Page 309 - Hardware Implementation of Finite-Field Arithmetic

An Example of Application—Elliptic Curve Cryptography 289

After some changes of variables, the elliptic curves can be
classified into five classes:

1. If the characteristic of K is not equal to 2 or 3 then the simplified
Weierstrass equation is

y = x + ax + b (10.3)
2
3
where a and b belong to K, and 4a + 27b ≠ 0.
2
3
2. If the characteristic of K is equal to 2, then a first simplified
Weierstrass equation is
2
3
y + xy = x + ax + b (10.4)
2
where a and b belong to K, and b ≠ 0. Such a curve is said to be
nonsupersingular.
3. If the characteristic of K is equal to 2, another simplified Weierstrass
equation is

y + cy = x + ax + b (10.5)
2
3
where a, b, and c belong to K, and c ≠ 0. Such a curve is said to be
supersingular.
4. If the characteristic of K is equal to 3, then a first simplified
Weierstrass equation is

y = x + ax + b (10.6)
2
3
2
where a and b belong to K, a ≠ 0 and b ≠ 0. Such a curve is said to be
nonsupersingular.
5. If the characteristic of K is equal to 3, another simplified Weierstrass
equation is
2
3
y = x + ax + b (10.7)
where a and b belong to K, and a ≠ 0. Such a curve is said to be
supersingular.
It has been demonstrated (Hasse theorem, [HMV04]) that the
number of points of E(L) belongs to the following interval:
q + 1 − 2q 1/2 ≤ #E(L) ≤ q + 1 + 2q 1/2 (10.8)
where q is the number of elements of L. Thus, for great values of q, the
number of points is approximately equal to the number of field
elements:

#E(L) ≅ q (10.9)
For practical applications the curves to be considered belong to
m
m
types 1, 2, or 3, that is, L = GF(p ) with p > 3 or L = GF(2 ).