Hastad's theorem

In this study, an attack on the El-Gamal encryption scheme ELG-LUC3-ECC is proposed, which is based on a third-order Lucas sequence over an elliptic curve finite field, using Hastad's and Julta's theorems.  Hastad's theorem was used to solve the problem of multivariate modular equations system, whereas Julta's theorem was used to find the solutions of multivariable modular equation.  As a result, the minimum amount of plaintext required for a successful attack may be determined.  Thus, similar attacks can be prevented if the quantity of plaintext remains within the appr