The RSA Algorithm: Easy Mechanics of Encryption and Decryption

RSA algorithm is a public key algorithm. It falls in the category of asymmetric cryptographic algorithm. It uses two keys private and public. The public key is known to all users in the network while the private key is kept secret and not shareable to all. It is named on its creates names -Rivest, Shamir and Adleman. It is based on the principle that it is easy to multiply two prime numbers but it is very hard to factor the product and get them back. This algorithm came into existence in the year 1977. The algorithm  works in the following mentioned steps:

Step1. Take two large prime numbers. Let them consider A and B, they should be of equal length. Now obtain their product(N).

         Therefore, N=A*B

Step 2. Subtract 1 from both and take their product as T

         Therefore, T=(A-1)*(B-1)

 Step 3. Choose the public key(E) which is a randomly chosen number such that it has no common factor with T.

 Step 4.Acquire the private key(D) as mentioned below:

        D = E^-1 mod T

Step5. The rule  for encrypting the block of plaintext M into cyphertext is as under:

      C = M^e mod N

This equation tells plaintext that M is raised to the power of E(public key) and then divided by N. The mod term equation mentioned in the fifth step indicates that the remainder of this division is sent as ciphertext C.

Step 6. Now the message which will be received from C at the receiver is decrypted to get the plaintext back by using the following equation:

M= C^D mod N

The overall encryption and decryption algorithm is illustrated in the below diagram:

Security of RSA

The RSA algorithm provides better security as it uses a large number of prime numbers A and B . And it is arduous even for a faster computer to break the code and to predict the product of prime numbers A and B. However as the computer improves with the passage of time, it decreases the time require to break the code. As a result, it also required to use of larger keys. However, then the time needed for encryption and decryption will also increase. The personal use key size is 768 bits . The key size for corporate usage is 1024 bits and 2048 bits for extremely valuable keys.

Example to Explain RSA algorithm

Select suitable values of A ,B,N,D and demonstrate the encryption and decryption procedures used in the RSA algorithm.

Select Two Prime Numbers

Let the two prime numbers A and B. Consider A=11 and B=23.

Evaluate N and T

N=A*B=11*23=253

T=(A-1)*(B-1)=10*22=220

Evaluate Public key (E) and Private key (D)

E(Public key ) should not have any factor other than 1 in common with T i.e. 220.

Thus, 220=2*2*55=2*2*5*11

Hence we can choose E=3

Now calculate D with the help of the following syntax:

D = E^-1 mod T

Therefore,D = 3^-1 mod 220

Next, we find the (multiple of 220+1)which is divisible by 3. Then we divide that number by 3 and select the quotient of this division as D.

Therefore ,(220*1)+1=220 not divisible by 3

(220*2)+1=441 not divisible by 3

Therefore, 441/3=147

or D=147

Carryout Encryption

Consider the letter 1 is to sent.

Therefore plaintext M=9,as F is the ninth alphabet.

Hence,    Ciphertext C=M^e mod N

C=9³ mod 221

Hence,C=729mod 221

orC=729/221

Q=3 remainder =66

Therefore Ciphertext C=66

Now, this number is sent to the receiver.

Carryout Decryption

Finally, let us calculate the decryption

Plaintext M= C^D mod N=66¹⁴⁷ mod 221=9.

Thus the original number is obtained.

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Advantages RSA Algorithm

• No need of sharing secret keys.
• Proof of owner’s authenticity.
• Faster encryption than DSA.
• Data can’t be modified in transit.

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