Some Results on Fermat's Theorem and Trial Division Method

Transcription

1 Some Results o Fermat's Theorem ad Trial Divisio Method Sajda Kareem Radi, Nagham Ali Hameed Some Results o Fermat's Theorem ad Trial Divisio Method Sajda Kareem Radi, Nagham Ali Hameed Mechaical Eg. Det., College of Egieerig / Mechaical Eg. Det., College of Egieerig /Al-Mustasiriya Uiversity, Baghdad, Iraq Al-Mustasiriya Uiversity, Baghdad, Iraq Receivig Date: Accet Date: Abstract A iteger > is called "Prime" if it has o other ositive divisors tha ad itself (withi the set of itegers), other wise is said to be a comosite". Prime umbers are very imortat i today's society. The methods that determie, if a articular iteger is rime or comosite, are called rimarily testig. This aer discusses ad writes the algorithms for rimarily testig. Also, ew theorems has bee obtaied for rimarily tests, ad bee used as a test as i the "trial divisio" method ad "Fermat's theorem". A comuter rogram has bee built, ad bee oeratig usig (" MATLAB 7"). Vol: 7 No:, Jauary 0 3 ISSN:

2 Some Results o Fermat's Theorem ad Trial Divisio Method -Itroductio: Prime umbers are rather old objects i mathematics; however, they did ot loose their fasciatio ad imortace. There have bee may tests of rimality ad algorithms to carry out these tests ad which are created throughout the years. The writte history of distiguishig rime umbers from comosites goes back to Sieve of Eratosthees who came u with the first recorded algorithm for rimality testig i the 3rd cetury BC. []. I the 7th cetury, mathematicias (Fermat, Legedre, Gauss, etc) cosidered rimality testig ad factorig to be some of the most imortat roblems i arithmetic. Their work laid the foudatio of a ew age i rimality testig, which bega i the 970. Miller i 976, Solovay- Strasse i 977,ad Rabid i 980 develoed efficiet algorithms for rimality testig ad factorig []. The first rimality tests were ot ru o the comuters. They were comuted by had. Today very large rime umbers are required ad are early imossible to write dow. The largest kow rime umber today is So it is obviously imractical to work with these umbers without usig comuters. Researchers ad mathematicias have bee strivig to fid a ucoditioally determiistic olyomial time algorithm to test for rimality. The first rogress was oly 48 made i 00. The larger rime ever foud without a comuter was 7, which has 44 digits (foud by Ferrier i 95). He used a desk calculator ad a variatio of Fermats Little Theorem. I the same year, usig a electroic comutig device, a rime with 79 digits was foud by Miller ad Wheeler. Today, the largest rime is over millio digits log [3]. I this aer, may of kow rimality testig methods have bee discussed usig roosed tests. These methods have bee imlemeted usig "Matlab 7" rogrammig laguage to obtai the result which could be evaluated. follows: - Tests for Primality: There are may differet rimality tests ad methods which ca be classified as.- Tests for Numbers of Secial Forms: These tests deal with umbers of secial form, such as: Vol: 7 No:, Jauary 0 4 ISSN:

3 Some Results o Fermat's Theorem ad Trial Divisio Method..- Mersee umbers iveted by the mok Mersee i (644) are defied by []. M Examle: If, M 047 Fermat umbers are defied by F + These tests are fast, simle, ad rovide rigorous roofs that their results are correct.these tests are as follows: Theorem : (Pei's Test): For, F is rime if ad oly if 3 ( F )/ (modf Proof: See [4] Theorem : (Lucas Lehmer's Test) : Let > be a rime, ad let ) M be the Mersee umber, the ad oly if M is rime if ad oly if M S divides (equivaletly, if S 0(mod) ) where the umbers (S) are give by the followig recurrece relatio S 4, Proof : See [5]. S S +,..- Determiistic Tests: These methods ca test ay umber of ay forms, but these methods are more theoretical tha ractical. Comared with robabilistic ( i sectio 3) rimality test methods, the outut results of determiistic are absolutely correct. I other words, whe a ositive odd umber is tested, the outut result has oly two ossible situatios either this umber is a rime or comosite [5]. These methods are as follows:..-lucas's Test: Let be a ositive iteger. If there exists a iteger a (mod ) ad for every rime factor q of < a < such that Vol: 7 No:, Jauary 0 5 ISSN:

4 Some Results o Fermat's Theorem ad Trial Divisio Method a q (mod) the is rime. If o such umber a exist, the is comosite [6]...- Theorem 3 (Wilso's Theorem ): This test roosed by Joh Wilso ad ublished by Edward Warig i 770. For ay ositive iteger, is rime if ad oly if ( )! (mod) Wilso's theorem is ot oly ecessary but also sufficiet for the rimality test. Proof: See [7]...3- Proth's Test: For k * + with k odd ad > k, if there exists a iteger α such that α ( ) (mod ) the is rime [5] Pockligto's Test: The Pockligto's Lemer rimality test devised by Hery Labour Pockligto ad Derrick Hery Lehmer to decide a give umber N is rime which is formulated as follows: - q is rime - 3- q / N a N (modn) gcd ( a ( N )/ q ad, N) The N is rime [8]. q> N.3- Probabilistic Tests:.3.- Miller- Rabi's Test: Give a ositive odd iteger ad let r s+, where s is odd umber. The follow the testig umbers: Vol: 7 No:, Jauary 0 6 ISSN:

5 Some Results o Fermat's Theorem ad Trial Divisio Method Choose a radom ositive iteger a with a. If a s (mod) or j a S (mod) for some 0 j r, the asses the test [5]..3.- Solovay - Strasse's Test : Let be a ositive iteger >, choose at radom, k > umbers a, < a<, such that gcd (a, ), ad comute a ( ) a (mod). () If () fails to hold for some a, the is comosite [9] Lehma's Test: Let be a iteger, choose a radom umber a less tha, the calculate ( ) a ( mod ),if a ( ) or (mod), is defiitely ot rime [0]. 3- Fermat's Little Primality Test: Let be a rime umber ad let a be ay iteger which is ot a multile of the a (mod) ( a ) I other word is multile of for every iteger a []. Examle: - P 5, a 3. The , which is a multile of (mod5) - P 6, a 5. The 55 35, ad 35 34, which Not a multile of 6. This examle shows that the theorem ca fail if is ot a rime umber. Results: The followig theorems are the roosed tests i this study. These theorems ca be used for testig as Fermat's theorem. Vol: 7 No:, Jauary 0 7 ISSN:

6 Some Results o Fermat's Theorem ad Trial Divisio Method Theorem 4: If is a odd rime, the + (mod). Theorem 5 : If is a odd rime, the I geeral. 3 + ( ) or ( ) ( mod ). Theorem 6 : If is a odd rime, the k ( k ) ± ( ) For ay rime k. Examle: Let k 5, the mod 5 ± ( ) mod For For 9 : mod 9. : 8 56 ± 00 mod. Therefore, 9 is a robable rime ad is a comosite. 4- Trial Divisio Method: Trial divisio as a rimality test is based uo the followig theorems: Theorem 7 : A odd iteger is rime if ad oly if is ot divisible by ay rime less tha or equal to. Proof : See [3] This method has the advatage of ot oly rovidig a roof of rimality of a rime, but of discoverig a o trivial factorizatio for comosite. The mai drawback of the trial divisio algorithm is that it takes too log if has o small rime factors []. Results: Vol: 7 No:, Jauary 0 8 ISSN:

7 Some Results o Fermat's Theorem ad Trial Divisio Method Before itroducig the formula for rimality testig which is obtaied i this study, it is rather to state the followig theorem : Theorem 8 : ( de Poligac's formula ) : If is a ositive iteger ad a rime, the the exoet of the biggest ower of that divides! is : Proof : S ee [3] work. e k k The followig two theorems are the tools for the rimality testig which is got i this Theorem 9: If is a iteger, the k k k k, If ad oly if is a rime. The followig theorem reduces the calculatio of dividig by the distict owers of to a sigle umber, which is. Theorem 0: If is a iteger, the If ad oly if is rime. Proof: If is a rime, the followig must be roved It is kow that is rime if ad oly if o rime Vol: 7 No:, Jauary 0 9 ISSN: q + r ad q+ ( r ), sice - < Therefore, q r + q r + q ( r ) ad + ad q ( r ) + By the defiitio of the greatest iteger, the followig is obtaied divides,,where 0 < r < ad 0 < r

8 Some Results o Fermat's Theorem ad Trial Divisio Method 30 Vol: 7 No:, Jauary 0 ISSN: q ad q O the other had, if () the 0 It is clear that : 0 k k k k if if (3) From () ad (3) it is cocluded that,, therefore must be a rime. Now if a iteger is to be tested usig theorem 5555 all the rimes that are less tha or equal to square root of must be listed, the ad are comuted if they are equal cotiue i this maer, ad comute ad util we reach k which is less tha or equal to, if for all these they are equal the must be rime otherwise it is a comosite. Examle: Let , is a rime umber. 5-imlemetatio

9 Some Results o Fermat's Theorem ad Trial Divisio Method This sectio itroduces the system imlemetatio of five rimality testig algorithms, which are as follows. - Algorithm for the Miller Rabi Primality Test - Iut > a odd iteger to be tested for rimality, t > - let - S. r such that r is odd 3 - Reeat from to t 4- Choose a radom iteger a, a 5- Comute x ar mod 6- If, ad x the j 7- While j s - ad x do the followig 8- comute x x mod 9- If x, the rit ( comosite ) : j j + 0- If x the rit ( comosite ) - Prit ( Prime ) Ed Ru : 9, a,s, r 9 9 rime - Algorithm for the Proth's Test - Iut ositive iteger - Iut odd iteger k < 3- Iut ositive iteger a 4 - Let k If a (mod) for some a the outut " is rime " 6- Outut P is ot rime. Ru : is 3 rime?, k 3, a rime 3 Algorithm for the Lucas Test - Iut odd iteger > to be tested for rimality,k Vol: 7 No:, Jauary 0 3 ISSN:

10 Some Results o Fermat's Theorem ad Trial Divisio Method - Factor - to its rime 3- L oo: reeat k times 4- Choose a radom iteger i the rage [, ] 5- If a ( mod ) the rit comosite otherwise 6 Loo : for all rime factors q of -: 7- If a(-) /q ( mod ) 8- If we did ot check this equality for all rime factors of the do ext loo 9- Otherwise rit rime otherwise do ext loorit comosite. Ru : is 5 rime? a 3, q 5 rime 4 - Algorithm for the Lucas- Lehmer Test - Iut : a Mersee umber - calculate P- 3- Use trial divisio to check if s has ay factors betwee ad sqrt of. If it does, the rit ( comosite)- 4- Let u For k to do the followig 6- Comute u (u ) mod 7- If u 0 the rit ( rime ) 8- rit " comosite " Ru : To test 7 3, u (( 4* 4 ) - ) mod rime 5- Algoithm for the Solovay- Strasse Test -Iut a odd iteger ad t - For I to t do the followig: 3-Choose a radom iteger a, a 4- Comute r a(-)/ (mod ) 5- If r, ad r -, the rit ( " comosite") 6- Comute the Jacobi symbol s ( a/ ) Vol: 7 No:, Jauary 0 3 ISSN:

11 Some Results o Fermat's Theorem ad Trial Divisio Method 7- If r s (mod ) the rit (" comosite" ) 8- Prit (" rime ") Ru : is 7 rime? a 4, r, s r s, 7 rime 6-Refereces: [] R.Slezeviciee,J.Steudig,S.Turskiee,"Recet Breakthrough i Primality Testig", Noliear Aalysis : Modellig ad Cotrol", vol.9, No., ( 7-84), 004. [] Shafi Goldwasser ad Joe Kilia.,"Primality Testig usig Ellitic Curves",Joural of ACM,vol.46,No.4,999. [3] Chelsea Arrigto, "Primality Testig ", 00. htt://rimes.utm.edu/otes/by_year.html [4] David M. Bressoud," Factorizatio ad Primality Testig ", Sriger- Verlag, New York,989. [5] Chia-Logw, Der- Chyua Lou, Te-Je Chag, "Comutatioal Reductio of Wilso's Primality Test for Moder Crytosystems ",Iformatica,vol.33,( ),009. [6] Sog Y.Ya, "Number Theory for Comutig", Sriger-Verlag, Berli, 000. [7] David M.Burto, " Elemetary Number Theory". Secod Editio, Web Publishers, 989. [8] Koblrtz, Neal, "A course i Number Theory ad Crytograhy", d Ed., Sriger, 994. [9] Al-Shamari A. S.S.," Aalytical Study of Some Public-key Crytosystems Deedig o Some Evaluatio Paramaters",M.Sc.Thesis,Uiv. of Tech.,Baghdad,995. [0] Hamdoo A.," Evaluatio of Primality Testig Algorithms ", i roceedig 5th Aual Al- Raffidiai Uiversity College, 999. [] Darre Glass, "Primality Testig ",004. Vol: 7 No:, Jauary 0 33 ISSN:

12 Some Results o Fermat's Theorem ad Trial Divisio Method [] Carl Pomerace, "Recet Develomet i Primality Testig", The Mathematical Itelligecer. Vol.3, No.3, PP.97-05, 98. [3] I.Nive., " A Itroductio to the Theory of Number ", Third editio, Joh Wiley ad Sos,New York, 97. Vol: 7 No:, Jauary 0 34 ISSN: