Chinese_remainder_theorem

https://en.wikipedia.org/wiki/Chinese_remainder_theorem

中国剩余定理

https://en.wikipedia.org/wiki/RSA_(cryptosystem)

The public key is (n = 3233, e = 17). For a padded ​​plaintext​​ message m, the encryption function is

 

The private key is (n = 3233, d = 413). For an encrypted ​​ciphertext​​ c, the decryption function is

 

For instance, in order to encrypt m = 65, we calculate

 

To decrypt c = 2790, we calculate

 

Both of these calculations can be computed efficiently using the ​​square-and-multiply algorithm​​ for ​​modular exponentiation​​. In real-life situations the primes selected would be much larger; in our example it would be trivial to factor n, 3233 (obtained from the freely available public key) back to the primes p and q. e, also from the public key, is then inverted to get d, thus acquiring the private key.

Practical implementations use the ​​Chinese remainder theorem​​ to speed up the calculation using modulus of factors (mod pq using mod p and mod q).

The values d_p, d_q and q_inv, which are part of the private key are computed as follows:

 

Here is how d_p, d_q and q_inv are used for efficient decryption. (Encryption is efficient by choice of a suitable d and e pair)