Pseudo-random number generators (PRNGs) play an important role to ensure the security and confidentiality of image cryptographic algorithms. Their primary function is to generate a sequence of numbers that possesses unpredictability and randomness, which is crucial for the algorithms to work effectively and provide the desired level of security. However, traditional PRNGs frequently encounter limitations like insufficient randomness, predictability, and vulnerability to cryptanalysis attacks. To overcome these limitations, we propose a novel method namely an elliptic curve genetic algorithm (ECGA) for the construction of an image-dependent pseudo-random number generator (IDPRNG) that merges elliptic curves (ECs) and a multi-objective genetic algorithm (MOGA). The ECGA consists of two primary stages. First, we generate an EC-based initial sequence of random numbers using pixels of a plain-image and parameters of an EC, that depart from traditional methods of population initialization. In our proposed approach, the image itself serves as the seed for the initial population in the genetic algorithm optimization, taking into account the image-dependent nature of cryptographic applications. This allows the PRNG to adapt its behavior to the unique characteristics of the input image, leading to enhanced security and improved resistance against differential attacks. Furthermore, the use of a good initial population reduces the number of generations required by a genetic algorithm, which results in decreased computational cost. In the second stage, we use well-known operations of a genetic algorithm to optimize the generated sequence by maximizing a multi-objective fitness function that is based on both the information entropy and the period of the PRNG. By combining elliptic curves and genetic algorithms, we enhance the randomness and security of the ECGA.