This study delves into the predictive capabilities of neural network models, specifically focusing on Recurrent Neural Networks (RNNs) and Long Short-Term Memory (LSTM) networks as well as the combination of both in a hybrid architecture, for forecasting the outputs of various pseudo-random number generators (PRNGs). The investigation extends across a diverse set of PRNG algorithms, including Linear Congruential Generator (LCG), Mersenne Twister (MT), Xorshift, and Middle Square. Through meticulous analysis, the study evaluates the accuracy of these models in predicting single and continuous outputs generated from the mentioned PRNGs. The research findings illuminate the superior predictive performance of hybrid models, attributed to their adeptness at capturing long-term dependencies, a crucial factor in decoding the complexities of PRNG sequences. Additionally, the impact of model optimization techniques, including dropout and L2 regularization, on enhancing predictive accuracy is thoroughly explored. This comprehensive examination not only underscores the potential of neural networks in identifying deterministic patterns within PRNG outputs but also offers valuable insights into optimal model selection and configuration. The implications of this work are significant, paving new avenues in cryptography and securing random number generation by highlighting the predictability of PRNGs under advanced neural network models.
In the ever-evolving landscape of machine learning, the ability to accurately predict future events based on sequential data stands as a cornerstone of numerous technological advancements and applications. From forecasting stock market trends to decoding human language, the significance of effective sequence prediction cannot be overstated. Central to this domain are Recurrent Neural Networks (RNNs) and Long Short-Term Memory (LSTM) networks, which have emerged as powerful tools in the machine learning arsenal for handling sequential data.
RNNs, known for their unique architecture that allows
information to persist, have been instrumental in modeling
time-dependent data [
This paper embarks on a comprehensive exploration of RNNs and LSTMs in the context of sequence prediction. We delve into the architectural intricacies of these models, their strengths and weaknesses, and their performance across various sequence prediction scenarios.
Our study is particularly focused on datasets generated by different Pseudo-Random Number Generators (PRNGs), offering a unique lens through which the capabilities of these models can be examined and understood.
Through rigorous experimentation and analysis, we aim to shed light on the nuances of sequence prediction and provide insights that could guide future applications and research in this fascinating area of machine learning.
Recent advancements in sequence prediction have been significantly influenced by the development and refinement of Recurrent Neural Networks (RNNs) and Long Short-Term Memory (LSTM) networks. These models have shown remarkable proficiency in handling sequential data, particularly in domains where understanding temporal dynamics is crucial.
0000-0002-2835-4279 (D. Proskurin); 0000-0002-3109-7971
(M. Iavich); 0000-0001-9036-6556 (T. Okhrimenko); 0000-0002-1247-9854
(O. Chukwukaelonma); 0000-0003-0123-5241 (T. Hryniuk)
© 2024 Copyright for this paper by its authors. Use permitted under
Creative Commons License Attribution 4.0 International (CC BY 4.0).
1. LSTM for Time Series Prediction: Studies have
demonstrated the effectiveness of LSTM models in time
series forecasting, a domain traditionally dominated by
statistical methods like ARIMA. Unlike these methods,
LSTMs can capture complex nonlinear relationships in time
series data [
2. RNNs in Natural Language Processing (NLP): RNNs
have been pivotal in advancing NLP. Their ability to process
sequential text data has led to breakthroughs in machine
translation, text generation, and sentiment analysis [
3. Sequence-to-Sequence Learning: The
sequence-tosequence learning framework, often implemented using
LSTMs, has revolutionized tasks like machine translation.
This approach involves training models on pairs of output
and output sequences, enabling the model to learn
mappings from one sequence to another. This framework
has been crucial in developing models that can translate
entire sentences with context, rather than translating on a
word-by-word basis [
4. Challenges and Limitations: Despite their successes,
RNNs and LSTMs are not without challenges. The vanishing
gradient problem in RNNs, where the model loses its ability
to learn long-range dependencies, has been partially
addressed by LSTMs but still poses limitations [
5. Future Directions: Ongoing research is exploring
more efficient and effective variants of RNNs and LSTMs,
such as attention mechanisms and Transformer models [
Neural networks are artificial intelligence models that
mimic human brain function [
Convolutional Neural Networks (CNNs) are similar to
standard artificial neural networks (ANNs) in that they use
neurons to improve themselves through learning [
Hybrid Neural Networks (HNNs), which integrate the
strengths of many neural networks, are becoming
increasingly popular in computer vision applications
including picture captioning and action identification.
However, there has been limited research on the effective
use of hybrid architectures for time series data, particularly
for trend forecasting purposes [
Recurrent Neural Networks (RNNs) represent a
paradigm shift in neural networks, specifically designed to
recognize patterns in sequences of data [
The core architecture of an RNN involves a hidden layer
where the activation at a given time step is a function of the
output at the same step and the activation of the hidden
layer at the previous step [
Long Short-Term Memory Networks, a special kind of
RNN, were developed to overcome the limitations of
traditional RNNs. LSTMs are adept at learning long-term
dependencies, thanks to their unique internal structure [
Forget Gate: Determines what information is discarded
from the cell state [
Output Gate: Updates the cell state with new
information from the current output [
Output Gate: Determines the next hidden state and
output based on the current output and the updated cell
state [
This architecture allows LSTMs to make more precise
decisions about what information to store, modify, and
output. As a result, LSTMs have been successfully applied
in various complex sequence modeling tasks, including
machine translation, speech synthesis, and even generative
models for music composition [
While both RNNs and LSTMs are designed for sequence
processing, the key difference lies in their ability to handle
long-term dependencies [
The choice between RNNs and LSTMs often boils down
to the specific requirements of the task at hand, the
complexity of the sequences involved, and the
computational resources available. LSTMs are generally
preferred for more complex tasks with longer sequences [
There are a large number of pseudorandom generators that
differ in their characteristics, construction methods, and
areas of possible application [
Description: The LCG is one of the oldest and simplest
PRNG algorithms [
Characteristics: The sequence generated by an LCG can
exhibit patterns due to its linear nature. These patterns,
while not immediately apparent, can be learned over time,
making it an interesting case for sequence prediction
models [
Description: The Mersenne Twister, specifically the
MT19937 variant, is known for its long period and
highquality outputs. It’s widely used in various applications due
to its reliability and speed [
Characteristics: MT generates sequences that are far
more complex and less predictable than LCG [
Description: Xorshift is a class of PRNGs that operates using
XOR (exclusive or) and bit-shifting operations [
Characteristics: Despite its simplicity, Xorshift can
produce high-quality random sequences [
Description: The Middle Square method is an older PRNG
technique that generates random numbers by squaring the
number and extracting the middle digits of the result [
Characteristics: This method is prone to quickly
converging to repetitive cycles or zeros, especially with
certain seed values [
In our study on “Predicting PRNG Output with Sequential Analysis”, we meticulously prepared a dataset to analyze the predictability of various Pseudorandom Number Generators (PRNGs), focusing on four widely recognized algorithms: Linear Congruential Generator (LCG) (Fig. 2), MiddleSquare (Fig. 4), Xorshift (Fig. 3), and Mersenne Twister (MT) (Fig. 1). Each of these PRNGs was chosen for its unique approach to generating sequences of pseudorandom numbers, providing a diverse test bed for our predictive models.
The dataset was generated using the following parameters to ensure consistency across all PRNGs: Sample Size: Each PRNG was used to generate a sequence of 10,000 numbers, with n = 10000, to create a substantial dataset for training and evaluation. Seed Value: A common seed value of 8956482 was applied to initialize each PRNG, ensuring that the starting point of the pseudorandom sequence was consistent across different generators. Word Size: For PRNGs where applicable, such as MiddleSquare, a word size of 8 bits was selected, balancing the need for computational efficiency with the desire for sequence complexity. Sequence Length: The output was segmented into sequences of length 10, which were then used as individual data points for the subsequent analysis. This sequence length was chosen to provide enough data for recognizing patterns without overwhelming the analytical models.
Once generated, the dataset was divided into three distinct sets to facilitate the training, testing, and validation of our predictive models:
Training Set: Used to train the models, allowing them to learn and adapt to the patterns inherent in the pseudorandom sequences generated by each PRNG.
Testing Set: Employed to assess the performance of the models on unseen data, providing an unbiased evaluation of their predictive capabilities.
Validation Set: Utilized during the model tuning phase to fine-tune parameters and prevent overfitting, ensuring that the models generalize well to new data.
This careful preparation and partitioning of the dataset were critical in establishing a robust foundation for our investigation into the predictability of PRNG outputs through sequential analysis. By standardizing the generation parameters and thoughtfully splitting the data, we aimed to create a fair and consistent testing environment for each of the predictive models applied in our study.
In our exploration of “Predicting PRNG Output with Sequential Analysis”, we employed a comprehensive approach by leveraging various neural network architectures. Each model was selected based on its ability to process sequential data, a core characteristic of PRNG outputs. Our analysis incorporated Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs), Long Short-Term Memory networks (LSTMs), and a custom Hybrid model, each designed to handle the intricacies of PRNG-generated data in distinct ways.
Application: Primarily used for single-value output prediction, CNNs are adept at identifying patterns within a fixed-size window of the sequence. This model excels at capturing local dependencies and spatial hierarchies in data, making it suitable for analyzing individual segments of the PRNG output.
Application: RNNs were employed for both single-value and continuous-value output predictions. Unlike CNNs, RNNs have a memory mechanism that allows them to process entire sequences of data, making them ideal for understanding the temporal dynamics and dependencies within PRNG outputs.
Application: Like RNNs, LSTMs were utilized for both single-value and continuous-value output predictions. LSTMs are a special kind of RNN capable of learning longterm dependencies. They are particularly effective in avoiding the vanishing gradient problem, enabling them to capture patterns over longer sequences of PRNG outputs.
Configuration: The Hybrid model represents an innovative approach, integrating the strengths of CNNs and LSTMs into a singular architecture. It comprises: • • •
the subsequence of the PRNG output.
LSTM Layer: To capture long-term dependencies and temporal patterns in the data, building upon the features extracted by the CNN layer.
Dense Layer: Serving as the output layer, it synthesizes the information processed by the CNN and LSTM layers to make predictions.
Application: Designed for versatility, the Hybrid model is equipped to handle both single-value and continuousvalue outputs, offering a robust solution for predicting PRNG outputs by leveraging the complementary strengths of convolutional and recurrent layers.
The strategic selection and configuration of these models underpin our analytical methodology. By employing a diverse array of architectures, each with its unique advantages, our study aims to comprehensively evaluate the predictability of PRNG outputs. The Hybrid model underscores our commitment to innovation, integrating multiple neural network paradigms to enhance predictive accuracy and insight into the sequential nature of PRNGgenerated data.
To rigorously assess the effectiveness of our models in predicting PRNG outputs, we employed a set of comprehensive evaluation metrics. These metrics are crucial for quantifying the accuracy of our predictions and facilitating a direct comparison between the different neural network architectures utilized in our study. Our evaluation framework is centered around the Mean Squared Error (MSE) and a specially devised Model Performance Score.
MSE serves as the cornerstone of our evaluation strategy. It calculates the average squared difference between the actual and predicted values, offering a precise measure of the prediction error’s magnitude. By squaring the errors, MSE gives more weight to larger errors, making it particularly sensitive to outliers and significant prediction inaccuracies.
In the context of predicting PRNG outputs, MSE provides a clear and direct measure of how closely the model’s predictions align with the actual sequence of numbers generated by the PRNGs. A lower MSE indicates higher prediction accuracy, reflecting a model’s ability to effectively capture and replicate the underlying patterns of the PRNG sequence.
Recognizing the need for a standardized metric that allows for an intuitive understanding of model performance, we introduced the Model Performance Score. This metric normalizes the MSE to a scale ranging from 0 to 1, where 0 represents the poorest performance (highest MSE) and 1 denotes perfect prediction accuracy (zero MSE).
The Model Performance Score is calculated by inversely scaling the MSE against a predetermined maximum error threshold. This approach ensures that the performance score is adjusted for the scale of the data and the expected variation in prediction accuracy, allowing for a fair comparison across different models and datasets.
This normalized score simplifies the interpretation of our results, providing a straightforward metric to gauge model effectiveness. It allows stakeholders to quickly assess the relative performance of each model in predicting PRNG outputs without delving into the complexities of raw MSE values.
Together, these evaluation metrics form the foundation of our analytical approach, enabling a nuanced analysis of model performance. MSE offers a detailed view of the prediction accuracy, while the Model Performance Score provides a high-level, comparative perspective. By incorporating both metrics, our study ensures a balanced and comprehensive evaluation of how well each neural network architecture can predict the seemingly unpredictable: output of pseudorandom number generators.
We conducted an extensive series of experiments to evaluate the predictive capabilities of various neural network configurations. These experiments were meticulously designed to explore the impact of different model parameters on the accuracy of PRNG output predictions. Below, we detail the variables involved in these experiments and highlight some critical observations related to model performance.
To systematically assess the effects of various hyperparameters on model performance, we tested a wide array of combinations, encompassing:
distinct characteristics in handling nonlinearities in the data.
Number of Neurons: The neuron counts tested were 8, 16, and 32. This range allowed us to explore the models’ capacity to learn and generalize from the data, balancing complexity with computational efficiency.
Epochs: All models were trained for [1000] epochs, providing ample opportunity for learning and convergence.
Model Layers: We varied the depth of the models
by testing configurations with [
Output Lengths: For continuous value prediction,
output lengths of [
One of the most notable findings from our experiments was the impact of dropout and L2 regularization techniques on model learning capabilities. Contrary to common practice in machine learning, where these techniques are employed to enhance model generalization and prevent overfitting, our experiments revealed that:
Models without dropout and L2 regularization demonstrated superior performance in learning and predicting PRNG outputs. The introduction of these regularization techniques led to models that were unable to adequately learn from the training data and, consequently, failed to predict accurately.
This observation suggests a unique aspect of predicting PRNG outputs: the data generated by PRNGs, while seemingly random, follows deterministic algorithms. The addition of regularization techniques, which are designed to introduce randomness and constraint to the learning process, may interfere with the model’s ability to capture the underlying deterministic patterns of PRNG sequences.
The results of these experiments provide valuable insights into the design and optimization of neural network models for predicting PRNG outputs. Specifically, they underscore the importance of tailoring model configurations to the specific characteristics of the data and the task at hand. In the context of PRNG prediction, minimizing external sources of randomness and constraint (e.g., through dropout and L2 regularization) appears to be crucial for enabling models to learn and replicate the deterministic patterns that govern PRNG behavior.
Our exhaustive investigation into predicting PRNG output through sequential analysis yielded compelling findings, elucidated through the analysis of the top-performing models for each PRNG. Here, we detail the significant outcomes for both single-output and continuous-output scenarios across different PRNGs: Xorshift, MT (Mersenne Twister), LCG (Linear Congruential Generator), and MiddleSquare.
For single-output predictions, our experiments have the following results.
Xorshift: The RNN model with 32 neurons, 5 layers, and the ReLU activation function emerged as the top performer, achieving a mean score of 0.9898 (Table 1). However, both Hybrid and CNN models came close to the same success rate suggesting that the Xorshift sequence characteristics are not particularly difficult to capture. 50% of all models were able to reach 90% success thresholds (Error! Reference source not found.).
Nevertheless, more improvement can get this number higher. MT: The CNN model with 8 neurons, 3 layers, and ReLU activation function led the pack with a mean score of 0.9832 (Table 2), indicating that CNN’s feature extraction capabilities are effective at decoding the MT’s output patterns.
28% of all models were able to reach 90% success thresholds (Fig. 6). relu relu relu relu relu relu relu relu relu 60% of all models were able to reach 90% success thresholds (Fig. 7). reaching a mean score of 0.9831 (Table 3). This underscores the Hybrid model’s robustness in capturing both local and long-range dependencies in LCG sequences.
Epochs 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 navigating the complex, squared calculations intrinsic to MiddleSquare algorithm. 64% of all models were able to reach 90% success thresholds (Fig. 8). These results underscore the nuanced relationship between PRNG algorithms and neural network architectures, suggesting that no single model architecture is universally superior. Instead, the optimal choice depends on the specific characteristics and mechanisms of the PRNG being predicted. While the models have been fine-tuned to achieve high predictive accuracy, the graphical analysis indicates that there is an inherent limitation to the exactness of these predictions. A further performance plot (Error! Reference source not found.) illustrates the correlation between the predicted and actual values of the PRNG sequence. The near-perfect linear alignment along the 45-degree line suggests that the model’s predictions are highly correlated with the actual PRNG outputs. The tight clustering of the points around this line demonstrates the model’s effectiveness in capturing the underlying pattern of the PRNG sequence. However, the slight deviation of points from the line implies that while the model can predict the general trend and distribution of the PRNG outputs, it cannot replicate the sequence with absolute precision.
The scatter plot showing predictions versus actual values (Fig. 10) for the Hybrid model using tanh activation, 16 neurons, and 3 layers reveals a close correspondence between predicted and actual values. However, the dispersion of points away from the line of perfect agreement (where predicted values equal actual values) suggests that while the model can approximate the PRNG’s output with high fidelity, it cannot achieve complete accuracy. The variance from the line of perfect prediction could be attributed to the deterministic yet complex nature of PRNGs, which inherently limits the predictability even with sophisticated models.
The continuous-output models demonstrated even higher predictive accuracy, with the Hybrid model configured for continuous predictions (Hybrid-C) achieving remarkable success.
For the MiddleSquare PRNG, the Hybrid-C model with tanh activation, 16 neurons, 3 layers, and an output length of 3 achieved a near-perfect mean score of 0.9955. Only 29% of all models were able to break the 90% success milestone (Fig. 11). models were able to break the 90% success threshold (Fig. 12). For the Xorshift PRNG, the Hybrid-C model with relu activation, 16 neurons, 2 layers, and an output length of 2 achieved a near-perfect mean score of 0.987906 (Table 7). Table 4 Xorshift, continuous output results 15% of all models were able to break the 90% success threshold (Fig. 13).
12% of all models were able to break the 90% success threshold (Fig. 14). The examination of continuous-output models reveals a notable enhancement in predictive performance compared to single-output models. This is particularly evident in the context of predicting sequences generated by the MiddleSquare PRNG.
The performance plot illustrating the correlation between predicted and actual values (Fig. 15) for the continuous-output model shows an even tighter linear alignment than the single-output model. This near-perfect correlation, along with a high success score of 0.9955, reflects the model’s exceptional predictive accuracy. The dense clustering of points along the diagonal suggests that the model can reliably predict the MiddleSquare PRNG’s output with high confidence, and such precision is indicative of the model’s ability to capture both the immediate and contextual dependencies within the PRNG’s sequence. The scatter plot for the continuous-output Hybrid model, which integrates CNN and LSTM architectures (Hybrid-C), showcases a substantial concentration of points closely aligned with the line of perfect prediction (Fig. 16). The model, employing tanh activation with 16 neurons across 3 layers, exhibits a remarkable ability to track the actual values throughout the sequence. This tight clustering indicates a substantial reduction in prediction errors and a strong alignment with the true PRNG sequence, suggesting a deeper understanding of the underlying patterns by the model. The continuous-output model’s superior performance, as evidenced by the closer proximity of predicted to actual values and the higher success score, highlights the benefit of utilizing sequential context in PRNG output prediction. The ability to forecast the sequence with a success score reaching 0.9955 marks a significant milestone, suggesting that models incorporating sequence history can more effectively decode the deterministic yet complex structure of PRNG outputs.
This analysis implies that continuous-output models hold great promise for applications where forecasting accuracy over sequences is critical. The insights gleaned from this research can inform the development of more secure PRNGs, capable of withstanding sophisticated sequential analysis. Future work will likely explore the expansion of this approach to more complex and higherdimensional sequences, potentially integrating additional layers of complexity and exploring the impact on model performance.
Our study’s findings highlight the nuanced nature of PRNG output prediction, with different models excelling for specific generators. This variation underscores the importance of model selection tailored to the characteristics of the PRNG being analyzed. For instance, the bestperforming model for the Xorshift generator might leverage its unique XOR and shift operations, whereas the optimal model for the Mersenne Twister (MT) would need to account for its complex bit manipulation and tempering techniques.
Remarkably, the single-output models consistently achieved a 98% success rate across various PRNGs, demonstrating a high level of accuracy in predicting the next output value based solely on a single preceding value.
This success rate is indicative of the models’ ability to decipher the underlying deterministic patterns that govern PRNG outputs.
Even more impressive, the continuous-output model, which utilizes sequences of values to predict subsequent outputs, reached a 99% success rate. This improvement suggests that incorporating more context in the form of continuous output sequences enables the models to better capture the PRNGs’ inherent algorithms, leading to more accurate predictions.
The success of our models in predicting PRNG outputs with such high accuracy has profound implications for the fields of cryptography and random number generation. While PRNGs are designed to produce sequences that are difficult to predict, our results suggest that advanced neural network models can uncover and exploit hidden patterns within these sequences. This finding calls for ongoing efforts to enhance the unpredictability and security of PRNGs, ensuring they remain robust against sophisticated analytical techniques. 10. Conclusions This research delves into the predictability of PRNGs using advanced neural network models. Our study demonstrates that tested architectures possess a remarkable ability to predict the outputs of various PRNGs, with enhanced accuracy observed in continuous output prediction scenarios showcasing a superior performance in capturing long-term dependencies within PRNG sequences, affirming their suitability for complex sequence prediction tasks. Our findings illuminate the nuanced dynamics of PRNG predictability and the potential vulnerabilities inherent within commonly used generators. By leveraging neural networks, we not only uncover the deterministic patterns masked as randomness but also push the boundaries of understanding in cryptographic security and random number generation.
Future research should explore the integration of more complex neural architectures and the application of these findings in real-world scenarios, such as secure communications and cryptographic key generation. The implications of our work suggest a pivotal shift towards more secure and unpredictable PRNG designs, bolstering the defenses against adversarial predictions and enhancing the integrity of cryptographic systems. 11. Future research directions These findings have significantly advanced our understanding of the capabilities and limitations of current PRNG technologies when subjected to advanced neural network-based predictive models. The high success rates achieved by these models, particularly the 99% success rate with continuous-output models, not only demonstrate the feasibility of predicting PRNG outputs but also underscore the intricate patterns that deterministic algorithms generate—patterns that sophisticated models can uncover.
This study opens several avenues for future research, aimed at both improving PRNG designs and developing more advanced predictive models:
Advanced PRNG Algorithms: There is a clear need for the development of new PRNG algorithms that incorporate mechanisms specifically designed to counteract the capabilities of neural network-based predictive models. Future research should focus on exploring algorithmic complexities that can more effectively obscure deterministic patterns.
Neural Network Enhancements: Our research has shown that certain neural network architectures are more adept at predicting PRNG outputs than others. Investigating the development of novel neural network models or hybrid architectures that can more efficiently process and predict complex sequences is an exciting frontier. This includes exploring deeper networks, attention mechanisms, and other advanced features that could further improve prediction accuracy.
Cross-Disciplinary Approaches: Combining insights from cryptography, machine learning, and complexity theory could yield innovative approaches to both PRNG design and predictive modeling. Interdisciplinary research might uncover new principles for creating sequences that are inherently more difficult to predict, as well as models that are more adept at understanding complex patterns.
Real-World Application Scenarios: Applying our findings to real-world scenarios, where PRNGs are used under various constraints and for different purposes, will be essential. This includes testing PRNGs in environments with high-security requirements, such as in blockchain technologies, secure communications, and digital signatures.
Ethical Considerations and Security Implications: As research progresses in predicting PRNG outputs, it is imperative to consider the ethical implications and potential security risks associated with disseminating advanced predictive models. Developing guidelines and best practices for responsible research and application in this area is crucial.
Enhancing PRNG security: The ability of neural networks to predict PRNG outputs with such accuracy highlights an urgent need for the cryptographic community to re-evaluate and enhance the design and implementation of PRNGs. Ensuring that PRNGs can withstand analysis by advanced predictive models is crucial for maintaining the security and integrity of cryptographic systems, which rely heavily on the unpredictability of these generators.
This work was supported by the Shota Rustaveli National Foundation of Georgia (SRNSFG) [NFR-22-14060] as well as the Ministry of Education and Science of Ukraine (grant №0122U002361 “Intelligent system of secure packet data transmission based on reconnaissance UAV”).