Algorithm Design and Implementation for a Mathematical Model of Factoring Integers

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1 IOSR Journal of Matheatics (IOSR-JM e-iss: , -ISS: X. Volue 3, Issue I Ver. VI (Jan. - Feb. 07, PP Algorith Design leentation for a Matheatical Model of Factoring Integers Jianhui I Deartent of Couter Science, Guangdong eusoft Institue, Foshan City, Guangdong Province, PRC, 5800 Abstract: Based on an aroxiate forula of factoring an odd coosite nuber, the article deduces a distribution for factors in big odd coosite nuber designs an algorith to ick u the factors. Matheatical deduction is resented in detail nuerical exerient is ade on soe big nubers. Exerient shows that the algorith is as efficient as the Pullard s Rho algorith for conventional nubers. Keywords: Integer factorization, Algorith design, nuerical exerients MSC 000: A5,0008 I. Introduction Factorization of integers has been an unsolved roble ever since the ancient tie. For the old trial aroach the Ferat aroach to odern aroach of nuber field sieve, huan being have tried colorful efforts to solve the roble, as suarized in article []. evertheless, a better resolution has been aealed fro both atheaticians researchers of inforation security. Conseuently, study of the roble never ceased. Recent years, literatures on the roble can freuently be seen in several occasions. In article [], Jongsoo Park Mathology Sys tried to do factorization using ultilication table; in article [3], W Aldrin, Wanabisi She Aywa, Cleohas Maende Geoffrey Muchiri Muketha raised a atheatical odel, naely a forula, to aroxiate factors of a coosite nuber. In article [4] [5], WAG built sieves of odd coosite nubers obtain aroaches to factorize large nubers via factorization of sall nubers. Articles [6] [7] ut a new aroach to analyze odd nubers by binary tree obtained new criterions for rie nubers factorization of odd nubers. It can see that, article [3] did not give an algorith or a detail rocedure to realize the atheatical odel though it resented a few secial sales to deonstrate the correctness of the forula. Meanwhile, an algorith called seuential searching algorith in article [7] is a little slower than the Pollard s rho algorith. Therefore, it is worth to have a try to an algorith that can ake the idea in article [3] realizable in a relative seed. This article ainly cobines the atheatical odel in article [3] with the seuential algorith in article [7]. An algorith is designed relative tests are ade. Exerients show that, the cobined algorith is as efficient as the Pullard s Rho algorith. II. Preliinaries. Sybols otations This article continues using the sybols notations that are given in article [7] unless secially coented.. eas ea (see in [3]. et be a large coosite integer of decial digit length l 5 difference d d d d d 0 ; then the rie factors are aroxiately ( (. ea (see in [7]. et (, be an odd coosite nuber such that (, (,, where are odd coried nubers that fit 3 ; let sybols (, (,0 (, resectively the leftost the rightost nodes on level + in the left branch of T (, ; let (, (, ( (, indicate resectively the first s s ultile-nodes left to left to (, (, nodes away fro (,0 (, ( odes (, (, ;, (, (, (, ( be (, (, ( be the node that is (, nodes away fro (,, be the id-node that is right to (, (, ; then it holds DOI: / Page

2 Seed Sieve of Odd Coosite ubers with Alications In Factorization of Integers ( There are exact nodes fro (, (, (, ( to (, exact nodes fro (, (, (, ( to (, ; (, (3 the distribution of (, (, (, (, (, ( on level + is as (, (,0, (,, (, (, (, (, figure illustrates. Fig. Distribution of Critical odes (> (, Range of s ultile node Range of s ultile node (, (,0 (, (, ( (, (, (, (, ( (, (, ( (, (, (, ea 3( see in [8] The floor function of a real nuber x, denoted by x that is defined by x x x satisfies ( x y x y x y ( x x x (3 x n x n for integer n. III. Theoretic Conclusions Proofs Corollary. et (, be an odd coosite such that (,, where are odd nubers that fit 3 ; then 6. Proof. By ea, it is easily to know, naely,. Siilarly, it yields (, That is Thus ote that, when > it yields Conseuently Since are both odd nubers, the corollary obviously holds. Corollary. et > (, be a large coosite integer of decial digit length l 5 (, (, difference d 0 ; then resectively are divisors of nodes in intervals [, ] (, (, [, (, ] R (,, where R (, I (, I DOI: / Page

3 Seed Sieve of Odd Coosite ubers with Alications In Factorization of Integers I (, ( 3 R 3 4 I (, ( 4 (, ( 3 R 3 (, ( Proof. By ea, are aroxiately calculated resectively by Therefore it yields d d (, ( d d (, ( (, ( d d (, ( d d 4 4 et D D ; fro 6, then it yields (, ( D (, ( 3 ( (, ( D (, ( 3 ( By ea 3, it yields (, ( D (, ( (, ( D (, ( 3 ow silify the ineualities ( (. ote that 3 (, ( 3 (, ( 3 (, ( 3 (, ( 3 (, ( 3 it knows Siilarly it yields (, ( 3 D (, ( D (, ( 3 4 (, ( D (, ( 3 ow consider D D are nuber of nodes left to the node translating R D D into I, I, I I R. (, (,, the corollary surely holds when (3 (4 (5 (6 (7 DOI: / Page

4 Seed Sieve of Odd Coosite ubers with Alications In Factorization of Integers IV. Algorith Design Exerients According to the corollaries that are roved in revious section, algorith to search search (SS for factorization of odd nubers can be designed exerients can be ade as follows. 4. Algorith Design ======== SS Algorith========== Inut: Odd coosite nuber (0,0 Ste. Calculate searching level: K log (0,0 ; Ste. Calculate: I ( 3, R K K 3 K 4 I (0,0 (, K K K 4 (0,0 ( 3, R K K 3 (0,0 ( ; K (0,0 (0,0 K Ste 3. Calculate reference node: ul K (, (0,0 ; K Ste 4. Calculate: I ul I, I R ul I R, I ul I, I R ul I R ; R R Ste 5. Search in [ I, I ],[ I, I ] the first odd nuber that has coon divisor with (0,0. ===============End of Algorith ============== 4. uerical Exerients To test the new algorith, nuerical exerients are ade on a Dell PC with.99ghz CPU 8G eories. For coarative urose, 0 big nubers are tested by both Pullard s Rho algorith the new algorith designed reviously the results are list in table. Seen fro figure, it knows that, the two algoriths are alost eually efficient. Table Exerient on Soe Big Integers s Factorization Couting tie( in seconds Pollard s Rho SSAlgorith = = = = = = = = = = = = = = = = = = = = Pullard'sRho SSAlgorith Fig. Pullard srho vs. SS Algorith DOI: / Page

5 Seed Sieve of Odd Coosite ubers with Alications In Factorization of Integers V. Conclusion It is necessary for a scientific researcher to ileent ake test of his theoretic idea. Coarisons or coarative study of kinds of different odels can ake it clear for a decision. For this urose, this article cobines the idea that was raised in article [3] the theory that was ut forward in article [7] realizes an aroach to factorize integers. The author hoe it could be a useful exloration a valuable reference in theoretic study technical develoent. Acknowledgent The research work is suorted by Foshan Bureau of Science Technology under rojects 06AG0065, 06AG AG0038. The author sincerely resents thanks to the all. References [] Sonal Sarnaik, Dinesh Gadekar Uesh Gaikwad, An overview to Integer factorization RSA in Crytograhy, ITERATIOA JOURA FOR ADVACE RESEARCH I EGIEERIG AD TECHOOGY,(9,-6(04 [] Jongsoo Park,Mathology Sys, "Prie Sieve Factorization Using Multilication Table", Journal of Matheatics Research,4(3:7-(0 [3] W Aldrin, Wanabisi She Aywa, Cleohas Maende Geoffrey Muchiri Muketha, "Factorization of arge Integers". International Journal of Matheatics Statistics Studies,(:39-44(03 [4] Xingbo WAG. Seed Sieve of Odd Coosite ubers with Alications In Factorization of Integers, OSR Journal of Matheatics (IOSR-JM, (5, Ver. VIII, 0-07(06 [5] Xingbo WAG, Factorization of arge ubers via Factorization of Sall ubers, Global Journal of Pure Alied Matheatics, ( 6, (06 [6] WAG Xingbo, Valuated Binary Tree: A ew Aroach in Study of Integers, International Journal of Scientific Innovative Matheatical Research (IJSIMR, 4(3, 63-67(06( DOI:0.043/ [7] Xingbo WAG, Genetic Traits of Odd ubers With Alications in Factorization of Integers, Global Journal of Pure Alied Matheatics,3(:38-333(07 [8] Wang Xingbo, A ean-value forula for the floor function on integers, Mathrobles Journal, (4,36-43(0 DOI: / Page