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The RSA Public-Key Encryption Algorithm

mod using private primes

Definition: The RSA is one of the popular public-key encryption algorithms.

Rivest et al published a method for obtaining digital signatures and public-key cryptosystems in 1978. In order to use the method, the encryption and decryption keys must be chosen as follows:

  1. Compute n as the product of two primes p and q: n = p×q. These two primes are very large and randomly selected primes.
  2. Compute f ( n ) = ( p –1)( q – 1).
  3. Select e such that e is relatively prime to f ( n ).
  4. Select d such that e×d = 1 mod ( f ( n )), where mod stands for modular operation.
  5. Choose ( e, n ) as the public key.
  6. Choose ( d, n ) as the private key.

In encryption and decryption, receiver‘s public key ( e R ,n R ) and private key ( d R , n R ) are used. Encryption is carried out by using m e = m e R mod n R , where m is the plaintext and m c is the ciphertext. Decryption is carried out by using m = ( m d R ) dR mod n R .

For authentication, sender‘s public key ( e s , n s ) and private key ( d s , n s ) are used. The signature is generated by using m c = m ds mod n s . The signature is authenticated by using m = ( m e ) e s mod n s .

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