# 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, 2021 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, 2021 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, 2021 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, 2021 at 12:52
• Okay, thanks! looking through the links. Mar 8, 2021 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)