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.

翻译：伪随机数生成器在确保图像密码算法的安全性和保密性方面发挥着重要作用。其主要功能是生成具有不可预测性和随机性的数字序列，这对于算法有效运作并提供所需的安全级别至关重要。然而，传统伪随机数生成器常面临随机性不足、可预测性以及易受密码分析攻击等局限性。为克服这些局限，我们提出一种新方法，即基于椭圆曲线遗传算法，构建图像相关伪随机数生成器，该方法融合了椭圆曲线与多目标遗传算法。椭圆曲线遗传算法包含两个主要阶段：首先，利用明文图像像素及椭圆曲线参数生成基于椭圆曲线的初始随机数序列，这不同于传统的种群初始化方法。在所提出的方法中，图像本身作为遗传算法优化中初始种群的种子，充分考虑了密码应用的图像相关性。这使得伪随机数生成器能根据输入图像的独特特性调整其行为，从而增强安全性并提高抵御差分攻击的能力。此外，良好的初始种群使用减少了遗传算法所需的世代数，进而降低了计算成本。第二阶段，我们采用遗传算法的经典操作，通过最大化基于信息熵与伪随机数生成器周期的多目标适应度函数，对生成的序列进行优化。通过结合椭圆曲线与遗传算法，我们提升了椭圆曲线遗传算法的随机性与安全性。