About special properties of the hidden structure of triangular numbers for immediate factorization
Artur Samojluk
University of Warmia and Mazury in OlsztynAbstract
The factorization problem belongs to a group of problems important in the security of information systems and cryptography. The article describes a new number factorization algorithm designed based on numerical experiments. We present an extension of number factorization using triangular numbers features. The described algorithm can be used to increase the security of key generation for the RSA algorithm.
Keywords:
factorization algorithm, RSA, cryptography, numerical experiments, triangular numbers, number theoryReferences
ADLEMAN L.M. 1991. Factoring numbers using singular integers. Proc. 23rd Annual ACM Symp. on Theory of Computing (STOC), New Orleans, p. 64-71. Google Scholar
ADLEMAN L.M. 2005. Algorithmic number theory and its relationship to computational complexity. Proceedings 35th Annual Symposium on Foundations of Computer Science, p. 88-113. https://doi.org/10.1109/SFCS.1994.365702. Google Scholar
ADLEMAN L.M., POMERANCE C., RUMELY R.S. 1983. On distinguishing prime numbers from composite numbers. Annals of Mathematic, 117: 173-206. Google Scholar
BACH E., SHALLIT J. 1996. Algorithmic number theory. Efficient algorithms. MIT Press, Cambrigde. Google Scholar
BONEH D. 1999. Twenty Years of attacks on the RSA Cryptosystem. Notices of the American Mathematical Society, 46(2): 203-213. Google Scholar
BREUR T. 1996. Statistical Power Analysis and the contemporary “crisis” in social sciences. Journal of Marketing Analytics, 4(2–3): 61–65. https://doi.org 10.1057/s41270-016-0001-3. Google Scholar
COHEN H. 1993. A Course in Computational Algebraic Number Theory. Springer-Verlag, Berlin, Heidelberg. Google Scholar
Command and Control in the Fifth Domain. 2012. Command Five Pty Ltd. Google Scholar
DIOPHANTUS. III n.e. Arithmetica. Google Scholar
DIXON J.D. 1981. Asymptotically Fast Factorization of Integer. Mathematics of Computation, 36(153): 255-260. Google Scholar
FAYYAD U., PIATETSKY-SHAPIRO G., SMYTH P. 1996. From Data Mining to Knowledge Dis-covery in Databases. AI Magazine, Norwich. Google Scholar
GAUSS C.F. 1966. Disquisitiones Arithmeticae By Arthur A. Clarke. Yale University Press, New Haven. Google Scholar
HASTIE T., TIBSHIRANI R., FRIEDMAN J. 2009. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New York. Google Scholar
KALISKI B. 2006. Growing Up with Alice and Bob: Three Decades with the RSA Cryptosystem. RSA Security. Google Scholar
LEHMAN R.S. 1974. Factoring Large Integers. Mathematics of Computation, 28(126): 637-646. Google Scholar
MARTÍN-LÓPEZ E., LAING A., LAWSON T. 2012. Experimental realisation of Shor’s quantum factoring algorithm using qubit recycling. Nature Photonics, 6: 773–776. arxiv.org. https://doi.org/10.1038/nphoton.2012.259. arXiv:1111.4147. Google Scholar
MOREE P., PETRYKIEWICZ I., SEDUNOVA A. 2018. A Computational History of Prime Numbers and Riemann Zeros. arXiv:1810.05244. https://doi.org/10.48550/arXiv.1810.05244. Google Scholar
NIVEN I., ZUCKERMAN H., MONTGOMERY H. 1991. An Introduction to The Theory of Numbers. John Wiley and Sons, Hoboken. Google Scholar
POLI R., LANGDON W.B., MCPHEE N.F. 2008. A Field Guide to Genetic Programming. Lulu Enterprises, Morrisville. Google Scholar
POLLARD J.M. 1975. A Monte Carlo method for factorization. Nordisk Tidskrift for Informationsbehandlung (BIT), 15: 331-334. Google Scholar
RIEMANN’S B. 1859. Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse. On the Number of Primes Less Than a Given Magnitude. Monatsberichte der Berliner Akademie, Berlin. Google Scholar
RIVEST R.L., SHAMIR A., ADLEMAN L. 1978. A Method for Obtaining Digital Signatures and Public-key Cryptosystems. Communications of the ACM, 21(2): 120–126. https://doi.org/10.1145/359340.359342. MIT Laboratory for Computer Science, Cambridge, Massachusetts. Google Scholar
RSA CyberCrime Intelligence Service. 2013. RSA Security, Rsa.com. Google Scholar
SAMOJLUK A. 2019. Algorithm of Geometric Factorization. Master thesis. Chair of Mathematics and Computer Science, Uniwersytet Warmińsko-Mazurski, Olsztyn. Google Scholar
SANJEEV A., BOAZ B. 2009. Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge. Google Scholar
SCHATZ D., BASHROUSH R., WALL J. 2017. Towards a More Representative Definition of Cyber Security. Journal of Digital Forensics, Security and Law, 12(2). Google Scholar
SEGARAN T., HAMMERBACHER J. 2009. Beautiful Data: The Stories Behind Elegant Data Solutions. O’Reilly Media, Sebastopol. Google Scholar
SIERPIŃSKI W. 1962. Liczby trójkątne (Triangle numbers). Biblioteczka Matematyczna, t. 12, Państwowe Zakłady Wydawnictw Szkolnych, Warszawa. Google Scholar
SIERPIŃSKI W. 1987. Wstęp do teorii liczb. Third Edition. Biblioteczka Matematyczna, t. 12, Państwowe Zakłady Wydawnictw Szkolnych, Warszawa. Google Scholar
SIERPINSKI W. 1988. Elementary Theory of Numbers. Państwowe Wydawnictwo Naukowe, Warszawa. Google Scholar
STEVENS T. 2018. Global Cybersecurity: New Directions in Theory and Methods. Politics and Governance, 6(2): 1–4. https://doi.org/10.17645/pag.v6i2.1569. Google Scholar
WILLIAMS H.C. 1982. A p+1 Method of Factoring. Mathematics of Computation, 39(159): 225-234. Google Scholar
YOST J. R. 2015. The Origin and Early History of the Computer Security Software Products Industry. IEEE Annals of the History of Computing, 37(2): 46–58. https://doi.org/10.1109/MAHC.2015.21. Google Scholar
University of Warmia and Mazury in Olsztyn
