# Elliptic curve in Binary Field implementation

For Elliptic curves defined over $$GF(2^n)$$, by adding any two points P and Q over $$GF(2^n)$$ we get the third point over $$GF(2^n)$$. In Elliptic Curve Digital Signature Algorithm (ECDSA) https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm , there is a usage of prime numbers, particularly for the calculation of multiplicative inverse. It works fine for the Weierstrass form of Elliptic curve over $$GF(P)$$.

Now, I want to implement ECDSA using a binary form of Elliptic curve (like Hessian curve), but every time I am doing it, it is giving the wrong answer. My question is, is it not possible to use any curve over $$GF(2^n)$$ for an implementation like this?

• Mar 8 at 12:40
• If you really want to implement it, you can use Stack Overflow to ask for help. In Cryptography.SE we are no interested in the implementation problems. As you can see a recent problem has been migrated to so Mar 8 at 12:50
• No, I was just asking logic-wise.... like is it possible? to use a binary curve for ECDSA where operations are done using prime numbers? Mar 8 at 12:51
• Also, the rational point of a curve defined over a field $E(K)$ form an abelian group where it is usual to define a Z-Module with scalar multiplication. The field for the underlying curve has only an effect on the security, the number of points, and efficient implementations. Therefore it is possible. Mar 8 at 12:52
• Okay, thanks! looking through the links. Mar 8 at 12:54

Yes it can. The only variation from the wikipedia article is in step 5 where it states to calculate $$r\equiv x_1\pmod{n}$$. This does not make sense when $$x_1$$ isbn element of a binary field. Instead binary field elements are converted to integers by a standards prescribed process (see ANSI X9.62-1984 section 4)