$k^{\prime}$ -Lucas Sequence for Elliptic Curve Cryptography

The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{k}^{\prime}$</tex> -Lucas sequence is a significant generalization of the classical Lucas sequence, exhibiting various intriguing properties and applications across mathematics and cryptography. The study of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{k}^{\prime}$</tex> -Lucas-like sequences on elliptic curves has gained attention recently, which offer unique structural advantages for cryptographic systems. An extension of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{k}^{\prime}$</tex> -Lucas sequence defined on elliptic curves (EC) is presented in this study, highlighting its recursive nature and algebraic characteristics. Additionally, we suggest an encryption scheme that makes use of this sequence, leveraging the complexities inherent in “Elliptic Curve Discrete Logarithm Problem (ECDLP)” [1]. Our findings indicate that this approach enhances security and opens new avenues for robust cryptographic protocols.