At first, the work was theoretical, but
eventually, researchers discovered that
elliptic curves are extremely useful for
If a browser wants to talk to a bank,
it needs to exchange keys to encrypt
the communication. Whitfield Diffie
and Martin Hellman, working at Stanford, invented an ingenious way to do
this. The problem is that to make their
method secure, you have to make the
numbers involved quite large, and
that makes it slow.
So in the 1980s, mathematicians
Victor Miller and Neal Koblitz suggested using elliptic curves to enable the
parameters to be smaller and the key
exchange to run faster. And now every
time you connect to Google, you are using this modified version of the Diffie-Hellman exchange. It is a remarkable
success story of mathematics—
something that for 1,800 years has only been
studied as an intellectual curiosity suddenly forms the underpinning of all secure network traffic.
Where does your own work come in?
Mathematician André Weil, who
studied elliptic curves in the 1960s,
discovered something called a pairing,
as a side tool in one of his proofs. As it
turns out, pairings are a godsend for
cryptographers. We already use elliptic
curves for key exchange. Now it turns
out they have this additional structure,
namely pairings, that enables new applications and this was recognized
by several researchers. In 2001, Matt
Franklin—who is a professor at UC
DAN BONEH, RECIPIENT of this year’s
ACM-Infosys Foundation Award,
discovered his passion for computers early; by the time he got to Princeton University, where he earned his
Ph.D., he had zeroed in on his line of
research: cryptography. Now a professor at Stanford University—and
head of the university’s Applied Cryptography Group—Boneh has made
groundbreaking contributions to the
field, pioneering the use of pairings
to solve cryptographic problems and
working on a range of broader computer security questions. Here, he
discusses his work.
What drew you to computer science?
I fell in love with computers at a
very early age. In high school, I learned
about the mathematics of cryptography and was completely taken by it.
It is a beautiful area involving elegant
mathematics, and at the same time,
the results are practical and used in
Can you take us through your work in
pairing-based cryptography, which is
among your best-known contributions
to the field?
Let’s start with some context. The
story begins 1,800 years ago with a
mathematician named Diophantus,
who lived in Alexandria and whose
work gave rise to many things we learn
in high school algebra. The ideas he
developed to solve one particular problem described something that is heavily studied in mathematics and called
an elliptic curve.
DOI: 10.1145/2800615 Leah Hoffmann
A Passion for Pairings
Dan Boneh on pairing-based cryptography, multilinear maps,
and how an 1,800-year-old “intellectual curiosity” became
the foundation of all secure network traffic.
[CONTINUED ON P. 127]
a beautiful area
at the same time,
the results are
practical and used in