Addition with bit-strings is controlled by an XOR
function.
4.1 An Example of an Elliptic Curve Group over
F2m
As a very small example, consider the
field F24, defined by using polynomial representation
with the irreducible polynomial f(x) = x4 + x + 1.
The
element g = (0010) is a generator for the field . The powers of g
are:
g0 = (0001) g1 = (0010) g2 =
(0100) g3 = (1000) g4 = (0011) g5 =
(0110)
g6 = (1100) g7 = (1011) g8
= (0101) g9 = (1010) g10 = (0111) g11 =
(1110)
g12 = (1111) g13 = (1101)
g14 = (1001) g15 = (0001)
In a true
cryptographic application, the parameter m must be large enough to
preclude the efficient generation of such a table otherwise the
cryptosystem can be broken. In today's practice, m = 160 is a suitable
choice. The table allows the use of generator notation (ge)
rather than bit string notation, as used in the following example. Also,
using generator notation allows multiplication without reference to the
irreducible polynomial
f(x) = x4 + x +
1.
Consider the elliptic curve y2 + xy = x3 +
g4x2 + 1. Here a = g4 and b =
g0 =1. The point (g5, g3) satisfies this
equation over F2m:
y2 + xy = x3 +
g4x2 + 1
(g3)2 +
g5g3 = (g5)3 +
g4g10 + 1
g6 + g8 =
g15 + g14 + 1
(1100) + (0101) = (0001) +
(1001) + (0001)
(1001) = (1001)
The fifteen points which
satisfy this equation are:
(1, g13) (g3,
g13) (g5, g11) (g6,
g14) (g9, g13) (g10,
g8) (g12, g12)
(1, g6)
(g3, g8) (g5, g3)
(g6, g8) (g9, g10)
(g10, g) (g12, 0) (0, 1)
These points are
graphed below:

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