# Real life application of Cyclic group [closed]

The encryption in Caesar cipher given by: $$E_k(P)\equiv P+k\,(\mathrm{mod}\,26)$$, where $$P$$ is the plain text and $$k$$ is the shift key. and The decryption in Caesar cipher given by: $$D_k(C)\equiv C-k\,(\mathrm{mod}\,26)$$, where $$C$$ is the cipher text.

Julius Caesar used this cipher to encrypt secret messages. If he had anything confidential to say, he wrote it in cipher, that is, by so changing the order of the letters of the alphabet, that not a word could be made out. If anyone wishes to decipher these, and get at their meaning, he must substitute the letter of the alphabet with substitution key.

Question: Since Julius Caesar used this cipher to encrypt secret messages in military purposes, can we say that, this this the application of the cyclic group $$(\mathbb{Z}_{26},+)$$ in real life or defense sector?

## closed as primarily opinion-based by user1729, Yanior Weg, Lee David Chung Lin, John Omielan, Lord Shark the UnknownMay 30 at 4:52

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

• Sure, why wouldn't we be able to say this? – Jack Crawford May 28 at 4:33
• Well. May be we can, but this application of this cyclic group can safely be characterized as outdated. OFDM on the other hand ... (to the extent it relies on the discrete Fourier transform on a cyclic group). – Jyrki Lahtonen May 28 at 5:14

## 1 Answer

Yes. You're right that Caesar's cipher uses modular arithmetic. (On $$\mathbb{Z}_{26}$$, or however many letters there were in Caesar's alphabet—I believe I/J and U/V/W were not distinguished in ancient Latin, so $$\mathbb{Z}_{23}$$.) You're also right that Caesar used this cipher for military applications—sending encrypted messages.

So, Caesar's cipher represents a practical application of cyclic group theory to military operations. It is one of many such applications throughout various cultures and throughout history, due to the importance of cryptography for military communications and the importance of group theory for cryptography. See also—much later— the ENIGMA machine.