The opponent possesses a string of ciphertext and has access to the encryption machinery as black box. What is the name of this attack? a quick example:

The encryption function: 
y = E(x) = (13x + 9)(mod 27), when the letters A–Z are taken to be the numbers 0–25 and the Space (punctuation) is the number 26.

The opponent has the ciphertext y = "NZC"

For each letter, he will cipher it again and again until he gets the starting value.

I'll show the letter 'N': 'N' = 13

E(13) = 16

E(16) = 1

E(1) = 22

E(22) = 25

E(25) = 10

E(10) = 4

E(4) = 7

E(7) = 19

E(19) = 13

So, the opponent knows that the letter 'T' = 19 was encrypted to get 'N'

Same goes to 'Z' = 25, the letter 'W' = 22 was encrypted to get 'Z'

and for 'C' = 2, the letter 'O' = 14 was encrypted to get 'C' 

From that, the opponent reveals the the plaintext: "TWO"

What is the name of this attack?

Thanks.
Actually, this specific attack against RSA is known as a 'cycling attack'; it works because RSA defines a permutation, and so by doing repeated encryptions a sufficient number of times, you'll always get back to where to started with eventually.

However, people have analyzed this attack, and shown that (with extremely high probability) the amount of work required by this attack (for a randomly chosen RSA modulus) takes more work than the known factorization methods.