Cryptography Reference
In-Depth Information
> Statistics:-Histogram(egos223, discrete = true, frequencyscale = absolute);
Statistics:-DataSummary(egos223);
[mean = 224.0000000, standarddeviation = 14.93318520,
skewness = 9.771282074 10ˆ{-16}, kurtosis = 1.999979619,
minimum = 195., maximum = 253., cumulativeweight = 49506.]
Let us look at the information provided by the example. As we have seen, the
number of points of the elliptic curves over
F
223
are integers in the interval [195,
253] and, moreover, all these integers are the order of some elliptic curve. We also
see that the plot is symmetric with respect to a vertical axis centered at the value
p
+
=
1
224 (which is the center of the interval), i.e., the number of curves of order
+
−
+
+
=
...
29 which
explains why the average value of the orders is precisely 224. This symmetry is a
consequence of the fact that if
y
2
p
1
t
is equal to the number of curves of order
p
1
t
for
t
1
x
3
=
+
ax
+
b
is an elliptic curve over
F
p
of
order
p
+
1
−
t
for some integer
t
, then taking its
quadratic twist
, namely the curve
y
2
x
3
u
2
ax
u
3
b
, where
u
∈ F
p
is a quadratic non-residue, induces a bijection
on the isomorphism classes of elliptic curves and the group of points of the twisted
curve has order
p
=
+
+
+
1
+
t
.
11.2.3 The Orders of Elliptic Curve Groups
An important thing we appreciate in the preceding example is that the orders of
the elliptic curve groups are all contained in a relatively small interval centered at
