# elliptic curve equation for F2^m field

what will be the elliptic curve equation for the following code after executing in sagemath. for the equation

E: y^2+xy=x^3+ax^2+b

R.<x> = PolynomialRing( GF(2) ) K.<x> = GF( 2**113, modulus = X^113 + X^9 + 1 ); a=003088250CA6E7C7FE649CE85820F7 b=00e8bee4d3e2260744188be0e9c723

E = EllipticCurve( K, [ 1,a, 0, 0, b ] );E

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The given code cannot be executed since:

• The modulus uses X, not defined in advance as a generator of the polynomial ring over $\mathbb F_2 =$ GF(2) .
• a, b should be elements of K (supposed to be already defined with characteristic two), but as they come there is no immediate coercion.

Note on the human readability. Using x as the name of the generator for K will be confusing when asking for the equation of the curve, which by default uses x,y as variables, so i will use u. So:

R.<X> = GF(2)[]
K.<u> = GF( 2**113, modulus = X^113 + X^9 + 1 );

a = K.fetch_int( 0x003088250CA6E7C7FE649CE85820F7 )
b = K.fetch_int( 0x00E8BEE4D3E2260744188BE0E9C723 )

E = EllipticCurve( K, [ 1, a, 0, 0, b ] )

print E
`

Which is now the question?

( 2017-12-19 06:18:31 -0500 )edit