The Elliptic Curve Digital Signature Algorithm (ECDSA)
8
Field Elements
. The nite eld
F
2
m
is comprised of all polynomials over
F
2
of degree less than
m
:
F
2
m
=
f
a
m
;
1
x
m
;
1
+
a
m
;
2
x
m
;
2
+
+
a
1
x
+
a
0
:
a
i
2f
0
1
gg
:
The eld element
a
m
;
1
x
m
;
1
+
a
m
;
2
x
m
;
2
+
+
a
1
x
+
a
0
is usually denoted by the bit string
(
a
m
;
1
a
m
;
2
:::a
1
a
0
) of length
m
, so that
F
2
m
=
f
(
a
m
;
1
a
m
;
2
:::a
1
a
0
):
a
i
2f
0
1
gg
:
Thus the elements of
F
2
m
can b e represented by the set of all binary strings of length
m
. The
multiplicative identity element (1) is represented by the bit string (00
:::
01), while the additive
identity element (0) is represented by the bit string of all 0's.
Field Operations
. The following arithmetic operations are dened on the elements of
F
2
m
when
using a polynomial basis representation with reduction polynomial
f
(
x
):
Addition:
If
a
=(
a
m
;
1
a
m
;
2
:::a
1
a
0
)and
b
=(
b
m
;
1
b
m
;
2
:::b
1
b
0
) are elemen
ts of
F
2
m
,
then
a
+
b
=
c
=(
c
m
;
1
c
m
;
2
:::c
1
c
0
), where
c
i
=(
a
i
+
b
i
) mod 2. That is, eld addition is
performed bitwise.
Multiplication:
If
a
=(
a
m
;
1
a
m
;
2
:::a
1
a
0
)and
b
=(
b
m
;
1
b
m
;
2
:::b
1
b
0
) are elements of
F
2
m
, then
a
b
=
r
=(
r
m
;
1
r
m
;
2
:::r
1
r
0
), where the p olynomial
r
m
;
1
x
m
;
1
+
r
m
;
2
x
m
;
2
+
+
r
1
x
+
r
0
is the remainder when the polynomial
(
a
m
;
1
x
m
;
1
+
a
m
;
2
x
m
;
2
+
+
a
1
x
+
a
0
)
(
b
m
;
1
x
m
;
1
+
b
m
;
2
x
m
;
2
+
+
b
1
x
+
b
0
)
is divided by
f
(
x
)over
F
2
.
Inversion:
If
a
is a non-zero elementin
F
2
m
, the
inverse
of
a
, denoted
a
;
1
, is the unique
element
c
2
F
2
m
for which
a
c
=1.
Example 2
(
Apolynomial basis representation of the nite eld
F
2
4
)Let
f
(
x
)=
x
4
+
x
+ 1 be the
reduction polynomial. Then the 16 elements of
F
2
4
are:
0 (0000) 1 (0001)
x
(0010)
x
+1 (0011)
x
2
(0100)
x
2
+1 (0101)
x
2
+
x
(0110)
x
2
+
x
+1 (0111)
x
3
(1000)
x
3
+1 (1001)
x
3
+
x
(1010)
x
3
+
x
+1 (1011)
x
3
+
x
2
(1100)
x
3
+
x
2
+1 (1101)
x
3
+
x
2
+
x
(1110)
x
3
+
x
2
+
x
+1 (1111)
Examples of the arithmetic op erations in
F
2
4
are:
(1101) + (1001) = (0100).
(1101)
(1001) = (1111) since (
x
3
+
x
2
+1)
(
x
3
+1) =
x
6
+
x
5
+
x
2
+ 1 and (
x
6
+
x
5
+
x
2
+
1) mod (
x
4
+
x
+1)=
x
3
+
x
2
+
x
+1.
(1101)
;
1
= (0100).
The element
=
x
= (0010) is a generator of
F
2
4
since its order is 15 as the following calculations
show:
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