Abstract
SURVEY OF ELLIPTIC CURVE CRYPTOGRAPHY AND ITS SECURITY APPLICATIONS
Bamarouf Mohamed*, Ahmed Asimi and Mbarek Lahdoud
ABSTRACT
Elliptic curve cryptography (ECC) over the finite fields, whose Elliptic curve cryp- tosystems are based on ECDLP (Elliptic Curve Discrete Logarithm Problem which is the problem of finding the positive number n given a point nP, where P is a point on the curve) for their security, is a powerful branch of cryptography (public key and secret key). The discrete logarithm (DL), in a finite field, is one of the NP −complete problems in number theory and it applies in several fields such as elliptic curves and cryptography. This problem has been raised by several authors such as Martin Hell- man, Tonelli Shanks, John M.Pollard, Adleman. Moreover, numerous methods have been proposed to solve it like Pohlig-Hellman algorithm, Baby-Step, Giant-Step algo- rithm, Rho-Pollard algorithm and Index computation algorithms. The ECC widely used in various security applications due to its efficiency, strong security properties and shorter keys (less-memory requirements and faster field arith- metic operations) such as authentification protocol design, key generation protocol, key exchange protoco, digital signatures, hash functions, security proofs in topical areas like cloud computing, blockchains, Internet of Things and Artificial Intelligence. Our aim in this paper is to present an extensive and careful study of elliptic curve cryptography (ECC) over finite fields and its security applications and also to discuss the arithmetic involved in elliptic curve and how these curve operations are crucial in determining the performance of cryptographic systems.
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