Public-key cryptography makes it possible to create digital signatures and do key negotiation, which is inevitable for today’s commercial and governmental online computer systems. Elliptic curve cryptography (ECC) is a preferred method to implement these services, due to its low computational complexity compared to other modular arithmetic systems such as the Rivest-Shamir-Adleman algorithm (RSA). This thesis investigates how the use of different curve representations impact the performance of the Elliptic Curve Discrete Signing Algorithm (ECDSA) and the Elliptic Curve Diffie-Hellman (ECDH) key-exchange. This thesis developes a point compression method for Hessian curves over both binary and odd prime characteristic fields, and an addition-subtraction formula for twisted Hessian curves. Using these mechnanisms, the thesis gives a full C implementation of twisted Hessian curves, binary Hessian curves, and binary Edwards curves as an extension to the open source library Miracl. To performance test these curve representations the thesis gives several random curves and presents a detailed test methodology. This methodology lets us compare the complexity of a scalar multiplication in the different curve representations as a number of field multiplications. Applying this methodology, the thesis compares the already implemented Weierstrass curves and twisted inverted Edwards curves in Miracl with the added curve representations. From these results, twisted inverted Edwards curves over odd prime characteristic fields clearly excel as the best performing curve representation for ECDSA-signature verification. For ECDSA-signature generation and ECDH key-exchange, projective Weierstrass over binary fields using Koblitz curves stands out as the best performing. Furthermore, the thesis evaluates the implementation of these curve representations in regards of simple side-channel attack protection.