# 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 + 2

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This is an answer. To this and all other many questions that involve(d) fE and fL.

You declare the elliptic curve E.

Independently, there appear R over K and the polynomials fE and fK. There is no connection to E. The question is if the rest in the division with rest of fE by fL has a specific property.

The answer is not yes, and not no, since the question is not well defined. (If the question is that the rest always has third degree. What does it mean always?)

P.S.

This answer should please not be taken as offensive, it is just an invitation to provide all (needed) framework, and to improve the communication. Note that on similar sites, like mathexchange and/or mathoverflow such questions get immediately downvotes.

Please give a reference, once for all times, to the involved structure. Which is the receipt, the definition, the framework to construct fE and fL, starting from an elliptic curve (and possibly some more data). Please provide a link. It is in my oppinion not fair to systematically give only the strictly minimal information for some programming issue. Potential helpers can not profit from the question, may have possibly a quick and simple, structural solution if the structure is present, and can construct similar examples to test their code.

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