# Koblitz curve

how to write sage code to find point on koblitz curve. how to write equation on koblitz curve.

Koblitz curve

how to write sage code to find point on koblitz curve. how to write equation on koblitz curve.

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Some examples may be enough to answer the two dot-ending questions.

There are two elliptic {\scshape Koblitz} curves over $\mathbb F_2$, the equations are $y^2 + xy = x^3 +ax^2+1$, $a$ is either zero or one. In sage, one can simply declare them and investigate points defined over bigger fields as follows:

```
F = GF(2) # general field with two elements, 0, 1
a = 0 # or we set a = 1 here, the following lines can be copied...
E = EllipticCurve( F, [ 1, a, 0, 0, 1 ] )
K.<u> = GF( 2^3 )
EK = EllipticCurve( K, [ 1, a, 0, 0, 1 ] )
# instead we may use EK = E.base_extend( K )
print "K :: %s" % K
print "EK :: %s" % EK
print "EK( K ) has %s elements" % EK.order()
print "The K-rational points of E are:"
for P in EK.points():
print P
```

This gives:

```
K :: Finite Field in u of size 2^3
EK :: Elliptic Curve defined by y^2 + x*y = x^3 + 1 over Finite Field in u of size 2^3
EK( K ) has 4 elements
The K-rational points of E are:
(0 : 1 : 0)
(0 : 1 : 1)
(1 : 0 : 1)
(1 : 1 : 1)
```

We may also work with bigger fields, for instance:

```
L.<v> = GF(2^37)
EL = E.base_extend( L )
print "ORDER %s for %s" % ( EL.order().factor(), EL )
P = EL.gens()[0] # pythonically first and only generator
# print "Generator:\n%s" % P
```

And we get:

```
ORDER 2^2 * 149 * 230603167 for Elliptic Curve defined by y^2 + x*y = x^3 + 1 over Finite Field in v of size 2^37
```

(The point P may also be printed by decommenting.)

The very first lines for the next genus would be in an example, dialog with the interpreter:

```
sage: K.<a> = GF(2^3)
sage: K
Finite Field in a of size 2^3
sage: R.<x> = PolynomialRing( K )
sage: C = HyperellipticCurve( x^5+1, x )
sage: C
Hyperelliptic Curve over Finite Field in a of size 2^3 defined by y^2 + x*y = x^5 + 1
sage: C.points()
[(0 : 1 : 0),
(0 : 1 : 1),
(a : a^2 + 1 : 1),
(a : a^2 + a + 1 : 1),
(a^2 : a^2 + a + 1 : 1),
(a^2 : a + 1 : 1),
(a + 1 : a^2 + 1 : 1),
(a + 1 : a^2 + a : 1),
(a^2 + a : a + 1 : 1),
(a^2 + a : a^2 + 1 : 1),
(a^2 + a + 1 : a + 1 : 1),
(a^2 + a + 1 : a^2 : 1),
(a^2 + 1 : a^2 + a + 1 : 1),
(a^2 + 1 : a : 1),
(1 : 1 : 1),
(1 : 0 : 1)]
```

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Asked: ** 2017-02-15 17:30:19 +0200 **

Seen: **360 times**

Last updated: **Feb 28 '17**

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