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Mathematics had always been perceived as an abstract field. You have numbers, symbols, concepts that seems to be the result of overthinking professors. This is, of course, far from the truth. Mathematics, especially applied mathematics, touches every aspect of human civilisation and life.
And in this specific case, may have saved human civilisation and millions of lives. |
"Now the question remains how much did [the cracking of the ENIGMA code] shorten the war. My own conclusion is that it shortened the war by not less than two years and probably by four years — that is the war in the Atlantic, the Mediterranean and Europe." - Sir Harry Hinsley, 'The Influence of ULTRA in the Second World War' (1993)
What is Cryptography?
Cryptography (also known as cryptology; comes from Greek κρυπτός, kryp-tos, "hidden, secret"; and γράφω, grá-phō, "I write", or -λογία,-logia, respectively)[1] is the practice and study of hiding information. It is sometimes called code, but this is not really a correct name. It is the science used to try to keep information secret and safe. Modern cryptography is a mix of mathematics, computer science, andelectrical engineering. Cryptography is used in ATM (bank) cards, computer passwords, and shopping on the internet.
When a message is sent using cryptography, it is changed (or encrypted) before it is sent. The method of changing text is called a "code" or, more precisely, a "cipher". The changed text is called "ciphertext". The change makes the message hard to read. Someone who wants to read it must change it back (or decrypt it). How to change it back is a secret. Both the person that sends the message and the one that gets it should know the secret way to change it, but other people should not be able to. Studying the cyphertext to discover the secret is called "cryptanalysis" or "cracking" or sometimes "code breaking".
Different types of cryptography can be easier or harder to use and can hide the secret message better or worse. Ciphers use a "key" which is a secret that hides the secret messages. The cryptographic method needn't be secret. Various people can use the same method but different keys, so they cannot read each other's messages. Since the Caesar cipher has only as many keys as the number of letters in the alphabet, it is easily cracked by trying all the keys. Ciphers that allow billions of keys are cracked by more complex methods.
Encryption and Decryption
Encryption allows information to be hidden so that it cannot be read without special knowledge (such as a password). This is done with a secret code or cypher. The hidden information is said to be encrypted.
Decryption is a way to change encrypted information back into plain text. This is the decrypted form. The study of encryption is called cryptography. Cryptanalysis can be done by hand if the cypher is simple. Complex cyphers need a computer to search for possible keys. Decryption is a field of computer science and mathematics that looks at how difficult it is to break a cypher.
Cryptanalysis is the study of methods for obtaining the meaning of encrypted information, without access to the secret information which is normally required to do so. Typically, this involves finding a secret key. In non-technical language, this is the practice of codebreaking or cracking the code, although "break" can also mean just a part of a complete solution.
Cryptography (also known as cryptology; comes from Greek κρυπτός, kryp-tos, "hidden, secret"; and γράφω, grá-phō, "I write", or -λογία,-logia, respectively)[1] is the practice and study of hiding information. It is sometimes called code, but this is not really a correct name. It is the science used to try to keep information secret and safe. Modern cryptography is a mix of mathematics, computer science, andelectrical engineering. Cryptography is used in ATM (bank) cards, computer passwords, and shopping on the internet.
When a message is sent using cryptography, it is changed (or encrypted) before it is sent. The method of changing text is called a "code" or, more precisely, a "cipher". The changed text is called "ciphertext". The change makes the message hard to read. Someone who wants to read it must change it back (or decrypt it). How to change it back is a secret. Both the person that sends the message and the one that gets it should know the secret way to change it, but other people should not be able to. Studying the cyphertext to discover the secret is called "cryptanalysis" or "cracking" or sometimes "code breaking".
Different types of cryptography can be easier or harder to use and can hide the secret message better or worse. Ciphers use a "key" which is a secret that hides the secret messages. The cryptographic method needn't be secret. Various people can use the same method but different keys, so they cannot read each other's messages. Since the Caesar cipher has only as many keys as the number of letters in the alphabet, it is easily cracked by trying all the keys. Ciphers that allow billions of keys are cracked by more complex methods.
Encryption and Decryption
Encryption allows information to be hidden so that it cannot be read without special knowledge (such as a password). This is done with a secret code or cypher. The hidden information is said to be encrypted.
Decryption is a way to change encrypted information back into plain text. This is the decrypted form. The study of encryption is called cryptography. Cryptanalysis can be done by hand if the cypher is simple. Complex cyphers need a computer to search for possible keys. Decryption is a field of computer science and mathematics that looks at how difficult it is to break a cypher.
Cryptanalysis is the study of methods for obtaining the meaning of encrypted information, without access to the secret information which is normally required to do so. Typically, this involves finding a secret key. In non-technical language, this is the practice of codebreaking or cracking the code, although "break" can also mean just a part of a complete solution.
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Turing, Alan Turing, Codebreaker
Alan Mathison Turing—an English mathematician, logician, and cryptanalyst—was a computer pioneer. Often remembered for his contributions to the fields of artificial intelligence and modern computer science (before either even existed), Turing is probably best known for what is now dubbed the “Turing Test.” It is a process of testing a machine’s ability to “think.” The basic premise of the Turing Test is that a human judge would be placed in isolation and have two conversations – one with a computer and one with another person – except the judge wouldn’t be told which was which. If the computer could fool the judge and carry on a conversation that is indistinguishable from that of the human, the computer is said to have passed the Turing Test. No computer has passed it yet. Less is known, however, about Turing’s intelligence work during WWII when he used his mathematical and cryptologic skills to help break one of the most difficult of German ciphers, ENIGMA. Solving An ENIGMA The ENIGMA machine was created for Germany by Arthur Scherbius in World War I. It is a cypher machine: a way of changing the letters of a message so that it appears to be randomly generated. Each time a letter is typed, it appears as another letter in the alphabet. However, the choices are anything but random. They are decided by a series of rotors which are set each day to a different starting set-up. Every press of a key turns the rotors to a new position. |
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In the early years of WWII, Turing worked at Britain’s code breaking headquarters in Bletchley Park. In addition to mathematicians, Bletchley Park also recruited linguists and chess champions, and attracted talent by approaching winners of a complex crossword puzzle tournament held by The Daily Telegraph.
Turing’s mathematical and logic skills made him a natural cryptanalyst. Whereas cryptographers write encryption systems, and cryptologists study them, cryptanalysts like Turing break them. In 1939, Turing created a method called “the bombe,” an electromechanical device that could detect the settings for ENIGMA, allowing the Allied powers to decipher German encryptions. Turing and his colleagues were also able to break the more complicated Naval ENIGMA system, which from 1941-1943 helped the Allies avoid German U-boats during the Battle of the Atlantic.
Poland was actually the first to realize that the solution to breaking ENIGMA would most likely be discovered by a mathematician. Polish cryptanalysts as early as 1932 could decode German ciphers and, by 1939, they were able to successfully decipher messages written with an earlier version of ENIGMA using a replica machine like “the bombe” that could emulate the way ENIGMA worked. When Poland was overrun by Germany in September 1939, the Polish as well as French cryptanalysts shared what they knew about ENIGMA with the UK, which allowed the cryptanalysts at Bletchley Park, including Turing, to finally crack the ENIGMA ciphers, as below. Go ahead, tinker with it and be amazed.
Turing’s mathematical and logic skills made him a natural cryptanalyst. Whereas cryptographers write encryption systems, and cryptologists study them, cryptanalysts like Turing break them. In 1939, Turing created a method called “the bombe,” an electromechanical device that could detect the settings for ENIGMA, allowing the Allied powers to decipher German encryptions. Turing and his colleagues were also able to break the more complicated Naval ENIGMA system, which from 1941-1943 helped the Allies avoid German U-boats during the Battle of the Atlantic.
Poland was actually the first to realize that the solution to breaking ENIGMA would most likely be discovered by a mathematician. Polish cryptanalysts as early as 1932 could decode German ciphers and, by 1939, they were able to successfully decipher messages written with an earlier version of ENIGMA using a replica machine like “the bombe” that could emulate the way ENIGMA worked. When Poland was overrun by Germany in September 1939, the Polish as well as French cryptanalysts shared what they knew about ENIGMA with the UK, which allowed the cryptanalysts at Bletchley Park, including Turing, to finally crack the ENIGMA ciphers, as below. Go ahead, tinker with it and be amazed.
Once the German messages were decrypted, the British began supplying the Office of Strategic Services (OSS, precursor to the CIA) with extensive information about foreign military, espionage, and sabotage activities. The most sensitive intelligence came from ULTRA—the code name applied to all intel coming from Bletchley Park, including the intercepts of German military messages sent with the ENIGMA machine. Because of the volume of the traffic and the overriding need for compartmentalization, the British insisted that the OSS set up a separate, extra-secure component to handle the material.
On March 1, 1943, OSS Director William J. Donovan created the Counterintelligence Division in the Secret Intelligence Branch—the first centralized, national-level counterintelligence office in US intelligence history and the predecessor of CIA’s Counterintelligence Staff and Counterintelligence Center.
Turing’s work at Bletchley Park was vital in ending the war. General Dwight D. Eisenhower told the British intelligence chief in July 1945 that ULTRA, “saved thousands of British and American lives and, in no small way, contributed to the speed with which the enemy was routed and eventually agreed to surrender.”
Turing would spend the majority of his career focused on what would eventually become modern day computing. He was posted to serve with the US Navy’s Cryptanalytic Section for several months in 1943, where he met and discussed mathematical models of communication and computation with Claude Shannon (the father of information theory). To this day, our communications networks are built on top of Shannon's ideas, while our computing devices, processors, and chips are built upon Turing's ideas. Turing’s contribution to modern computing was so significant that the prestigious A.M. Turing Award—sometimes known as the “Nobel Prize” of Computer Science—is named after him.
For a dramatisation of Turing's work at Bletchley Park, I highly recommend The Imitation Game, starring Benedict Cumberbatch (of Star Tek: Into Darkness and Sherlock Holmes fame) as Turing, and Keira Knightley as fellow codebreaker and fiancee Joan Clarke.
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Ponder this
What makes ENIGMA so complicated? Nothing about it is random, and everything about the machine can be mathematically rationalised, even the letter-switching part. Why is the real reason it's so difficult to crack?
Modern cryptography now uses public and private keys, large prime numbers as encryption elements. Is it possible to crack it in the old fashioned "brute force", trial-and-error way as ENIGMA was? Why, or why not?
Discuss
Mathematics is probably amongst the most enigmatic of the sciences. It's applications, although extensive, are usually unseen and even less understood by the layman. Applied mathematics is a specific field that cryptanalysis belongs too, along with statistics, actuarial sciences, and finance.
How pervasive is mathematics in your daily life? Give specific examples, knowledge, skills, goods or services. Debate the prioritisation, or lack thereof, of mathematics in Malaysian academic life, and why it is so?
Mathematics is probably amongst the most enigmatic of the sciences. It's applications, although extensive, are usually unseen and even less understood by the layman. Applied mathematics is a specific field that cryptanalysis belongs too, along with statistics, actuarial sciences, and finance.
How pervasive is mathematics in your daily life? Give specific examples, knowledge, skills, goods or services. Debate the prioritisation, or lack thereof, of mathematics in Malaysian academic life, and why it is so?
Further readings
Alan Turing, a mathematician and pioneer in the computing sciences and the head codebreaker in the British effort of solve the ENIGMA code.
Cipher Machines, a more comprehensive coverage on how the ENIGMA machine works.
History of Cryptography, at the Engineering and Technology History Wiki.
Applied mathematics, the field of which cryptanalysis belongs to.
