<![CDATA[This article develops a conceptual framework that connects quotient field construction, kurva eliptik curve group structures, and symbolic operations in Elliptic Curve Cryptography (ECC), enabling the mathematical mechanisms of ECC to be analyzed explicitly before being implemented over finite fields. Although practical implementations of ECC employ finite fields as the basis for arithmetic operations, understanding quotient field construction from an integral domain remains important as the mathematical foundation of kurva eliptik curve operations. This article investigates the algebraic structure of the quotient field and its application to kurva eliptik curve theory, focusing on point addition and the verification of the Abelian group properties of the point set . This approach allows every algebraic operation to be represented symbolically (symbolic computation), enabling kurva eliptik curve point operations to be analyzed more explicitly. Such an understanding bridges the concepts of abstract algebra and kurva eliptik curve group structures. Therefore, the study of quotient fields provides a conceptual framework that supports the theoretical understanding of ECC. This article aims to provide a deeper understanding of the mathematical foundations underlying the use of kurva eliptik curves in cryptography as a step toward developing a conceptual framework for explaining ECC.]]>
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