Cryptography Reference
In-Depth Information
with
y y
xx
+
λ
=
1
2
+
1
2
where P 1 ¹ - P 1 . Then 2 P 1 = ( x 3 , y 3 ),
2. Point doubling. Let
Px
(, )
y
EF
(
,
1
1
1
m
2
where
b
2
2
=++=+
λλ
x
a
x
(3.32)
3
1
2
x
1
and
2
=++
λ
(3.33)
y
x
xx
3
1
3
3
P
+¥=¥+ = for all
P
P
Î
3. Identity.
PEF
(
)
1
1
1
1
m
2
then (, ) (,
++=¥ The point
4. Negatives. If
Px y
(, )
EF
(
),
xy
xx
y
)
.
1
m
2
xx + is denoted by - P 1 and is called the negative of P 1 ; note that - P 1 is
indeed a point in
(,
)
Also, - ¥=¥ .
EF
(
).
m
2
3.6 Why Elliptic Curve Cryptography?
One of the main reasons for using elliptic curves in cryptography is because of its
reduction in key size, while providing the same level of security as RSA. For example,
a 313-bit key size in elliptic curve systems will provide the same level of security as a
512-bit key size in RSA. Hence, it leads to faster computations as well as reduced stor-
age space, processing power, and bandwidth.
3.7 Elliptic Curve Discrete Log Problem
Let E be an elliptic curve defined over a finite field with q elements ( F q ), such that
M     F q of order n . Let
Îá ñ and
NP
Ml =
(3.34)
l Î- . Hence, integer l is called the discrete logarithm of M to
the base N , where l = log N M . (Note: The value of l is very large in real cryptographic
applications.)
In the area of solving hard problems, we normally gauge in terms of the size of the
problem. For example, in the case of RSA, the hardness exists in factorizing the length
of the modulus. In the case of elliptic curve crypto systems, the hardness exists in work-
ing with the number of points q in the group.
[0,
1]
where l Z and
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