Cryptography Reference
In-Depth Information
Now consider the surface
y − x
2
+
xz
+6
x
+
z
=0
.
This surface contains the parabola
y
=
x
2
6
x, z
= 0. The inclusion of the
parabola in the surface corresponds to a surjective ring homomorphism
−
R
[
x, y, z
]
/
(
y − x
2
+
xz
+6
x
+
z
)
R
[
x, y
]
/
(
y
+6
x − x
2
)
−→
f
(
x, y, z
)
−→
f
(
x, y,
0)
.
We also have a surjective map on the algebraic objects representing the tan-
gent spaces
{ax
+
by
+
cz}
mod (
y
+6
x
+
z
)
−→ { ax
+
by}
mod (
y
+6
x
)
corresponding to the inclusion of the tangent line to the parabola in the tan-
gent plane for the surface at (0
,
0
,
0). In this way, we can study relations
between geometric objects by looking at the corresponding algebraic objects.
Wiles works with rings such as
O
p
[[
x
]]
/
(
x
2
− px
), where for simplicity we
henceforth assume that
O
p
is the
p
-adic integers and where
O
p
[[
x
]] denotes
power series with
p
-adic coecients. The zeros of
x
2
− px
are 0 and
p
,sothis
ring corresponds to the geometric object
S
1
:
•
•
0
p
The tangent space is represented by the set obtained by looking only at the
linear terms, namely
{
ax
|
a
∈O
p
}
mod (
px
). Since
a
1
x ≡ a
2
x
mod
px
⇐⇒
a
1
≡ a
2
(mod
p
)
,
the tangent space can be identified with
Z
p
.
As another example, consider the ring
p
3
)), which
O
p
[[
x
]]
/
(
x
(
x
−
p
)(
x
−
corresponds to the geometric object
S
2
:
•
•
•
p
3
0
p
The tangent space is
Z
p
4
.
There is an inclusion
S
1
⊂
S
2
, which corresponds to the natural ring ho-
momorphism
p
3
))
−→ O
p
[[
x
]]
/
(
x
2
O
p
[[
x
]]
/
(
x
(
x
−
p
)(
x
−
−
px
)
.
The map on tangent spaces is the map from
Z
p
4
to
Z
p
that takes a number
mod
p
4
and reduces it mod
p
.
Now consider the ring
O
p
[[
x, y
]]
/
(
x
2
−px, y
2
−py
). In this case, we are look-
ing at power series in two variables, and two power series are congruent if their
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