The Mathematics of Cryptography
During World War II, the Germans devised the Enigma machine, which used a code so clever that the Allies would have had to sort through thousands of possible decryptions before hitting the right one. To try to break the Enigma code, the British government hired a number of mathematicians, including Alan Turing, whose mathematical ideas are fundamental to all modern-day computers. Turing was able to use these ideas to crack the Enigma code, making a crucial contribution to the defeat of Germany.
Fifty years after Turing’s time, the area of cryptology—the devising and breaking of codes for secret communications—has developed into a sophisticated field of study utilizing a great deal of higher-level mathematics. The government’s main center for cryptology, the National Security Agency, is the nation’s largest employer of mathematicians.
As a mathematician at the NSA, Linda Shields works on a wide variety of problems in cryptology. “Mathematics comes into play because almost all of today’s schemes for disguising communications are based on mathematical properties,” Linda explains. She uses many different branches of mathematics in her work, such as abstract algebra, probability, and statistics. “Many of our problems do not come as mathematics problems—we must find the mathematics in them ourselves, and it is hard to predict in advance which areas of mathematics will be the most fruitful.”
Typically, the projects Linda has worked on start out as very difficult problems “with no solution in sight and no hint as to the direction to go.” Trading ideas with coworkers, she tries to probe the problem in as many ways as possible. Computers are important experimental tools in her work, and the NSA has access to some of the most powerful in the world. “If we are lucky, we begin to really understand the problem and its mathematical structure,” she says. “Then the fun begins, which is trying to take advantage of that structure to construct a solution.”
After completing bachelor’s and master’s degrees in mathematics from Indiana University, Linda came to work at NSA. For most of her career, her job has involved identifying projects for herself and others. She directed research teams for six years and for the past three years has managed a larger research organization within NSA. “I work with other managers to suggest research areas, set general directions, and, where possible, to work on specific problems myself,” she explains.
Linda enjoys the teamwork at NSA. “It is especially rewarding to see several years of effort, by different people in different organizations, suddenly come together to produce a spectacular new result. Working on a team effort like that has been the most satisfying aspect of my career.”
This brochure was published in 1991, so some information may be out-of-date.