# Common Classical Ciphers

When studying cryptography, the first thing every student learns is some historical ciphers. There are way too many of those ciphers to name them all. So my question is: What are the most important classical ciphers (every cryptographer should have heard about)?

Cryptography as we know it today dates from the Renaissance, in a certain sense, in a mathematical sense. --Whitfield Diffie

If you look at introductory cryptography texts, you will usually see some of the same ciphers, methods, and cryptographic tools covered in a chapter on classical cryptography:

The Scytale, a tool to perform a transposition cipher

The Polybius Square, for fractionating plaintext characters

The Caesar cipher, a keyed substitution cipher

Codebooks, a book for storing cryptographic codes

The tabula recta, a square table that defines a polyalphabetic cipher

della Porta's bigraphic substitution, a kind of polygraphic substitution

Homophonic substitution, mapping plaintext letters to more than one ciphertext letter

Vigenère cipher, a kind of polyalphabetic substitution

Vigenère autokey cipher, a keyword is used to make the keystream and the original plaintext

Columnar transposition, writing out a message in rows and columns to transpose it

The one-time pad, an encryption technique and model. The one-time pad is not a cipher by today's standards.

Playfair cipher, a digram substitution cipher

The bifid and trifid ciphers, the bifid uses a Polybius square and transposition; the trifid uses fractionation and transposition

The ADFGX and ADFGVX ciphers, ciphers used by Germany in World War I. ADFGVX used a Polybius square for fractionation, columnar transposition, and alphabetical transposition

The affine cipher, a monoalphabetic substitution cipher

VIC cipher, a complex cipher using a lagged Fibonacci generator, columnar transpositions, a straddling checkerboard, and mod 10 chain addition--all constructed from memorized information.

The Hill cipher, which opens a new dimension, that of a polygraphic substitution cipher based on algebra.

## What is very likely going to be covered in an introductory textbook:

Caesar cipher- monoalphabetic substitution
Vigenère cipher- polyalphabetic substitution
Affine cipher
Hill cipher


## What is hardly likely to be covered:

Straddling checkerboards (fractionation)
VIC cipher
Homophonic substitution with unusual symbols
Beaufort cipher
Nihilist cipher
Beale cipher


Importantly, Whitfield Diffie points out that a critical advance in cryptography was made during the Renaissance by Leon Battista Alberti (1404 – 1472): the distinction between a cryptographic key and a cryptographic system. (1) Before the Renaissance, this distinction was difficult to make because the systems were very simple. For example, in using a codebook, all of the expense is put into the codebook, the secret piece, not into the system of looking up plaintext and writing ciphertext. The Renaissance innovation was a method that resisted cryptanalysis by shifting the expense to the public piece, the system, making the secret part cheap. (2) As a keen student of the history of cryptography, Diffie relates specific early progress in classical cryptography to one of his own advances, this being the sort of thing that might be good for a cryptographer to understand.

EDIT:

As another way to answer the question, I looked at several introductory university-level textbooks to see what topics they covered vis-à-vis classical cryptography. Monoalphabetic and polyalphabetic refer to general discussion:

1. Alberti’s original contributions to cryptography can be found in his “On Writing in Ciphers” in Chapters 13-23. These are discussed by Lionel March in “The Mathematical Works of Leon Battista Alberti”, Spinger, 2010.

2. Whitfield Diffie, Information Security—Before & After Public-Key Cryptography, YouTube, video 1BJuuUxCaaY.

• Thank you very much for your answer! But why did you not include the Enigma into your list? Aug 10 '21 at 9:58
• @Titanlord Good question! The electromechanical rotor machines, and even a non-electrical rotor device such as the M-209, are not thought of as belonging to classical cryptography. The Enigma machines had what amounts to an IV, and the three-rotor German Army/Air Force Enigma had about $2^{67}$ settings. That strength is way above what any classical cipher system (except the OTP) could muster. Aug 11 '21 at 7:11
• Classical ciphers are often considered to be the same as can be done with pen and paper. Rotor machines are a separate category, where the Enigma is just the most famous one, but not the only one.
– tylo
Aug 16 '21 at 10:23
• That is a wonderful table you've provided, outlining what authors you'd want to be familiar with to study those techniques. Would you be able to provide a bibliography of those authors? For example, I've decided that I'd like to read up on the works of Schneier (Applied Cryptography - Amazon), Baumslag (A Course in Mathematical Cryptography? - Amazon) and Vaudenay (?) Aug 27 '21 at 16:30
• @Daniel Bragg Yes, when I have some time, I will do exactly that. Aug 27 '21 at 17:00

It's disputable that historic ciphers are necessary knowledge to understand how modern cryptography works. Towards this, the historic Kerckhoffs's principles (especially the second: the key shall be the only secret) are much more necessary IMHO.

On the other hand, historic ciphers and their pitfalls are useful to understand how attacking cryptography works, and how cryptography should not be.

Short list of common and interesting historic ciphers:

• Caesar's cipher, which modern description can be: symbols belong to a finite group, key is a symbol, encryption is addition of the key in the group, decryption reverses that by adding the opposite of the key in the group. This formulation covers modular addition (the most usual Caesar's cipher), ROT13, and XOR with the key.
• The affine cipher: symbols belong to a finite ring, key is two symbols $$a$$, $$b$$ with $$a$$ invertible in the ring, encryption is $$x\mapsto a\cdot x+b$$ in the ring. This formulation covers the ring $$\mathbb Z_{26}$$ (the most usual affine cipher), and (adding grouping of symbols) the Hill cipher.
• Monoalphabetic substitution, where key is a bijection of the symbol set (the above are a special case of monoalphabetic substitution).
• Vigenère's cipher, which extends Caesar's cipher using a key that changes cyclically. That extension principle is also applicable to any of the above (though the result would not be called Vigenère's cipher).
• Aug 6 '21 at 21:03

I can't expand on the breadth of the other answers, but instead I'd like to focus specifically on one. It stands out from all the rest as it's a 'classical' cipher, yet is still in use. And it's use is expanding.

The one time pad has been in constant use for over a hundred years, and going from strength to strength. It is the logical extension of the Vernam cipher, but may have been invented 35 years earlier by a Sacramento banker named Frank Miller (see Frank Miller: Inventor of the One-Time Pad).

Notwithstanding, there is much to be learned from it's key generation, management and key reuse issues. And notions of entropy and measurement thereof. Especially when one considers the number of questions on this site that misconstrue it, or try to improve it.

Furthermore, there is extensive on-going research and investment across the world in solving the key distribution issue in the form of quantum key distribution networks. Some now claim to have achieved ~ 1 Gbit/s of sifted key material over distances exceeding 20,000 km! All for an over hundred year old cipher.