Math that helped solve Fermat's theorem now safeguards the digital world

Pierre de Fermat, the 17th-century mathematician whose last theorem, solved in the 1990s, informs elliptic curve cryptography. Photo / 123RF

Defenses against digital snoopers keep getting stronger. Encryption is what keeps communications safe when you use Signal and other messaging apps, make online financial transactions, buy and sell cryptocurrencies like Bitcoin and trust that private information in your Apple iPhone will stay private.

While a variety of end-to-end encryption techniques seek to protect the flows of information from spies and eavesdroppers, one of the most powerful and ubiquitous is elliptic curve cryptography, invented in 1985. The method's underlying math helped solve the famous riddle of Fermat's last theorem and was promoted by the charitable foundation of James M. Vaughn Jr., an heir to oil riches. In the 1970s and 1980s, Vaughn funded experts who pursued knotty questions of mathematics that were assumed to have no practical value.

Vaughn's funding of Fermat studies backed the investigation of elliptic curves as a possible solution. The obscure branch of mathematics turned out to beget a new generation of powerful ciphers — in particular, elliptic curve cryptography.

In his 2009 autobiography, "Random Curves," Neal I. Koblitz, a University of Washington mathematician who aided Vaughn and was one of two inventors of the technique, described its "biggest friend" as the National Security Agency. An arm of the Pentagon, the NSA works to strip governments of their secrets while concealing its own. It relies heavily on elliptic curve cryptography.

In an interview, Vaughn said NSA officials sent math experts to the conferences he sponsored.

"They always had people there," he recalled.

Of course, digital thieves are trying to undo the decades of encryption strides with new kinds of spyware and cyberweapons. Public encryption has become so powerful that the hackers often try to seize control of smartphones and steal their data before it has been scrambled and securely transmitted.

In public talks, Andrew Wiles, an Englishman who solved the Fermat puzzle, has seldom spoken of cryptography. In 1999, however, he touched on the topic at the Massachusetts Institute of Technology in describing recent math advances.

Wiles now teaches at the University of Oxford, which in 2013 opened a $100 million building named after him. Officials from Britain's equivalent of the NSA — the Government Communications Headquarters, or GCHQ — are no strangers to the Andrew Wiles Building.

In 2017, for instance, two officials from GCHQ gave talks there. They were Dan Shepherd, a researcher who helped uncover a major vulnerability in a proposed cipher, and Richard Pinch, the agency's head of mathematics.

Now retired from government service, Pinch describes his interests as not only elliptic curves but Fermat studies that bear on cipher development.

Written by: William J. Broad
© 2022 THE NEW YORK TIMES