The Abel prize — considered the 'Nobel' prize of mathematics — has been awarded to John Tate, recently retired from the University of Texas at Austin, for his work on algebraic number theory, the mathematical discipline that deals with connections between whole numbers and lies at the heart of Internet security.

Established in 2002, the Abel Prize is presented annually by the King of Norway and carries a cash award of US$1 million.

"Number theory knits together the subtle and strange properties of whole numbers in a beautiful way," says mathematician Ian Stewart at the University of Warwick in Coventry, UK. "Tate has really made himself the master of number theory."

Tate's work in number theory helped to crack one of the most famous challenges in mathematics: proving Fermat's Last Theorem. The theorem states that you cannot find three positive integers a, b and c that satisfy the equation a^{n} + b^{n} = c^{n} for any integer value of n greater than two. The theorem was solved in 1995 by Andrew Wiles of Princeton University in New Jersey, thanks in part to Tate's research into the rules obeyed by 'elliptic curves' — curves generated by a particular family of equations in number theory.

"Fermat's Last Theorem is simple to state, but took 350 years to prove, using the machinery of number theory developed by Tate," says Stewart.

Long thought to be one of the purest forms of pure mathematics — as it had little real-world relevance — number theory has now become vitally important for securely encoding data to be transmitted across the Internet. "If you go back to the 1950s, most mathematicians would have agreed that number theory wasn't particularly useful — some thought that a vice and some a virtue — but then along came the computer," says Stewart.

## King of the code

One important method for ensuring secure transmission across the Internet uses encryption keys based on 200-digit numbers that are the multiple of two prime numbers. Thanks to Tate's developments in number theory, algorithms can easily generate such numbers for encoding purposes, says Stewart. However, there are no algorithms to perform the reverse operation — working out the constituent primes of the 200-digit number — making it impossible to for hackers to crack the codes. "Try to find the prime factors of a 200-digit number with pencil and paper — or even with a computer program — and it would take longer than the age of the Universe," says Stewart.

ADVERTISEMENT

Tate's work is also at the heart of error-correcting codes that allow corrupted digital information to be reconstructed. "When you're driving along, listening to music and hit a bump, the reason your CD doesn't skip is thanks to these error-correcting codes," says Stewart. "It's also the reason that the messages that you send on your mobile phones aren't garbled by all the other radio signals flying through the air."

Tate is a popular choice for the Abel award, says mathematician Helge Holden at the Norwegian University of Science and Technology in Trondheim. "Tate's achievements in number theory go right back to his doctoral thesis, which became famous, and span more than 60 years during which his name has been given to many different theorems in the field," Holden says. "This is a well-deserved award for lifetime achievement."