Page 16
GEC 1: Recommended Elliptic Curve Domain Parameters Ver. 0.5
3
Recommended Elliptic Curve Domain Parameters over
F2m
This section speciﬁes the elliptic curve domain parameters over
F 2m recommended in this document.
The section is organized as follows. First Section 3.1 describes relevant properties of the recommended
parameters over
F 2m. Then Section 3.2 speciﬁes recommended 113-bit elliptic curve domain parameters
over
F 2m, Section 3.3 speciﬁes recommended 131-bit elliptic curve domain parameters over
F 2m, Sec-
tion 3.4 speciﬁes recommended 163-bit elliptic curve domain parameters over
F 2m, Section 3.5 speciﬁes
recommended 193-bit elliptic curve domain parameters over
F 2m, Section 3.6 speciﬁes recommended
233-bit elliptic curve domain parameters over
F 2m, Section 3.7 speciﬁes recommended 239-bit elliptic
curve domain parameters over
F2m, and Section 3.8 speciﬁes recommended 283-bit elliptic curve domain
parameters over
F 2m.
3.1
Properties of Elliptic Curve Domain Parameters over
F2m
Following SEC 1 [10], elliptic curve domain parameters over
F 2m are a septuple:
T
=
(m; f
(x);a ;b ;G;n ;h )
) deﬁned by the equation:
consisting of an integer m specifying the ﬁnite ﬁeld
F 2m, an irreducible binary polynomial f
(x) of degree
m specifying the polynomial basis representation of
F 2m, two elements a ;b
2
F 2m specifying an elliptic
curve E
(F 2m
;
+x:y
= x3
+a :x2
E : y2
+b in
F 2m
a base point G
=
(xG
;yG
) on E
(F 2m
)=n.
), a prime n which is the order of G, and an integer h which is the
cofactor h
= #E
(F 2m
When elliptic curve domain parameters over
F 2m are speciﬁed in this document, m is represented directly
as an integer, f
(x) is represented directly as a polynomial, and the remaining components are represented
as octet strings converted using the conventions speciﬁed in SEC 1 [10].
Again following SEC 1 [10], elliptic curve domain parameters over
F 2m must have:
m
2
f113 ;131 ;163 ;193 ;233 ;239 ;283
;409 ;571 g
:
Furthermore elliptic curve domain parameters over
F 2m must use the reduction polynomials listed in
Table 3 below.
This restriction is designed to encourage interoperability while allowing implementers to supply efﬁcient
implementations at commonly required security levels.
Here recommended elliptic curve domain parameters are supplied at each of the following sizes:
m
2
f113 ;131 ;163 ;193 ;233 ;239 ;283 g
:
No recommended elliptic curve domain parameters are currently supplied at the larger sizes:
m
2
f409 ;571 g;