Quadratic Ciphers - Introduction to Cryptography with Java Applets - page 182

then compute n = 6911 6947 = 48010717. The value of n can be made known to anyone,

and in fact is necessary for encryption.
We wish to encipher the message

SHOOT NOW GEEK

which we will regroup into blocks of four letters each,

SHOO TNOW GEEK

then convert the characters into their numerical equivalents. Leading zeros are important:

18071414 19131422 06040410.

Notice that should our messages not be evenly divisible into blocks of size 4, we should
use some type of padding scheme. We proceed to encrypt the first block:

C

18071414 2

1339280 (mod 48010717)

This residue is the first ciphertext block. The second block we encrypt as follows:

C

19131422 2

22338923 (mod 48010717)

and the third as:

C

6040410 2

40412478 (mod 48010717)

The transmitted enciphered message is the sequence of numbers

01339280 22338923 40412478.

Now, if you have done the job right and haven't told anyone about the two secret num-
bers, p = 6911, and q = 6947, you should be the only individual able to decrypt. (Of course,
in this example, n = 48010717 is easily factorable into n = 6911 6947; in reality we would
use a much larger block size, and much larger primes.) To decrypt the first enciphered block,
you must solve the congruence

2 (mod 48010717)

1339280 P

for P . Using the Chinese Remainder Theorem, we derive the four roots

→ P 18071414 (mod 48010717)

P 16274554 (mod 48010717)

P 29939303 (mod 48010717)

P 31736163 (mod 48010717).