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: Next