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3. Curves over Binary Fields

For each field degree m, a pseudo-random curve is given, along with a Koblitz curve. The pseudo-random curve has the form

E: y² + xy = x³ + x² + b,

and the Koblitz curve has the form

E_a: y² + xy = x³ + ax² + 1

where a = 0 or 1.

For each pseudorandom curve, the cofactor is f = 2. The cofactor of each Koblitz curve is f = 2 if a = 1 and f = 4 if a = 0.

The coefficients of the pseudo-random curves, and the coordinates of the base points of both kinds of curves, are given in terms of both the polynomial and normal basis representations discussed in 1.3.

For each m, the following parameters are given:

Field Representation:

• The normal basis type T
• The field polynomial (a trinomial or pentanomial)

Koblitz Curve:

• The coefficient a
• The base point order r
• The base point x coordinate G_x
• The base point y coordinate G_y

Pseudo-random curve:

• The base point order r

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