# A Construction Method for the Random Factor-Based G Function
## 1 Introduction  
As the communication spectrum becomes increasingly crowded, secure communication to avoid interference and interception has become a hot topic recently [1; 2]. Spread-spectrum technology expands the frequency spectrum of the signal to a wider range, making it less susceptible to interference or interception during transmission. It has been widely used in military and civilian secure communication [3; 4]. The differential frequency-hopping technique is a specific form of spread-spectrum communication. It changes the signal frequency in continuous time intervals, which provides more flexibility and unpredictability in the communication signal [5; 6; 7]. Since differential frequency-hopping technology offers high interference resistance and security, this wireless communication technology has extensive application potential in both military and civilian fields [8]. In recent years, machine learning research has rapidly developed in multiple fields, becoming an important driving force for technological progress and innovation [9; 10; 11]. However, in differential frequency hopping communication systems, the construction method of the G function remains a fundamental and critical aspect [12; 13]. A large number of construction methods have been proposed by scholars around the world. [PERSON] et al. [14] proposed a method for controlling the G function by cascading multiple pseudo-random number generators and dynamically adjusting the period, number, delay, and other parameters of the generators. The performance of the sequence improves with an increase in the number of pseudo-random number generators; however, the optimal number of generators has yet to be determined. Furthermore, [PERSON] et al. [15] proposed a method for generating high-order differential frequency-hopping sequences, which entails a trade-off between greater computational and storage resources and the randomness and security of the frequency-hopping sequence. Concurrently, this scheme requires the incorporation of atrusted entity to authenticate the user cluster, thereby exacerbating the overall complexity of the system. [PERSON] et al. (2016) developed an encryption scheme for pseudo-random sequences using logical chaotic mapping, which improved two-dimensional uniformity but had a negative impact on other system performance. Meanwhile, [PERSON] et al. (2017) proposed an enhanced differential frequency hopping (EDFH) framework that integrates training signals into traditional transmission processes and designs a mixed signal matched filter (CMF) to optimize the receiver's processing of user and interference signals. [PERSON] et al. (2018) proposed a G function design scheme based on iterative decomposition. This scheme differs from the previous one in that it modifies the frequency transfer relationship over different time periods by decomposing the frequency pair set. However, this approach encounters a challenge in accurately decoding the de-mapping algorithm, which in turn increases the bit error rate. The existing G function is constrained by predefined conditions, resulting in limited flexibility in generating frequency-hopping sequences. To address this limitation, this paper proposes the Random Factor-Based G function. By introducing a random factor, the method dynamically adjusts the position of frequencies within subsets, breaking the fixed frequency-hopping pattern. This approach enhances system adaptability and improves anti-interference capabilities. The main contributions of this paper can be summarized as follows:  
1. We propose the RFGF, which significantly enhances the two-dimensional continuity of frequency-hopping sequences and improves the anti-interference capabilities of communication systems. 2. The RFGF increases the unpredictability of frequency-hopping sequences by introducing random disruptions within each subset's hopping patterns. This approach ensures that frequency selections are more random rather than sequentially ordered. 3. The effectiveness of RFGF is demonstrated via simulations.