Computer and Information Security Handbook, 4th Edition

11. Elliptic Curve Cryptosystems

For simplicity, we shall restrict our attention to ECs over Z_p, where p is a prime greater than 3. We mention, however, that ECs can be defined more generally over any finite field [4]. An EC E over Z_p is defined by an equation of the form

(47.1)

where a, b ∈ Z_p, and 4a³ + 27b² ≠ 0 (mod p), together with a special point O called the point at infinity. The set E(Z_p) consists of all points (x, y), x ∈ Z_p, y ∈ Z_p, which satisfy the defining Eq. (47.1), together with O.

An Example

Let p = 23 and consider the EC E: y² = x³ + x + 1, defined over Z₂₃. (In the notation of Eq. (47.1), we have a = 1 and b = 1.) Note that 4a