Cryptography Reference
In-Depth Information
Z
2
× Z
2
and
Z
4
, respectively. Since these groups are not isomorphic, we see that
these curves are not isomorphic either.
Exercise 11.12
Show that the elliptic curves
y
2
x
3
b
and
y
2
x
3
a
x
b
=
+
ax
+
=
+
+
a
,
defined over
F
5
are isomorphic if and only if either they are equal or
a
=
b
. Using this, compute the isomorphism classes of elliptic curves
b
=
0 and
b
=−
over
F
5
.
F
5
given by the equations
y
2
=
Exercise 11.13
Show that the elliptic curves over
x
3
1 and
y
2
x
3
+
=
+
2 are not isomorphic but their groups of points are isomorphic
(both are isomorphic to
Z
6
).
Example 11.7
Let us now do a larger example in order to appreciate how the orders
of the groups of rational points are distributed. Consider the prime
p
=
223. We
compute the list of all elliptic curves over
F
223
but we do not print the list as it is too
long:
> ecl223 := EllipticCurvesList(223):
F
223
is 223
2
−
=
The number of elliptic curves over
223
49506:
> nops(ecl223);
49506
Let us compute the orders of the groups of these curves, without printing them:
> egos223 := EllipticGroupOrder
∼
(ecl223):
The maximum and minimum values of the orders are:
> min(egos223); max(egos223);
195
253
The number of curves of each order, starting with 195 and ending with 253 is:
> map(x -> ListTools:-Occurrences(x, egos223), [$195 .. 253]);
[222, 592, 111, 888, 222, 1110, 481, 444, 666, 888, 999, 444, 555, 2220, 444, 1332,
444, 888, 666, 1110, 777, 1998, 666, 666, 703, 1776, 333, 1332, 999, 1554, 999,
1332, 333, 1776, 703, 666, 666, 1998, 777, 1110, 666, 888, 444, 1332, 444, 2220,
555, 444, 999, 888, 666, 444, 481, 1110, 222, 888, 111, 592, 222]
We see that all the integers in the interval [195, 253] are the order of some of these
elliptic curves as none of the values in the preceding list is 0. To better appreciate
how the orders are distributed, we may plot a histogram of these data, along with a
summary of statistical data:
