Elliptic curve cryptography is recognized as one of the most advanced an‎d efficient cryptographic methods, playing a crucial role in securing digital communications. Among these, Montgomery curves, first introduced by Montgomery to enhance the Lenstra factorization method, have found widespread application in modern cryptography. These curves, particularly when using the Montgomery ladder, enable efficient an‎d side-channel-resistant execution of Elliptic Curve Diffie-Hellman (ECDH). However, the inherent limitations in the addition law of Montgomery curves necessitate practical implementations to map them to twisted Edwards curves for utilizing efficient complete addition laws. In this thesis, extended Montgomery coordinates are introduced, an‎d non-multiplicative mappings between extended twisted Edwards coordinates an‎d extended Montgomery coordinates are presented. This approach allows the transfer of complete addition laws from twisted Edwards curves to Montgomery curves without requiring additional multiplications. Furthermore, using a scaling technique, the addition laws for twisted Edwards curves have been refined, leading to improved computational efficiency an‎d reduced processing costs. These advancements significantly enhance elliptic curve cryptography performance, making Montgomery curves a more attractive option for lightweight cryptographic applications resistant to security attacks.

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عمران احمدي درويشوند , نيلوفر احمدي پور