The public key cryptosystem is fundamental in safeguard communication in cyberspace. This paper described a new cryptosystem analogous to El-Gamal encryption scheme, which utilizing the Lucas sequence and Elliptic Curve. Similar to Elliptic Curve Cryptography (ECC) and Rivest-Shamir-Adleman (RSA), the proposed cryptosystem requires a precise hard mathematical problem as the essential part of security strength. The chosen plaintext attack (CPA) was employed to investigate the security of this cryptosystem. The result shows that the system is vulnerable against the CPA when the sender decrypts a plaintext with modified public key, where the cryptanalyst able to break the security of the proposed cryptosystem by recovering the plaintext even without knowing the secret key from either the sender or receiver.

The security of LUC type cryptosystems was investigated. In this study, chosen message attack was employed to analyze the security of LUC, LUC3 and LUC4,6 cryptosystems. The cryptanalyst invades the system by obtaining a signature without the sender’s consent, and use it to break the system. Finding shows that LUC4,6 cryptosystem is more resilient against chosen message attack compare with LUC and LUC3 cryptosystems.

Elliptic Curve Cryptosystem based on second order Lucas sequenceis a cryptosystem using elliptic curves over finite fields as a mask and incorporate with second order of Lucas sequence. The security of the Elliptic Curve Cryptography cryptosystem depends on the discrete logarithms. In this cryptosystem, Lucas sequence is employed to compute the ciphertext or recover the plaintext. The Elliptic Curve Cryptosystem based on second order Lucas sequence is vulnerable when the bit of the decryption key, $d$ flips by using fault based attack.

LUC-type cryptosystems are asymmetric key cryptosystems based on the Lucas sequence that is extended from RSA. The security challenge is comparable to RSA, which is based on the intractability of factoring a large number. This paper analysed the security of LUC, LUC3, and LUC4,6 cryptosystems using a common modulus attack.For a common modulus attack to be successful, a message must be transmitted to two distinct receivers with the same modulus. The strengths and limitations of the LUC, LUC3, and LUC4,6 cryptosystems when subjected to a common modulus attack were discussed as well. The results reveal that the LUC4,6 cryptosystem provides greater security than the LUC and LUC3.

The current research aims to provide a viable numerical method for solving difficult engineering and science problems which are in the form of higher order ordinary differential equations. The proposed method approximates these ordinary differential equations using Newton-Gregory backward difference polynomial in predictor–corrector mode. The predictor–corrector algorithm is then fitted with a variable order step size algorithm to reduce computational cost. The variable order stepsize algorithm allows the method to predetermine the preferred level of accuracy with the added advantage of less computational cost. The method is subsequently programmed with a two-point block formulation which can be altered for parallel programming. This research also discusses order and stepsize strategies of the variable order stepsize algorithm. Stability and convergence estimations of the method are also established. Numerical results obtained will validate the accuracy and efficiency of the method using various types of linear and nonlinear higher order ordinary differential equations