Some properties of continued fraction expansions related to certain sequence of integers and its applications in elliptic curve cryptography

In this paper, we consider the sequence of integers ri, si, ti ∈ ℤ of regular continued fraction (RCF) expansions generated from the extended Euclidean algorithm. These sequences always satisfy ri = sia + tib where ri is a remainder whereas si and ti arising from the extended Euclidean algorithm are equal, up to sign, to the convergent of the continued fraction expansion of a/b. We discuss the behavior of these sequences and provide their full proof in detail with their simple calculation of sample. Throughout this work, we deal with the concept of Euclidean algorithm and extended Euclidean algorithm together with continued fraction algorithm. These algorithms are involved in the improvement of computational efficiency of the elliptic curve cryptography (ECC). Henceforth from that, we tend to associate these sequences in ECC. Last but not least, we found that the value of integers (ri, si, ti) satisfy various properties in RCF which then used to solve the shortest vector problem in representing point multiplications in ECC, namely the Gallant, Lambert & Vanstone (GLV) integer decomposition method and the integer sub decomposition (ISD) method.