THE RSA ALGORITHM
• To this day the RSA together with the AES algorithm is
the mostly used algorithm in commercial systems.
It is used:
-to protect web traffic, in the SSL protocol
(Security Socket Layer),
-to guarantee email privacy and authenticity in
PGP (Pretty Good Privacy)
-to guarantee remote connection in SSH (Secure
Shell).
-Furthermore it plays an important role in the
modern payment systems through SET protocol (Secure
Electronic Transaction).
• RSA has been used in most digital data, information and
telephone security applications.

25. ADVANTAGES OF RSAADVANTAGES OF RSA
• Very fast, very simple encryption and verification.
• Easier to implement than Elliptical curve
cryptography(ECC).
• Easier to understand.
• Widely deployed, better industry support.
DISADVANTAGES OF RSADISADVANTAGES OF RSA
• Very slow key generation.
• Slow decryption, which is slightly tricky to implement
securely.
• Two-part key is vulnerable to GCD attack if poorly
implemented.

26. CONCLUSIONCONCLUSION
• RSA is the most popular public-key cryptosystem available today. It can be
used for both encryption and authentication. The security of RSA is
related to the assumption that factoring is difficult. An easy factoring
method or some other feasible attack would break RSA.
• RSA is built into current operating systems by Microsoft, Apple, Sun, and
Novell. In hardware, RSA can be found in secure telephones, on Ethernet
network cards, and on smart cards. The estimated installed base of RSA
encryption engines is around 20 million, making it by far the most widely
used public-key cryptosystem in the world.
• Finally, DES is much faster than RSA. In software, DES is generally at least
100 times as fast as RSA. In hardware, DES is between 1,000 and 10,000
times as fast, depending on the implementation. Implementations of RSA
will probably narrow the gap a bit in coming years, as there are growing
commercial markets, but DES will get faster as well.