On the Point at Infinity of Elliptic Curves and a New Group Structure on the Quadratic Curve x^2

Abstract

Elliptic curves play a fundamental role in contemporary cryptography due to their deep algebraic properties and their ability to provide high levels of security with relatively small parameter sizes. The theoretical and practical foundations of elliptic curve cryptography (ECC) were established through independent contributions by Koblitz and Miller, whose pioneering work demonstrated that the group structure induced by elliptic curves could be effectively employed in public-key cryptographic schemes. Since then, ECC has become a cornerstone of modern cryptographic protocols, including widely deployed mechanisms such as the Elliptic Curve Digital Signature Algorithm (ECDSA). In this study, we re-examine the algebraic and geometric significance of the point at infinity. Rather than treating this element as an abstract artifact, we introduce an alternative geometric interpretation by associating the identity with the line at infinity. This perspective preserves the classical group law while offering a more intuitive and geometrically transparent understanding of the identity element within the elliptic curve framework. Motivated by the chord-and-tangent addition law on elliptic curves, we further extend these ideas to a non-elliptic setting by defining a new binary operation on the quadratic curve . Consequently, the resulting algebraic system forms an abelian group. The proposed approach not only enhances the conceptual clarity of the group law underlying elliptic curves but also illustrates how similar algebraic constructions can arise in alternative geometric contexts. These findings provide a theoretical framework that may inspire future explorations of cryptographic primitives beyond classical elliptic curve models.Elliptic curves play a fundamental role in contemporary cryptography due to their deep algebraic properties and their ability to provide high levels of security with relatively small parameter sizes. The theoretical and practical foundations of elliptic curve cryptography (ECC) were established through independent contributions by Koblitz and Miller, whose pioneering work demonstrated that the group structure induced by elliptic curves could be effectively employed in public-key cryptographic schemes. Since then, ECC has become a cornerstone of modern cryptographic protocols, including widely deployed mechanisms such as the Elliptic Curve Digital Signature Algorithm (ECDSA). In this study, we re-examine the algebraic and geometric significance of the point at infinity. Rather than treating this element as an abstract artifact, we introduce an alternative geometric interpretation by associating the identity with the line at infinity. This perspective preserves the classical group law while offering a more intuitive and geometrically transparent understanding of the identity element within the elliptic curve framework. Motivated by the chord-and-tangent addition law on elliptic curves, we further extend these ideas to a non-elliptic setting by defining a new binary operation on the quadratic curve . Consequently, the resulting algebraic system forms an abelian group. The proposed approach not only enhances the conceptual clarity of the group law underlying elliptic curves but also illustrates how similar algebraic constructions can arise in alternative geometric contexts. These findings provide a theoretical framework that may inspire future explorations of cryptographic primitives beyond classical elliptic curve models.

Keywords

• Elliptic curves
• Point at infinity
• Quadratic curves
• Group law
• Geometric interpretation

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