Chapter 3 Cryptography 71
Cryptography Overview
• The order n of P
P is sometimes called the base point.
The Cofactor
We mentioned previously that the prime number n that is the order of P must evenly
divide the order of the elliptic curve. That is, we know that the number h =#E(F
q
)/n is
an integer. We call h the cofactor, and set it as our last parameter:
• The cofactor h =#E(F
q
)/n
Summary of Elliptic Curve Terminology
Table 3-2 lists the elliptic curve parameters and gives a short description of each
parameter. For a brief description,refer to the previous sections in this chapter; for a
detailed discussion, see [13], [14], and [19] in “Related Documents” on page xx.
Table 3-2 Elliptic Curve Parameters
Notation Name Description
F
q
base field Either:
F
p
: {0,1,...,p–1} with arithmetic mod p
or
F
2
m : strings of m bits. Addition is bitwise XOR,
multiplication exists, but has no quick description
a, b coefficients of the curve a and b are elements of F
q
. They determine an
equation, which depends on the base field:
For F
p
:y
2
=x
3
+ax+b
For F
2
m:y
2
+ xy = x
3
+ax
2
+b
P point of prime order
or
base point
(x
P
,y
P
)
The pair x
P
, y
P
satisfies the curve equation.
n order of P The smallest nonzero number such that P added
to itself n times is the zero point, Ο, on the curve.
n is prime.
h cofactor The order of the curve divided by the order of P:
#E(F
q
)/n
