About special properties of the hidden structure of triangular numbers for immediate factorization

• Autorzy: Samojluk Artur

Identyfikatory

DOI

10.31648/ts.7278

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

factorization algorithm; RSA; cryptography; numerical experiments; triangular numbers; number theory

• Wydawca: Wydawnictwo Uniwersytetu Warmińsko-Mazurskiego w Olsztynie
• Czasopismo: Technical Sciences / University of Warmia and Mazury in Olsztyn
• Rocznik: 2022
• Tom: nr 25(1)
• Strony: 35--57
• Opis fizyczny: Bibliogr. 33 poz., rys.

Twórcy

autor

Samojluk Artur

• Katedra Metod Matematycznych Informatyki, Wydział Matematyki i Informatyki, Słoneczna 54, 10-710 Olsztyn

• https://orcid.org/0000-0001-5822-2210

Bibliografia

• ADLEMAN L.M. 1991. Factoring numbers using singular integers. Proc. 23rd Annual ACM Symp. on Theory of Computing (STOC), New Orleans, p. 64-71.
• 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.
• ADLEMAN L.M., POMERANCE C., RUMELY R.S. 1983. On distinguishing prime numbers from composite numbers. Annals of Mathematic, 117: 173-206.
• BACH E., SHALLIT J. 1996. Algorithmic number theory. Efficient algorithms. MIT Press, Cambrigde.
• BONEH D. 1999. Twenty Years of attacks on the RSA Cryptosystem. Notices of the American Mathematical Society, 46(2): 203-213.
• 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.
• COHEN H. 1993. A Course in Computational Algebraic Number Theory. Springer-Verlag, Berlin, Heidelberg.
• Command and Control in the Fifth Domain. 2012. Command Five Pty Ltd.
• DIOPHANTUS. III n.e. Arithmetica.
• DIXON J.D. 1981. Asymptotically Fast Factorization of Integer. Mathematics of Computation, 36(153): 255-260.
• FAYYAD U., PIATETSKY-SHAPIRO G., SMYTH P. 1996. From Data Mining to Knowledge Dis-covery in Databases. AI Magazine, Norwich.
• GAUSS C.F. 1966. Disquisitiones Arithmeticae By Arthur A. Clarke. Yale University Press, New Haven.
• HASTIE T., TIBSHIRANI R., FRIEDMAN J. 2009. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New York.
• KALISKI B. 2006. Growing Up with Alice and Bob: Three Decades with the RSA Cryptosystem. RSA Security.
• LEHMAN R.S. 1974. Factoring Large Integers. Mathematics of Computation, 28(126): 637-646.
• 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.
• 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.
• NIVEN I., ZUCKERMAN H., MONTGOMERY H. 1991. An Introduction to The Theory of Numbers. John Wiley and Sons, Hoboken.
• POLI R., LANGDON W.B., MCPHEE N.F. 2008. A Field Guide to Genetic Programming. Lulu Enterprises, Morrisville.
• POLLARD J.M. 1975. A Monte Carlo method for factorization. Nordisk Tidskrift for Informationsbehandlung (BIT), 15: 331-334.
• 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.
• 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.
• RSA CyberCrime Intelligence Service. 2013. RSA Security, Rsa.com.
• SAMOJLUK A. 2019. Algorithm of Geometric Factorization. Master thesis. Chair of Mathematics and Computer Science, Uniwersytet Warmińsko-Mazurski, Olsztyn.
• SANJEEV A., BOAZ B. 2009. Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge.
• SCHATZ D., BASHROUSH R., WALL J. 2017. Towards a More Representative Definition of Cyber Security. Journal of Digital Forensics, Security and Law, 12(2).
• SEGARAN T., HAMMERBACHER J. 2009. Beautiful Data: The Stories Behind Elegant Data Solutions. O’Reilly Media, Sebastopol.
• SIERPIŃSKI W. 1962. Liczby trójkątne (Triangle numbers). Biblioteczka Matematyczna, t. 12, Państwowe Zakłady Wydawnictw Szkolnych, Warszawa.
• SIERPIŃSKI W. 1987. Wstęp do teorii liczb. Third Edition. Biblioteczka Matematyczna, t. 12, Państwowe Zakłady Wydawnictw Szkolnych, Warszawa.
• SIERPINSKI W. 1988. Elementary Theory of Numbers. Państwowe Wydawnictwo Naukowe, Warszawa.
• 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.
• WILLIAMS H.C. 1982. A p+1 Method of Factoring. Mathematics of Computation, 39(159): 225-234.
• 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.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)