The Eternal Battle of Symbol Creation and Destruction: 80 Years of the Enigma Machine | Coffee and theories | Sciences


The year 2023 marks the 80th anniversary of a key event during World War II: the decoding of the Enigma machine by the Allied side (thanks to the outstanding work of mathematician Alan Turing, among others). This device was used by the Germans to secretly transmit military plans, so its decryption greatly influenced the course of the war, at the same time as it represented a remarkable advance in encryption. The course of the discipline is largely determined by constant conflict: while some seek to transmit information securely through encryption, others will attempt to crack these codes to gain access to the information. This produces better and better systems.

The first zeros to appear were the monosyllabic ones. Its process is very simple: each letter of the original message is changed by another letter to produce an encrypted message. For example, every ‘e’ is changed to ‘g’. The problem with this system is that the frequency of the letters of the original text is transmitted to the ciphertext. Thus, if there are several “g” and “q” in the encrypted message, they will likely correspond to the letters “e” and “a”, the two most common letters in Spanish-language text. With this idea, developed in the frequency analysis technique, we can start decoding. the story gold bugWritten by Edgar Allan Poe, the story is intertwined with a detailed explanation of the method.

The letter is repeated
The letter ‘c’ is repeated more than other letters, such as ‘g’, which is why it is identified as one of the most common letters, in this case, the letter ‘a’.

In response to this weakness, polyalphabetic substitution ciphers emerged. In them, the character is not always encoded to the same character, as the substitution rule changes throughout the text. In classic examples of these systems, such as the Vigenère cipher, few substitution rules are used and it is easy to group the ciphertext characters according to the rule obtained with each one. Once the characters are separated into groups, each of them is a ciphertext using a single-character substitution cipher, so frequency analysis can be applied.

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Although the Enigma machine performed the same type of polyalphabetic cipher, it was a huge advance over the classic techniques. It has three rotors that create contact between the keyboard and the light board. When a key is pressed, the encoded letter lights up and the configuration of the rotors changes, so that a different base is used for the next letter. The specifics of the internal mechanism means that a large number of different rules are used, which makes the code extremely difficult to crack: it is only possible if the initial configuration of the machine is known. Small system problems, combined with the mathematical and computational developments of several years of work and a dose of luck, finally allowed the Enigma to be deciphered.

3-rotor Enigma machine.
3-rotor Enigma machine.WIKIMEDIA COMMON

One of these weaknesses was the need to pre-distribute information about the hardware configuration, with the risk that it could be intercepted. So the difficulty of exchanging keys securely was the next problem to be solved in order to make the systems more robust. The answer didn’t come until the 1970s, with the Diffie-Hellman protocol.

Let’s say that Antonio and Beatriz want to encrypt their communication, for which they will need to agree on a secret key. This protocol eliminates the need to meet in person to do so. The basic mathematical operation is often explained as if it were a mixture of different paints. Both agents decide on a commonly used color, which need not be kept a secret. And each one secretly chooses a color known only to himself. By mixing their colors with the common paint, they get two new different colors, which will be the information they send.

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If someone intercepts the information, they’ll have a hard time knowing your secret colors: even if they know the “generic” color, separating the mixture to determine what colors it makes up is a very expensive process. However, when they receive it, they can add their secret color again, thus obtaining a combination of the same three colours, which will be their common secret. In order for the interceptor to achieve the same final mix, it must separate one of the submitted mixes and add the resulting special color to the other. Since this is not possible, and because it is very expensive, the protocol is secure.

A visual explanation of the Diffie-Hellman algorithm.
A visual explanation of the Diffie-Hellman algorithm.

In fact, this process is not done with paint, but with mathematics. Its security depends on a quick process of arithmetic on the computer, exponential in standard arithmetic, but it is very difficult to undo knowing only the result. While the usual exponent follows a clear pattern, the exponent in regular arithmetic is very difficult to predict and therefore undo. This is known as the discrete logarithm problem.

Usual exponential vs. exponential in modular arithmetic.
Usual exponential vs. exponential in modular arithmetic.

But before a shared key can be agreed upon, the participants in the connection must be able to verify that the other person is not a fraud. For this, authentication certificates are used. Once this is done, you can use the Diffie–Hellman method to agree on a secret key, and send encrypted information at the end. The actual operations are more detailed than described here, but they are essential components, and are essential for secure communication on the Internet. Although the field is constantly evolving, these protocols have been completely secure for many years. However, there will never be a shortage of people trying to crack it, so the crypto battle will never end.

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Spain’s Laura Castilla And Javier Penafiel Thomas is a pre-doctoral researcher at the Complutense University of Madrid and the Supreme Council for Scientific Research, respectively, and members of the Institute of Mathematical Sciences

Coffee and theories is a section dedicated to mathematics and the environment in which it was created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the most recent developments in this discipline, share points of convergence between mathematics and other social and cultural expressions, and remember those who marked its development and knew how to turn coffee into theories. The name evokes Hungarian mathematician Alfred Rennie’s definition: “A mathematician is a machine that turns coffee into theorems.”

Edition and format: Ágata A. Timón G Longoria (ICMAT).

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