rsa - algorithm by muthugomathy and meenakshi shetti of git college
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PUBLIC KEY CRYPTOGRAPHYRSA ENCRYPTION
ALGORITHM
Meenakshi ShettiMuthu 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
DECRYPTION
ENCRYPTION
EVE
ALICE BOB
TRAP DOOR –ONE WAY FUNCTION
EASY
HARD
46 mod 12 ≡10312345mod 17 ≡ 3910135
BASE
EXPONENET
MODULUS
REMAINDER
memod N ≡ c
EASY
HARD
memod N ≡ c
?emod N ≡ c
memod N ≡ cemod N- public key
m- message C -remainder
cd mod N ≡ mmedmod N ≡ m
me mod N ≡ c
e- encryptiond - decryption
STEP 1 -> PRIME FACTORIZATION
STEP 2 -> PHI FUNCTION
STEP 3-> EULER’S THEOREM
For computation of e and d
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 thanɸ[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 rules1) 1k=1mk*ɸ[N]= 1 mod NWe multiply eponent ɸ[N] by any number k,
the solution is still 12) 1*m=mm*mk*ɸ[N]= m mod Nmk*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 NWhere d= k*ɸ[N]+1
e
Meaning d is ALICE’s private key .
It is the trap door which will perform undo operation
EVE
ALICE BOBP1=53
P1=59N= 53* 59
ɸ[N]=52*58
e=3d=2*(3016)+1 3d=2011
N=3127
e=3N=3127
him=him=89
893 mod 3127=1394
e=3
N=3127
c=1394
ɸ[N]=3016
d=2011
13942011 mod 3127 = 89
cd mod N = m
m=89m=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.
ADVANTAGES1. 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.
DISADVANTAGES1. 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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