CHAPTER 3

Elliptic Curve Arithmetic

Cryptographic mechanisms based on elliptic curves depend on arithmetic involving the points of the curve. As noted in Chapter 2, curve arithmetic is defined in terms of underlying field operations, the efficiency of which is essential. Efficient curve operations are likewise crucial to performance.

Figure 3.1 illustrates module framework required for a protocol such as the Elliptic Curve Digital Signature Algorithm (ECDSA, discussed in §4.4.1). The curve arithmetic not only is built on field operations, but in some cases also relies on big number and modular arithmetic (e.g., τ -adic operations if Koblitz curves are used; see §3.4). ECDSA uses a hash function and certain modular operations, but the computationally-expensive steps involve curve operations.

Elliptic Curve Digital Signature Algorithm

(Protocols, Chapter 4)

Random number	Big number and	Curve arithmetic
generation	modular arithmetic	(Chapter 3)

Field arithmetic (Chapter 2)

Figure 3.1. ECDSA support modules.