Relationship between Lucas Sequences and Gaussian Integers in Cryptosystems

Both Gaussian integers and Lucas sequences have been applied in cryptography. This paper presents the mathematical relationship between Lucas sequences and Gaussian integers. It also explores the complexity of Discrete Logarithm Problem (DLP) for Gaussian integers modulo prime by reducing it to Lucas Sequences DLP and real integer DLP. We demonstrate that the algorithms based on the Gaussian Integer DLP have advantages over the corresponding algorithms based on real integer DLP or Lucas Sequences DLP. Numerical examples are provided.