Cryptography is crucial for information security, from the Caesar cipher-named after Julius Caesar- to modern cryptographic algorithms like RSA. These systems have evolved to protect data storage, processing, and transmission over networks. Today, privacy and confidentiality in online operations depend on encryption algorithms; for example, RSA uses the HyperText Transfer Protocol Secure (HTTPS) to secure data transmission, whether personal or financial. A 2022 study by Entrust showed that 62% of companies use encryption strategies, reflecting a 19% increase since 2018, and highlighting the importance of continued research on these algorithms [1].

Cryptanalysis studies methods for deciphering encrypted messages without having the key. The factorization of semi-prime numbers is a vital approach to compromising the security of public-key algorithms like RSA. The security of RSA relies on the difficulty of factoring very large composite numbers into their prime factors. For example, if asked to factor the number 15, one can easily identify 3 and 5 as its prime factors. However, for a number as large as RSA-2048, for example https://acortar.link/c48jLQ, which has 617 digits, finding its prime factors is extremely difficult, even for a non-quantum supercomputer. Advances in factorization techniques, such as the Multiple Polynomial Quadratic Sieve (MPQS) and the General Number Field Sieve (GNFS), require longer encryption keys to maintain security.

This study analyzes cryptanalysis in the factorization of semi-prime numbers through a systematic mapping of the literature. Factorization methods for numbers with more than 10 digits were identified, including the use of artificial intelligence. Although parallel processing has improved execution times, the complexity remains high. Therefore, it is important to explore how emerging technologies can assist. The document is organized as follows: methodology, results and discussions, and conclusions.

It is important to clarify that the search and selection of articles began by entering the search string in Google Scholar, which was also the data source that yielded the most results. For other sources, most of the articles found were not selected due to EC2. As a result, as shown in Table 7, the total number of selected articles that met at least one IC and did not meet any EC was 15, except for 2 articles selected through backward snowballing. A total of 309 articles were excluded. The compendium of primary articles resulting from the search is presented in Table 8 along with their references, so that readers can explore them further. The results of the quality assessment are presented in Table 9, highlighting that only 2 articles (A6, A13) achieved the highest rating with 9 points each. It is also noteworthy that most articles scored above 1 point, with 5 articles scoring 5 points.

Table 7

#	Data source	Found	Relevant	Primary Selected
1	Google Scholar	246	42	15
2	ACM Digital Library	2	0	0
3	IEEE Xplore	1	0	0
4	Science Direct	5	0	0
5	Scopus	2	0	0
6	Springer Link	68	3	0
7	Backward snowballing	2	2	2
TOTAL		326	47	17

Table 8

Id	Title	No. of citations	Year	Reference
A1	An Empirical Comparison of the Quadratic Sieve Factoring Algorithm and the Pollard Rho Factoring Algorithm	0	2021	[7]
A2	Parallel structured gaussian elimination for the number field sieve	2	2021	[8]
A3	Fermat factorization using a multi-core system	3	2020	[9]
A4	Factoring integers witd a brain-inspired computer	9	2018	[10]
A5	An improved parallel block Lanczos algoritdm over GF (2) for integer factorization	12	2017	[11]
A6	Use of SIMD-based data parallelism to speed up sieving in integer-factoring algorithms	13	2017	[12]
A7	Efficiency of RSA key factorization by open-source libraries and distributed system architecture	3	2017	[13]
A8	Fast cryptanalysis of RSA encrypted data using a combination of mathematical and brute force attack in distributed computing environment	2	2017	[14]
A9	Impact of Optimization and Parallelism on Factorization Speed of SIQS	2	2016	[15]
A10	Strategy of Relations Collection in Factoring RSA Modulus	0	2016	[16]
A11	A parallel line sieve for the GNFS Algorithm	1	2014	[17]
A12	SIMD-Based Implementations of Sieving in Integer-Factoring Algorithms	1	2013	[18]
A13	An integrated parallel GNFS algorithm for integer factorization based on Linbox Montgomery block Lanczos method over GF (2)	13	2010	[19]
A14	Nios II hardware acceleration of the epsilon quadratic sieve algorithm	3	2010	[20]
A15	Application of bio-inspired algoritdm to tde problem of integer factorization	23	2010	[21]
A16	Parallel Multiple Polynomial Quadratic Sieve on Multi-core Architectures	6	2007	[22]
A17	Integer Factorization by a Parallel GNFS Algorithm for Public Key Cryptosystems	18	2005	[23]

Table 9

Article	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12	Total
A1	-1	0	1	0	-1	1	1	0	1	0	-1	-1	0
A2	1	0	-1	0	1	0	1	0	0	-1	-1	0	0
A3	1	1	-1	1	1	1	1	0	0	0	0	0	5
A4	-1	1	-1	1	1	1	0	0	0	0	1	0	-1
A5	1	0	-1	1	1	1	1	0	0	-1	1	1	5
A6	1	1	-1	1	1	1	1	0	1	1	1	1	9
A7	0	1	-1	1	0	1	1	0	0	-1	-1	0	1
A8	1	1	-1	1	1	1	1	0	1	-1	-1	0	4
A9	1	1	-1	0	1	1	1	0	1	0	-1	0	4
A10	1	1	-1	0	1	1	1	1	1	0	0	-1	5
A11	1	1	-1	1	1	0	1	0	1	-1	0	1	5
A12	0	1	-1	0	1	1	1	0	1	0	0	0	4
A13	1	1	-1	1	1	1	1	0	1	1	1	1	9
A14	0	1	-1	-1	1	0	0	0	1	0	-1	0	0
A15	-1	0	-1	-1	1	0	1	0	1	1	1	1	3
A16	1	1	-1	1	1	1	1	0	1	0	-1	0	5
A17	1	1	-1	1	1	1	1	0	1	0	0	1	7

In this study, which covers 17 research articles dedicated to the factorization of large numbers using various methods and/or techniques to address the computational challenge of performing time-costly operations, it was evident that most approaches focus on the implementation of parallel processing techniques, representing 45% of the studies. This approach particularly leverages the performance of various processors to accelerate processes such as sieving or solving sparse matrices, which are critical steps in cryptanalysis algorithms like QS, MPQS, and NFS. This approach showed significant improvements in algorithm performance compared to their sequential implementations.

The study provides a diverse perspective on the strategies used by researchers in the field of cryptanalysis, such as code optimization, low-level instructions, AI techniques, and distributed processing. By evaluating the performance of various algorithms and factorization tools, these approaches enrich the research field and should continue to be explored as alternatives to the majority of proposed solutions for accelerating factorization processes. Most of these approaches demonstrated performance improvements that help reduce the factorization times of some algorithms.

Table 11

A1	The study faced challenges related to time constraints, the complexity of large numbers, variability in prime factors, and the comparison of factorization algorithm performance. These factors influence the difficulty of efficiently addressing the factorization of large numbers.
A2	The study faced challenges in concurrency and memory management when implementing a parallel algorithm for NFS and overcame these obstacles through solutions such as localized row allocation and memory optimization techniques.
A3	In the context of this study, the researchers encountered obstacles when employing high-performance computing to accelerate the Fermat Factorization (FF) algorithm. These difficulties included the task of breaking the problem into equally sized parts, efficient communication between processors, and the interdependencies present in some solution steps. Additionally, a lack of previous research on parallelizing the FF algorithm, particularly from the perspective of parallel computing, was identified.
A4	This study encountered difficulties such as peak collisions and synchronization issues in a neuromorphic approach. The factor base partitioning strategy also presented challenges. These complexities highlight the need for further research to overcome these difficulties in factoring large numbers.
A5	The study faced challenges in factoring due to the exponential time required to find the prime factors of a large semiprime number. Issues such as matrix and vector multiplications were encountered in the parallel Block Lanczos algorithm. Additionally, communication and synchronization stages in the parallel implementation took significant time. These challenges were addressed by redesigning the algorithm to improve efficiency.
A6	In this study, challenges such as computational complexity, parallelizing the sieving in factorization algorithms using SIMD, and reducing data packing overhead using online and lattice sieving were faced. Intel SSE2 and AVX instructions were used to overcome these difficulties, achieving acceleration of 5-40%.
A7	This study addressed challenges in choosing the appropriate open-source factorization library for large numbers on a computer cluster. They compared the efficiency of libraries such as Msieve, GGNFS, and CADO-NFS, noting that Msieve was not optimal for sieving but showed improvements when combined with GGNFS. In the cluster implementation, unexpected increases in time were observed, possibly due to inefficient distribution. Additionally, they faced difficulties in estimating the time required to factor large RSA numbers
A8	The challenge in factoring large numbers is the computational complexity in finding their prime factors. Given the lack of efficient algorithms for this task, especially for numbers resulting from the multiplication of large primes, the process consumes significant time and resources. In the study, the lack of numbers with known factors limited the generalization of results, and scalability faced obstacles due to the exponential increase in computational requirements with larger numbers. In summary, complexity, lack of data, and scalability marked the difficulties in this study.
A9	Factoring large numbers involves increasing computational complexity with the size of the number, which requires more time. This study identified challenges in implementing and optimizing the SIQS algorithm for this task. Factoring numbers with more than 70 digits became slow due to its complexity. Memory management was improved with a modified storage approach. The sieving phase, the longest part, faced difficulties with polynomial calculations and candidate selection. The search for optimal performance for different numbers was addressed by time and memory analysis and adjustments. In summary, the study tackled challenges related to computational complexity, memory management, and optimization of the SIQS algorithm for factoring large numbers.
A10	The difficulties addressed in this study include the computational complexity of factoring large numbers, the challenge of finding prime factors, the need for efficient algorithms and hardware designs, and the limitations of real-world systems.
A11	In the study of the parallel implementation of the GNFS algorithm for factoring large integers, key obstacles were identified, including the time consumed by the sieving step, inefficiencies in previous implementations due to excessive communication and load imbalance, and the goal of parallelizing all steps of the algorithm. These difficulties were addressed in the study to improve the efficiency and acceleration of the parallel implementation of GNFS in the factoring of large integers.
A12	The researchers faced difficulties in trying to reduce the overhead of packing and unpacking associated with SIMD-based parallelism. While they achieved some acceleration in index calculations, performing it in the subtraction phase proved challenging. It was observed that the frequent data exchange between regular registers and SIMD hindered the benefits of data parallelism. Surprisingly, there were no attempts at SIMD-based parallelization in sieving algorithms found in the literature.
A13	The main challenge in factoring large numbers is the time complexity of algorithms like the General Number Field Sieve (GNFS). Although GNFS can factor numbers with more than 140 digits, it takes several months; the slowest part is the sieving and solving the linear system using the Montgomery Block Lanczos method, which is time-consuming. The challenge is to find more efficient and faster methods for these stages to factor large numbers in a reasonable amount of time.
A14	The central challenge in factoring large numbers is the increasing computational effort as the numbers grow, as calculating polynomials and performing sieving tasks take more time with larger numbers. This study faced similar difficulties, highlighting the increased computational effort, the need to compute various polynomials and perform sieving, and the constraints of hardware and software platforms in terms of performance and portability.
A15	The primary factor in factoring large numbers is the intensification of computational effort as the numbers grow, since polynomial computations and sieving become more prolonged with larger numbers. This study faced similar difficulties, highlighting the increase in computational effort, the need to calculate various polynomials and perform sieving, and the limitations regarding performance and portability of hardware and software platforms.
A16	In this study, challenges related to the unpredictability of data dependencies, irregular data access patterns, and variation in parameters in symbolic computations were identified, thus making it difficult to predict memory and processor usage patterns. Transitioning from shared-memory to distributed computing required redesigning the existing software. Synchronization between threads during the sieving stage introduced delays and communication overhead, and the choice between MPI (Message Passing Interface) with shared memory or multiple threads depended on the length of the number.
A17	The study presented difficulties during the implementation of the GNFS algorithm. Among these, the high memory demand for larger composite numbers was highlighted, thus posing a challenge in terms of memory usage, even for numbers with fewer than 116 digits. Additionally, synchronization issues were noted, as each processor performs sieving on different pairs and the master node waits for the slave nodes to finish, which can affect efficiency. Furthermore, the current difficulty in factoring integers larger than 116 digits was mentioned due to memory requirements and parameter selection that impact the efficiency of the GNFS algorithm.

There are several challenges surrounding research on fast factorization of semi-prime numbers. One of the most significant is the increase in computational effort required as the numbers to be factored grow in bit length. This complexity is further intensified by the volume of operations and results that need to be stored in memory. One of the most relevant challenges in the context of parallel processing is synchronization between processors and memory management, as inadequate control or optimization can impact the factorization process.

Finally, it is worth noting that despite recent advances in computing and mathematics, it is still not possible to factorize numbers as large as a 2048-bit RSA key within a reasonable time frame. The largest number factored to date is RSA-250 in February 2020 [24], an 829-bit number that would have taken about 2700 years to factorize with a single core. However, by utilizing parallel and distributed processing, it was factorized in a few months using multiple machines around the world. Given that this is an exponential problem, researchers in [24] estimate that a 1024-bit number would be 200 times harder to factor. Studies in quantum technology, such as [25] suggest that with a quantum computer, factorization could be achieved in polynomial time, leading to significantly shorter times compared to current algorithms, which have exponential complexity. This poses a challenge for current information security, prompting the development of quantum cryptographic systems [26].