Secure Key Exchange over an Insecure Medium with Public Key Cryptography - Implementing SSL/TLS Using Cryptography and PKI - page 133

way into TLS in the past few years. TLS 1.2 introduced support for Elliptic-Curve
Cryptography ( ECC ) in 2008. Although it hasn't, at the time of this writing, found
its way into any commercial TLS implementations, it's expected that ECC will
become an important element of public-key cryptography in the future. I explore
the basics of ECC here — enough for you to add support for it in the chapter
9, which covers TLS 1.2 — but overall, I barely scratch the surface of the fi eld.
ECC — elliptic-curves in general, in fact — are complex entities. An elliptic-
curve is defi ned by the equation y 2 b. a and b are typically fi xed

and, for public-key cryptography purposes, small numbers. The mathematics
behind ECC is extraordinarily complex compared to anything you've seen so
far. I won't get any deeper into it than is absolutely necessary.
Figure 3-5 shows the graph of y 2

x 3 ax

x 3 ax b, the ellip tic curv e defi ned by

x 3 ax has no solu-

a 1, b 0. Notice the discontinuity between 0 and 1;

tions between 0 and 1 because x 3 x < 0.

3

2

1

0

− 1

− 2

− 3

− 3 − 2 − 1 0 1 2 3

Figure 3-5: Elliptic curve with a = 1, b = 0

Cryptographic operations are defi ned in terms of multiplicative operations
on this curve. It's not readily apparent how one would go about “multiplying”
anything on a curve, though. Multiplication is defi ned in terms of addition,
and “addition,” in ECC, is the process of drawing a line through two points
and fi nding it's intersection at a third point on the curve as illustrated in
Figure 3-6.