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PUBLIC KEY CRYPTOGRAPHY 
RSA ENCRYPTION ALGORITHM 
Meenakshi Shetti 
Muthu Gomahty V
CONTENTS 
• CRYPTOGRAPHY 
• WHAT IS A KEY ? 
• PRIVATE KEY CRYPTOGRAPHY 
• PUBLIC KEY CRYPTOGRAPHY 
• RSA ALGORITHM 
• ADVANTAGES 
• DISADVANTAGES 
• REFERENCES
CRYPTOGRAPHY 
•It’s a greek word which means hidden secret in 
writing 
•Cryptography is the practice and study of 
techniques for secure communication in the 
presence of third parties(called adversaries).
What is a “key”? 
A key is a piece of information (a parameter) that 
determines the functional output of a 
cryptographic algorithm or cipher.
PRIVATE KEY CRYPTOGRAPHY 
• Also called as Symmetric-key algorithms 
• They are a class of algorithms for cryptography that 
use the same cryptographic keys for both encryption 
of plaintext and decryption of ciphertext.
Public key cryptography 
• Also known as asymmetric cryptography 
• Refers to a cryptographic algorithm which requires two separate keys, one 
of which is secret (or private) and one of which is public.
Non secret ENCRYTION USING LOCK 
ALICE BOB
ENCRYPTION 
DECRYPTION
EVE 
ALICE BOB
TRAP DOOR –ONE WAY FUNCTION 
EASY 
HARD
EXPONENET 
REMAINDER 
12345 
mod 17 ≡ 135903 
346 mod 12 ≡10 
BASE 
MODULUS
memod N ≡ c
EASY 
memod N ≡ c 
HARD 
?emod N ≡ c
memod N ≡ c 
emod N- public key 
C -remainder 
m- message
me mod N ≡ c 
cd mod N ≡ m 
medmod N ≡ m 
e- encryption 
d - decryption
For computation of e and d 
STEP 1 -> PRIME FACTORIZATION 
STEP 2 -> PHI FUNCTION 
STEP 3-> EULER’S THEOREM
Multiplication of two extra large 
numbers are easy to compute. 
But prime factorization of a 
number is the hardness of the 
problem . 
Prime factorization is what used 
to build the trap door
STEP 1 -> PRIME FACTORIZATION 
P1 – 150 digits long 
P2 – 150 digits long 
P1 * P2 = N 
N- 300 digits long
STEP 2 -> PHI FUNCTION 
- breakability of a number 
Given a number N – it output’s how many integers are 
less than or equal to N that do not share a common 
factor with N 
ɸ[8] = 1 
2 
3 
4 
5 
6 
7 
8 
ɸ[8] = 1 
2 
3 
4 
5 
6 
7 
8 
We want to find ɸ[8] , 
we look at all integers 
from 1 to 8 , then we 
count how many 
integers does not 
share a factor greater 
than 1 
ɸ[8] = 4
• In the case of ɸ of a prime number – 
As prime numbers does not share common 
factor of any number greater than 
ɸ[P]=P-1 
i.e, ɸ[7] = 1 
2 
3 
4 
5 
6 
7 
As none of them share a common 
factor with 7 
ɸ[7] = 7-1 
ɸ[7] = 6
ɸ[N] is also multiplicative 
ɸ[A*B] = ɸ[A] * ɸ[B] 
= (A-1) * (B-1) 
ɸ[N] = ɸ[P1] * ɸ[P2] 
ɸ[N] = (P1-1) * (P2-1) 
77=7*11 
ɸ[7] = ɸ[7] * ɸ[11] 
ɸ[7] = (7-1) * (11-1) = 6 * 10 =60
STEP 3-> EULER’S THEOREM 
- Relation between the phi function and modular 
exponentiation 
mɸ[N]= 1 mod N 
Pick 2 numbers that do not share a common factor 
m=5, n=8 
5ɸ[8]= 1 mod 8 
54= 1 mod 8 
625=1 mod 8
Modify this equation using 2 simple rules 
1) 1k=1 
mk*ɸ[N]= 1 mod N 
We multiply eponent ɸ[N] by any number k, 
the solution is still 1 
2) 1*m=m 
m*mk*ɸ[N]= m mod N 
mk*ɸ[N]+1= m mod N
We now have an equation to find e and d which depends 
on ɸ[N] 
mk*ɸ[N]+1= m mod N 
me*d= m mod N 
Where d= k*ɸ[N]+1 
e 
Meaning d is ALICE’s private key . 
It is the trap door which will perform undo operation
EVE 
N=3127 
ALICE BOB 
P1=53 
d=2011 
P1=59 
N= 53* 59 
N=3127 
ɸ[N]=52*58 
e=3 
d=2*(3016)+1 
3 
d=2011 
e=3 
hi 
m=hi 
m=89 
893 mod 3127=1394 
c=1394 
ɸ[N]=3016 
cd mod N = m 
13942011 mod 3127 = 89 
m=89 
m=hi 
c=1394
• Any one wth N, e and c can find d if and only if they know 
the prime factorization of N 
• If N is large enough it requirs 100 to 1000 years to find 
factorize 
• It is the most widely used public key cryptography 
algorithm and most copied software in the history 
• Every internet user is using RSA whether they realise on 
the hardness of prime factorization which results in deep 
question of distribution of prime numbers.
APPLICATIONS 
• When it comes to assymetric cryptography the most 
popular and widely used application that comes to 
anyone's mind is PGP. PGP stands for “Pretty Good 
Privacy” and is the standard public key cryptography 
application used today. In the examples of this project 
we chose to use PGP Desktop. The reason for this 
choice is that PGP Desktop is easier to use than other 
text-based versions of PGP such as gnuPGP. PGP 
Desktop provides us with a very intuitive GUI 
accessible from the Windows Start Menu ,the PGP 
taskbar icon and from Windows explorer (shell 
integration). So from now on, every time we mention 
PGP, we will be referring to the PGP Desktop version.
ADVANTAGES 
1. Convenience: It solves the problem of distributing the key for encryption. 
2. Provides for message authentication: Public key encryption allows the use 
of digital signatures which enables the recipient of a message to verify that 
the message is truly from a particular sender. 
3. Detection of tampering: The use of digital signatures in public key 
encryption allows the receiver to detect if the message was altered in transit. 
A digitally signed message cannot be modified without invalidating the 
signature. 
4. Provide for non-repudiation: Digitally signing a message is akin to 
physically signing a document. It is an acknowledgement of the message and 
thus, the sender cannot deny it.
DISADVANTAGES 
1. Public keys should/must be authenticated: No one can be absolutely sure that a 
public key belongs to the person it specifies and so everyone must verify that their public 
keys belong to them. 
2. Slow: Public key encryption is slow compared to symmetric encryption. Not feasible for 
use in decrypting bulk messages. 
3. Uses up more computer resources: It requires a lot more computer supplies 
compared to single-key encryption. 
4. Widespread security compromise is possible: If an attacker determines a 
person's private key, his or her entire messages can be read. 
5. Loss of private key may be irreparable: The loss of a private key means that all 
received messages cannot be decrypted
REFERENCES 
1. Frederick J. Hirsch. "SSL/TLS Strong Encryption: An 
Introduction". Apache HTTP Server. Retrieved 
2013-04-17.. The first two sections contain a very 
good introduction to public-key cryptography. 
2. N. Ferguson; B. Schneier (2003). Practical 
Cryptography. Wiley. ISBN 0-471-22357-3. 
3. J. Katz; Y. Lindell (2007). Introduction to Modern 
Cryptography. CRC Press. ISBN 1-58488-551-3. 
4. A. J. Menezes; P. C. van Oorschot; S. A. 
Vanstone (1997). Handbook of Applied 
Cryptography. ISBN 0-8493-8523-7.
THANK YOU

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RSA - ALGORITHM by Muthugomathy and Meenakshi Shetti of GIT COLLEGE

  • 1. PUBLIC KEY CRYPTOGRAPHY RSA ENCRYPTION ALGORITHM Meenakshi Shetti Muthu Gomahty V
  • 2. CONTENTS • CRYPTOGRAPHY • WHAT IS A KEY ? • PRIVATE KEY CRYPTOGRAPHY • PUBLIC KEY CRYPTOGRAPHY • RSA ALGORITHM • ADVANTAGES • DISADVANTAGES • REFERENCES
  • 3. CRYPTOGRAPHY •It’s a greek word which means hidden secret in writing •Cryptography is the practice and study of techniques for secure communication in the presence of third parties(called adversaries).
  • 4. What is a “key”? A key is a piece of information (a parameter) that determines the functional output of a cryptographic algorithm or cipher.
  • 5. PRIVATE KEY CRYPTOGRAPHY • Also called as Symmetric-key algorithms • They are a class of algorithms for cryptography that use the same cryptographic keys for both encryption of plaintext and decryption of ciphertext.
  • 6. Public key cryptography • Also known as asymmetric cryptography • Refers to a cryptographic algorithm which requires two separate keys, one of which is secret (or private) and one of which is public.
  • 7. Non secret ENCRYTION USING LOCK ALICE BOB
  • 10. TRAP DOOR –ONE WAY FUNCTION EASY HARD
  • 11. EXPONENET REMAINDER 12345 mod 17 ≡ 135903 346 mod 12 ≡10 BASE MODULUS
  • 13. EASY memod N ≡ c HARD ?emod N ≡ c
  • 14. memod N ≡ c emod N- public key C -remainder m- message
  • 15. me mod N ≡ c cd mod N ≡ m medmod N ≡ m e- encryption d - decryption
  • 16. For computation of e and d STEP 1 -> PRIME FACTORIZATION STEP 2 -> PHI FUNCTION STEP 3-> EULER’S THEOREM
  • 17. Multiplication of two extra large numbers are easy to compute. But prime factorization of a number is the hardness of the problem . Prime factorization is what used to build the trap door
  • 18. STEP 1 -> PRIME FACTORIZATION P1 – 150 digits long P2 – 150 digits long P1 * P2 = N N- 300 digits long
  • 19. STEP 2 -> PHI FUNCTION - breakability of a number Given a number N – it output’s how many integers are less than or equal to N that do not share a common factor with N ɸ[8] = 1 2 3 4 5 6 7 8 ɸ[8] = 1 2 3 4 5 6 7 8 We want to find ɸ[8] , we look at all integers from 1 to 8 , then we count how many integers does not share a factor greater than 1 ɸ[8] = 4
  • 20. • In the case of ɸ of a prime number – As prime numbers does not share common factor of any number greater than ɸ[P]=P-1 i.e, ɸ[7] = 1 2 3 4 5 6 7 As none of them share a common factor with 7 ɸ[7] = 7-1 ɸ[7] = 6
  • 21. ɸ[N] is also multiplicative ɸ[A*B] = ɸ[A] * ɸ[B] = (A-1) * (B-1) ɸ[N] = ɸ[P1] * ɸ[P2] ɸ[N] = (P1-1) * (P2-1) 77=7*11 ɸ[7] = ɸ[7] * ɸ[11] ɸ[7] = (7-1) * (11-1) = 6 * 10 =60
  • 22. STEP 3-> EULER’S THEOREM - Relation between the phi function and modular exponentiation mɸ[N]= 1 mod N Pick 2 numbers that do not share a common factor m=5, n=8 5ɸ[8]= 1 mod 8 54= 1 mod 8 625=1 mod 8
  • 23. Modify this equation using 2 simple rules 1) 1k=1 mk*ɸ[N]= 1 mod N We multiply eponent ɸ[N] by any number k, the solution is still 1 2) 1*m=m m*mk*ɸ[N]= m mod N mk*ɸ[N]+1= m mod N
  • 24. We now have an equation to find e and d which depends on ɸ[N] mk*ɸ[N]+1= m mod N me*d= m mod N Where d= k*ɸ[N]+1 e Meaning d is ALICE’s private key . It is the trap door which will perform undo operation
  • 25. EVE N=3127 ALICE BOB P1=53 d=2011 P1=59 N= 53* 59 N=3127 ɸ[N]=52*58 e=3 d=2*(3016)+1 3 d=2011 e=3 hi m=hi m=89 893 mod 3127=1394 c=1394 ɸ[N]=3016 cd mod N = m 13942011 mod 3127 = 89 m=89 m=hi c=1394
  • 26. • Any one wth N, e and c can find d if and only if they know the prime factorization of N • If N is large enough it requirs 100 to 1000 years to find factorize • It is the most widely used public key cryptography algorithm and most copied software in the history • Every internet user is using RSA whether they realise on the hardness of prime factorization which results in deep question of distribution of prime numbers.
  • 27. APPLICATIONS • When it comes to assymetric cryptography the most popular and widely used application that comes to anyone's mind is PGP. PGP stands for “Pretty Good Privacy” and is the standard public key cryptography application used today. In the examples of this project we chose to use PGP Desktop. The reason for this choice is that PGP Desktop is easier to use than other text-based versions of PGP such as gnuPGP. PGP Desktop provides us with a very intuitive GUI accessible from the Windows Start Menu ,the PGP taskbar icon and from Windows explorer (shell integration). So from now on, every time we mention PGP, we will be referring to the PGP Desktop version.
  • 28. ADVANTAGES 1. Convenience: It solves the problem of distributing the key for encryption. 2. Provides for message authentication: Public key encryption allows the use of digital signatures which enables the recipient of a message to verify that the message is truly from a particular sender. 3. Detection of tampering: The use of digital signatures in public key encryption allows the receiver to detect if the message was altered in transit. A digitally signed message cannot be modified without invalidating the signature. 4. Provide for non-repudiation: Digitally signing a message is akin to physically signing a document. It is an acknowledgement of the message and thus, the sender cannot deny it.
  • 29. DISADVANTAGES 1. Public keys should/must be authenticated: No one can be absolutely sure that a public key belongs to the person it specifies and so everyone must verify that their public keys belong to them. 2. Slow: Public key encryption is slow compared to symmetric encryption. Not feasible for use in decrypting bulk messages. 3. Uses up more computer resources: It requires a lot more computer supplies compared to single-key encryption. 4. Widespread security compromise is possible: If an attacker determines a person's private key, his or her entire messages can be read. 5. Loss of private key may be irreparable: The loss of a private key means that all received messages cannot be decrypted
  • 30. REFERENCES 1. Frederick J. Hirsch. "SSL/TLS Strong Encryption: An Introduction". Apache HTTP Server. Retrieved 2013-04-17.. The first two sections contain a very good introduction to public-key cryptography. 2. N. Ferguson; B. Schneier (2003). Practical Cryptography. Wiley. ISBN 0-471-22357-3. 3. J. Katz; Y. Lindell (2007). Introduction to Modern Cryptography. CRC Press. ISBN 1-58488-551-3. 4. A. J. Menezes; P. C. van Oorschot; S. A. Vanstone (1997). Handbook of Applied Cryptography. ISBN 0-8493-8523-7.