ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 25 Jul 2018 02:51:34 +0200Elliptic curve arithmetichttps://ask.sagemath.org/question/43123/elliptic-curve-arithmetic/the below code is performed mod arithmetic of two polynomial fE and fL over prime field P= 5 and the extension is 5^2. the mod of two polynomial is always third degree polynomial. My fE polynomial is always 3p degree and fL polynomial is p degree. my question is whatever p(large p = 112 bit ,128 bit 160 bit) I will take my mod is always third degree. is it because of any polynomial property.
p=5
A=4
B=4
print"\n p=",p
print"\n A=",A
print"\n B=",B
F = GF(p)
E = EllipticCurve( F, [A,B] );E
S.<a> = PolynomialRing( F )
K.<a> = GF( p**2);#K.modulus#, modulus=W^2+W+1 )
print "\n Modulus of K is =", K.modulus()
R.<z> = PolynomialRing( K, sparse=True )
fE=z^15 + (4*a + 4)*z^11 + 2*z^10 + (a + 3)*z^7 + z^6 + (2*a + 2)*z^5 + z^3 + 2*z^2 + (3*a + 4)*z + 1
fL=(4*a*z^5 + (a + 4)*z + 3);fL
f1= (fE%fL).monic;f1
z^3 + (3*a + 4)*z^2 + 3*z + 3*a + 2Mon, 23 Jul 2018 16:19:18 +0200https://ask.sagemath.org/question/43123/elliptic-curve-arithmetic/Comment by slelievre for <p>the below code is performed mod arithmetic of two polynomial fE and fL over prime field P= 5 and the extension is 5^2. the mod of two polynomial is always third degree polynomial. My fE polynomial is always 3p degree and fL polynomial is p degree. my question is whatever p(large p = 112 bit ,128 bit 160 bit) I will take my mod is always third degree. is it because of any polynomial property. </p>
<p>p=5
A=4
B=4
print"\n p=",p
print"\n A=",A
print"\n B=",B</p>
<p>F = GF(p)
E = EllipticCurve( F, [A,B] );E</p>
<p>S.<a> = PolynomialRing( F )
K.</a><a> = GF( p**2);#K.modulus#, modulus=W^2+W+1 )
print "\n Modulus of K is =", K.modulus()</a></p><a>
<p>R.<z> = PolynomialRing( K, sparse=True )</p>
<p>fE=z^15 + (4<em>a + 4)</em>z^11 + 2<em>z^10 + (a + 3)</em>z^7 + z^6 + (2<em>a + 2)</em>z^5 + z^3 + 2<em>z^2 + (3</em>a + 4)<em>z + 1
fL=(4</em>a<em>z^5 + (a + 4)</em>z + 3);fL
f1= (fE%fL).monic;f1
z^3 + (3<em>a + 4)</em>z^2 + 3<em>z + 3</em>a + 2</p>
</a>https://ask.sagemath.org/question/43123/elliptic-curve-arithmetic/?comment=43136#post-id-43136Duplicate of [Ask Sage question 43122](https://ask.sagemath.org/question/43122).Wed, 25 Jul 2018 02:51:34 +0200https://ask.sagemath.org/question/43123/elliptic-curve-arithmetic/?comment=43136#post-id-43136