Tag: Public Key Cryptosystem

Public Key Cryptography | RSA Algorithm Example

Computer Networks

Cryptography in Network Security-

 

Before you go through this article, make sure that you have gone through the previous article on Cryptography.

 

We have discussed-

  • Cryptography is a method of storing and transmitting data in a particular form.
  • Cryptography techniques are-

 

 

In this article, we will discuss about Asymmetric Key Cryptography.

 

Asymmetric Key Cryptography-

 

In this technique,

  • Sender and receiver use different keys to encrypt and decrypt the message.
  • It is called so because sender and receiver use different keys.
  • It is also called as public key cryptography.

 

Working-

 

The message exchange using public key cryptography involves the following steps-

 

 

Step-01:

 

At sender side,

  • Sender encrypts the message using receiver’s public key.
  • The public key of receiver is publicly available and known to everyone.
  • Encryption converts the message into a cipher text.
  • This cipher text can be decrypted only using the receiver’s private key.

 

Step-02:

 

  • The cipher text is sent to the receiver over the communication channel.

 

Step-03:

 

At receiver side,

  • Receiver decrypts the cipher text using his private key.
  • The private key of the receiver is known only to the receiver.
  • Using the public key, it is not possible for anyone to determine the receiver’s private key.
  • After decryption, cipher text converts back into a readable format.

 

Advantages-

 

The advantages of public key cryptography are-

  • It is more robust.
  • It is less susceptible to third-party security breach attempts.

 

Disadvantages-

 

The disadvantages of public key cryptography are-

  • It involves high computational requirements.
  • It is slower than symmetric key cryptography.

 

Number of Keys Required-

 

To use public key cryptography,

  • Each individual requires two keys- one public key and one private key.
  • For n individuals to communicate, number of keys required = 2 x n = 2n keys.

 

Asymmetric Encryption Algorithms-

 

The famous asymmetric encryption algorithms are-

 

 

  1. RSA Algorithm
  2. Diffie-Hellman Key Exchange

 

In this article, we will discuss about RSA Algorithm.

 

RSA Algorithm-

 

Let-

  • Public key of the receiver = (e , n)
  • Private key of the receiver = (d , n)

 

Then, RSA Algorithm works in the following steps-

 

Step-01:

 

At sender side,

  • Sender represents the message to be sent as an integer between 0 and n-1.
  • Sender encrypts the message using the public key of receiver.
  • It raises the plain text message ‘P’ to the eth power modulo n.
  • This converts the message into cipher text ‘C’.

 

C = Pe mod n

 

Step-02:

 

  • The cipher text ‘C’ is sent to the receiver over the communication channel.

 

Step-03:

 

At receiver side,

  • Receiver decrypts the cipher text using his private key.
  • It raises the cipher text ‘C’ to the dth power modulo n.
  • This converts the cipher text back into the plain text ‘P’.

 

P = Cd mod n

 

NOTE-

 

‘e’ and ‘d’ must be multiplicative inverses modulo Ø(n)

 

After decryption, receiver must have-

P = Cd mod n

P = (Pe mod n)d mod n

P = Ped mod n

For this equation to be true, by Euler’s Theorem, we must have-

ed = 1 mod Ø(n)

OR

ed = kØ(n) + 1

Thus, e and d must be multiplicative inverses modulo Ø(n).

 

Steps to Generate Public Key And Private Key-

 

An individual can generate his public key and private key using the following steps-

 

Step-01:

 

Choose any two prime numbers p and q such that-

  • They are different.
  • They are very large.

 

Step-02:

 

Calculate ‘n’ and toilent function Ø(n) where-

  • n = p x q
  • Ø(n) = (p-1) x (q-1)

 

Step-03:

 

Choose any value of ‘e’ such that-

  • 1 < e < Ø(n)
  • gcd (e, Ø(n)) = 1

 

Step-04:

 

Determine ‘d’ such that-

 

 

  • You already know the value of ‘e’ and Ø(n).
  • Choose the least positive integer value of ‘k’ which gives the integer value of ‘d’ as a result.
  • Use trial and error method.
  • Start substituting different values of ‘k’ from 0.

 

PRACTICE PROBLEMS BASED ON RSA ALGORITHM-

 

Problem-01:

 

In a RSA cryptosystem, a participant A uses two prime numbers p = 13 and q = 17 to generate her public and private keys. If the public key of A is 35, then the private key of A is _______.

 

Solution-

 

Given-

  • Prime numbers p = 13 and q = 17
  • Public key = 35

 

Step-01:

 

Calculate ‘n’ and toilent function Ø(n).

 

Value of n,

n = p x q

n = 13 x 17

∴ n = 221

 

Toilent function,

Ø(n) = (p-1) x (q-1)

Ø(n) = (13-1) x (17-1)

∴ Ø(n) = 192

 

Step-02:

 

  • We are already given the value of e = 35.
  • Thus, public key = (e , n) = (35 , 221)

 

Step-03:

 

Determine ‘d’ such that-

 

 

Here,

  • The least value of ‘k’ which gives the integer value of ‘d’ is k = 2.
  • On substituting k = 2, we get d = 11.

 

Thus, private key of participant A = (d , n) = (11, 221).

 

Problem-02:

 

In the RSA public key cryptosystem, the private and public keys are (e, n) and (d, n) respectively, where n = p x q and p and q are large primes. Besides, n is public and p and q are private. Let M be an integer such that 0 < M < n and f(n) = (p-1)(q-1).

 

Now consider the following equations-

I. M’ = Me mod n and M = (M’)d mod n

II. ed ≡ 1 mod n

III. ed = 1 mod f(n)

IV. M’ = Me mod f(n) and M = (M’)d mod f(n)

 

Which of the above equations correctly represent RSA cryptosystem?

  1. I and II
  2. I and III
  3. II and IV
  4. III and IV

 

Solution-

 

Clearly, Option (B) is correct.

 

To gain better understanding about RSA Algorithm,

Watch this Video Lecture

 

Next Article- Diffie Hellman Key Exchange Algorithm

 

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