"Group Theory and the Enigma"
By Doug Peterson
In the 1960 movie, Sink the Bismarck, Allied Forces in World War II are chasing the crippled German "supership," the Bismarck. At one point, the hero is asked why he thinks the Bismarck is heading for Brest, France. The hero responds that he simply had to make a hard decision. And Brest was the most likely destination.
In reality, the "hard decision" to track the Bismarck to Brest was based on much more than gut feelings. The British knew the Bismarck was heading for Brest because they had broken messages that had been enciphered with the Enigma - the famed German code machine.
John Lawrence, professor of mathematics at the University of Waterloo in Canada, brought an Enigma machine to the University of Illinois in April of 2001.
He also unraveled the tale of how a young Polish mathematician, Marian Rejewski, did what many thought was impossible - crack the Enigma code. Lawrence's visit, co-sponsored by CSL and the Department of Mathematics, was part of CSL's Golden Anniversary series of lectures.
The Enigma machine resembles a glorified typewriter, but with some crucial differences. When an operator types a letter on the Enigma keyboard, the machine goes through a complicated series of permutations. For instance, if you type the letter "b," the machine might tell you to encipher it as "x." But the next time you type the letter "b," it might tell you to encipher it as "d."
By pressing down on a key, a circuit is completed, Lawrence explained. The circuit pathway runs from the keyboard to the plugboard, through a series of three rotors to a reflector, then back through the rotors and plugboard. Along this path, the letter is changed several times. And when the circuit is finally completed, a letter lights up on the lampboard. This is the ciphertext letter.
In actual use, the settings for the Enigma machine were different every day and the rotors were switched around on a regular basis - making the machine even more difficult to crack.
But Rajewski was undaunted. According to Lawrence, he solved a system of six equations and demonstrated that it was possible to mathematically retrieve the daily settings for the machine.
Thanks to Rajewski's work, the Poles were able to read a large portion of the German's enciphered messages beginning in the early 1930s. Just before the Germans invaded Poland, the Poles revealed their breakthrough to the French and British, enabling the Allies to carry on the deciphering work in places such as Britain's Bletchely Park.
"But why didn't the German military become aware that the messages were in fact being deciphered?" asked Lawrence. "To some extent, there was a certain amount of arrogance involved. They thought it was simply impossible to break the Enigma."
But fate also helped conceal the Allies' breakthrough from the Germans, he said.
In explaining how fate played a role, Lawrence noted that the Germans were continually improving the Enigma - switching the rotors more often and even adding rotors to the machine. After the Germans added a fourth rotor to the Enigma machine, there was a period in which the British were no longer able to decipher messages being sent by German U-Boats.
During this blackout period, the Germans were likely to have greater success sinking ships in the North Atlantic. It wouldn't take much for the Germans to figure out that their increased success was due to the addition of a fourth rotor - tipping them off that their previous Enigma messages had been an open book for the Allies.
But the Germans never put it all together. Why?
As it turned out, the United States had just entered the war during the blackout period when the British couldn't read Enigma messages, Lawrence said. As a result, the Germans diverted U-Boats from the North Atlantic to the East Coast of North America, where they had a field day picking off unescorted American ships.
The Germans attributed their increased success to the Americans' lack of a convoy system to protect shipping. They didn't connect the success to the addition of a fourth rotor to the Enigma machine. So the secret of the breaking of the Enigma remained safe.
"The Germans never thought that if you set up some mathematical equations with a couple of theorems that you could actually figure out the settings and internal wiring of the Enigma," Lawrence said. "They never suspected that there was a fundamental weakness in the system."
The Enigma, like the Bismarck, was vulnerable.
By Doug Peterson
In the 1960 movie, Sink the Bismarck, Allied Forces in World War II are chasing the crippled German "supership," the Bismarck. At one point, the hero is asked why he thinks the Bismarck is heading for Brest, France. The hero responds that he simply had to make a hard decision. And Brest was the most likely destination.
In reality, the "hard decision" to track the Bismarck to Brest was based on much more than gut feelings. The British knew the Bismarck was heading for Brest because they had broken messages that had been enciphered with the Enigma - the famed German code machine.
John Lawrence, professor of mathematics at the University of Waterloo in Canada, brought an Enigma machine to the University of Illinois in April of 2001.
He also unraveled the tale of how a young Polish mathematician, Marian Rejewski, did what many thought was impossible - crack the Enigma code. Lawrence's visit, co-sponsored by CSL and the Department of Mathematics, was part of CSL's Golden Anniversary series of lectures.
The Enigma machine resembles a glorified typewriter, but with some crucial differences. When an operator types a letter on the Enigma keyboard, the machine goes through a complicated series of permutations. For instance, if you type the letter "b," the machine might tell you to encipher it as "x." But the next time you type the letter "b," it might tell you to encipher it as "d."
By pressing down on a key, a circuit is completed, Lawrence explained. The circuit pathway runs from the keyboard to the plugboard, through a series of three rotors to a reflector, then back through the rotors and plugboard. Along this path, the letter is changed several times. And when the circuit is finally completed, a letter lights up on the lampboard. This is the ciphertext letter.
In actual use, the settings for the Enigma machine were different every day and the rotors were switched around on a regular basis - making the machine even more difficult to crack.
But Rajewski was undaunted. According to Lawrence, he solved a system of six equations and demonstrated that it was possible to mathematically retrieve the daily settings for the machine.
Thanks to Rajewski's work, the Poles were able to read a large portion of the German's enciphered messages beginning in the early 1930s. Just before the Germans invaded Poland, the Poles revealed their breakthrough to the French and British, enabling the Allies to carry on the deciphering work in places such as Britain's Bletchely Park.
"But why didn't the German military become aware that the messages were in fact being deciphered?" asked Lawrence. "To some extent, there was a certain amount of arrogance involved. They thought it was simply impossible to break the Enigma."
But fate also helped conceal the Allies' breakthrough from the Germans, he said.
In explaining how fate played a role, Lawrence noted that the Germans were continually improving the Enigma - switching the rotors more often and even adding rotors to the machine. After the Germans added a fourth rotor to the Enigma machine, there was a period in which the British were no longer able to decipher messages being sent by German U-Boats.
During this blackout period, the Germans were likely to have greater success sinking ships in the North Atlantic. It wouldn't take much for the Germans to figure out that their increased success was due to the addition of a fourth rotor - tipping them off that their previous Enigma messages had been an open book for the Allies.
But the Germans never put it all together. Why?
As it turned out, the United States had just entered the war during the blackout period when the British couldn't read Enigma messages, Lawrence said. As a result, the Germans diverted U-Boats from the North Atlantic to the East Coast of North America, where they had a field day picking off unescorted American ships.
The Germans attributed their increased success to the Americans' lack of a convoy system to protect shipping. They didn't connect the success to the addition of a fourth rotor to the Enigma machine. So the secret of the breaking of the Enigma remained safe.
"The Germans never thought that if you set up some mathematical equations with a couple of theorems that you could actually figure out the settings and internal wiring of the Enigma," Lawrence said. "They never suspected that there was a fundamental weakness in the system."
The Enigma, like the Bismarck, was vulnerable.
John Lawrence
Professor, Department of Pure Mathematics
University of Waterloo, Ontario, Canada
Presentation Title: "Group Theory and
the Enigma"
Delivered:
April 6, 2001
Professor, Department of Pure Mathematics
University of Waterloo, Ontario, Canada
Presentation Title: "Group Theory and
the Enigma"
Delivered:
April 6, 2001
