## 3 Proposed Design  
One Omega network is designed to generate a strong cryptographic pseudorandom DNA-based key. The design consists of biological DNA-based operations in which pseudorandom substitution and permutation operations are modified based on biological DNA, as discussed in Section 1. The modifications of pseudorandom substitution and per-mutation operations are shown in Tables 3 and 4. Table 3 presents the DNA XOR rules, which we denoted with the symbol (*). Table 4 presents the complementary mapping rules, which we denoted with the symbol (#). Table 5 presents the mRNA mapping rules, which we denoted with the symbol (# and \(\sigma\)). Our proposed design of an Omega network consists of three rows and four columns, as shown in Figure 1. We denoted it as \(\text{B}_{\text{row, column}}\). \(\text{B}_{2,3}\), \(\text{B}_{2,4}\), and \(\text{B}_{3,4}\) are based on DNA XOR rules. \(\text{B}_{1,2}\), \(\text{B}_{1,4}\), and \(\text{B}_{3,3}\) are based on mRNA mapping rules, referring to the first letter followed by the second. \(\text{B}_{1,1}\), \(\text{B}_{2,1}\), and \(\text{B}_{3,2}\) are based on mRNA mapping rules, which refer to the second letter followed by the first letter. \(\text{B}_{1,3}\), \(\text{B}_{2,2}\), and \(\text{B}_{3,1}\) are based on complementary mapping rules. The operations are arranged randomly, and each operation is equally balanced, appearing in the design in order to produce cryptographically pseudorandom outputs. Firstly, two 128-bit binary string sperm and string ovum are generated. String sperm and string ovum is a union using XORing binary, which produces a string zygote. Three 128-bit random strings are generated, which are taken from Pi [31] and which we denote as random string \(a\), random string \(b\), and random string \(c\). The string zygote and random strings are passed to the Omega network. Firstly, the string zygote and random string \(b\) are XORed as the input B\({}_{1,\,1}\). Random string \(a\) and random string \(c\) are XORed as the input B\({}_{1,\,2}\). Random string \(a\) and random string \(b\) are XORed as the input B\({}_{1,\,3}\). String zygote and random string \(c\) are XORed as the input B\({}_{1,\,4}\). These have completed the first row, four columns of the Omega network. Secondly, the output of B\({}_{1,\,1}\) and B\({}_{1,\,3}\) are XORed as the input to B\({}_{2,\,1}\). The output of B\({}_{1,\,1}\) and B\({}_{1,\,2}\) are XORed as the input to B\({}_{2,\,2}\). The output of B\({}_{1,\,2}\) and B\({}_{1,\,4}\) are XORed as the input to B\({}_{2,\,3}\). The output of B\({}_{1,\,3}\) and B\({}_{1,\,4}\) are XORed as the input to B\({}_{2,\,4}\). These have completed the second row, four columns of the Omega network. Next, the output of B\({}_{2,\,1}\) and the output of B\({}_{2,\,4}\) are XORed as the input to B\({}_{3,\,1}\). The out-put of B\({}_{2,\,1}\) and the output of B\({}_{2,\,3}\) are XORed as the input to B\({}_{3,\,2}\). The output of B\({}_{2,\,2}\) and the output of B\({}  
the Omega network. Lastly, the output of \(\text{B}_{3,\,1}\) is XORed with \(\text{B}_{3,\,4}\); the length is 128 bits. The output \(\text{B}_{3,\,2}\) is XORed with \(\text{B}_{3,\,2}\); the length is 128 bits. Both strings are concatenated as 256 bits, yielding what is best known as a cryptographic pseudorandom DNA-based key. The complete steps are presented in Algorithm 1. Here, we provide a test case of the proposed Omega network design using four inputs and their corresponding outputs. The four inputs are string sperm, string ovum, and three random strings. The output is a cryptographically pseudorandom key.