Understanding elliptic curves contributed to solving mathematical problems in number theory that had been unsolved for centuries. Elliptic curves were also used in solving one of the millennial problems, which is Fermat's last theorem. They are also connected with many hypotheses and problems in mathematics that have yet to be solved. Elliptic curves defined over finite fields are widely used in public key cryptography, since they have proven to be groups that have the best properties for implementing the Diffie-Hellman protocol. This article provides an overview of the theoretical assumptions that are necessary for the development of cryptographic algorithms based on elliptic curve cryptography, which includes defining elliptic curves, defining the properties of arithmetic operations on elliptic curves used in cryptography with reference to curves defined over finite fields.

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