Primitive polynomials selection method for pseudo-random number generator

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1 Joural of hysics: Coferece Series AER OEN ACCESS rimitive polyomials selectio method for pseudo-radom umber geerator To cite this article: I V Aiki ad Kh Alajjar 08 J. hys.: Cof. Ser View the article olie for updates ad ehacemets. This cotet was dowloaded from I address o 03/07/08 at 7:39

2 IO ublishig IO Cof. Series: Joural of hysics: Cof. Series (08) 0003 doi :0.088/ /944//0003 rimitive polyomials selectio method for pseudo-radom umber geerator I V Aiki, Kh Alajjar Kaza Natioal Research Techical Uiversity amed after A.N.Tupolev-KAI, Kaza, 40, Russia. aikiigor777@mail.ru, khkaza@mail.ru Abstract. I this paper we suggested the method for primitive polyomials selectio of special type. This kid of polyomials ca be efficietly used as a characteristic polyomials for liear feedback shift registers i pseudo-radom umber geerators. The proposed method cosists of two basic steps: fidig miimum-cost irreducible polyomials of the desired degree ad applyig primitivity tests to get the primitive oes. Fially two primitive polyomials, which was foud by the proposed method, used i pseudoradom umber geerator based o fuzzy logic (FRNG) which had bee suggested before by the authors. The sequeces geerated by ew versio of FRNG have low correlatio magitude, high liear complexity, less power cosumptio, is more balaced ad have better statistical properties.. Itroductio Nowadays high quality pseudoradom umbers have a critical importace to may scietific applicatios. seudoradom umber geerators based o liear feedback shift registers (LFSRs) [] are amog the fastest log-period geerators curretly available. They require less hardware for implemetatio, have more higher speed of operatios ad produce pseudoradom sequeces with good statistical properties. I papers [,3] we suggested ew type of pseudoradom umber geerator based o fuzzy logic - FRNG. The structure of this geerator is preseted o Figure. Figure. Geeral structure of proposed FRNG. Cotet from this work may be used uder the terms of the Creative Commos Attributio 3.0 licece. Ay further distributio of this work must maitai attributio to the author(s) ad the title of the work, joural citatio ad DOI. ublished uder licece by IO ublishig Ltd

3 IO ublishig IO Cof. Series: Joural of hysics: Cof. Series (08) 0003 doi :0.088/ /944//0003 This geerator ivolves several LSFRs, buffers ad fuzzy o-liear fuctio with two fuzzy variables f 0 (umber of oes) ad f - f (the differece betwee umber of two cosecutive oes (00) ad umber of two cosecutive zeros (00)), ad a group of fuzzy if-the rules. I [3] we evaluated the quality of suggested geerator ad showed that it passed basic statistical tests from NIST ad Diehard budles. Moreover, this geerator showed better quality tha some other well-kow geerators. Nevertheless, it is extremely importat to ivestigate crucial parameters of the suggested geerator ad choose the best oes to improve the quality ad the efficiecy of geerated pseudo-radom umbers sequeces. Oe of the basic parameters are characteristic polyomials of LFSRs which are used i FRNG. It's well kow that a biary sequece geerated by LFSR possesses good statistical properties if its characteristic polyomial is primitive. For example, the period T of such sequece (is called M- sequece) is -, where is the degree of the polyomial. M-sequeces are used for obtaiig uiformly distributed radom umbers [4] ad they widely used i practice, for example i modelig or cryptography. The quality of M-sequeces grows with icreasig the degree of the characteristic polyomial of LFSR, so it is very importat i practice to obtai primitive polyomials with big degrees. There are several published tables with primitive biary polyomials. Watso i [5] gives oe primitive of degree for every <00. Stahke i [6] gives a primitive with a miimum umber of ozero coefficiets (triomial or petaomial) for every 68. Oe primitive petaomial for every degree M j, 8 j 7, is also preseted i [7] (here M j deotes the jth Mersee expoet, the prime for which ( Mj -) is also prime). 5,9,7-weight primitive polyomials of degree up to 800 over GF() are preseted i [8]. Aside from uiform distributio, good F -liear geerators are also required to have characteristic polyomials (x) whose umber of ozero coefficiets is ot too far from the half of degree, i.e., approximately / [9]. I particular, geerators for which (x) is a triomial or a petaomial, which have bee widely used i the past, should be avoided. It is kow, that these geerators have differet statistical weakesses [0]. I this paper we propose a method of fidig good primitive biary polyomials. These polyomials have good statistical properties, high diffusio capacity (due to relatively high umber of o-zero coefficiets) ad low power cosumptio. The proposed method icludes two mai stages: Fidig miimum-cost irreducible polyomials of the desired degree. Applyig primitivity tests to fid primitive polyomials. This paper is orgaized as follows: i the sectio we describe the step related with fidig miimum-cost irreducible polyomials of the desired degree, i the sectio 3 we describe the step related with applyig primitivity tests to get oly primitive polyomials, i the sectio 4 we describe some experimetal results, i Coclusio we give the basic results of the research.. Fidig miimum-cost irreducible polyomials of the desired degree It was proved i [8], that characteristic polyomial f(x) of special type ca be used to costruct a - stage stadard LFSR with miimum-cost (cocerig the umber of -iput XOR gates eeded to implemet the LFSR). These polyomials ca be described by the geeral expressio (): b b b )( )...( ) x () m f ( x ) ( arameters of these polyomials should satisfy the followig coditios (): b b b,, b b... b b, b b... b b,b b, 3 m m m () We will use this type of polyomials to fit the requiremets of high efficiecy regardig power cosumptio which directly depeds o the umber of -iput XOR gates eeded to implemet it i hardware (I our case it's equal to m+). These types of polyomials have also a high diffusio capacity which depeds o the umber of o-zero coefficiets, which should be ot too far from the half of degree /. This umber is defied as t m. For example if the the most suitable

4 IO ublishig IO Cof. Series: Joural of hysics: Cof. Series (08) 0003 doi :0.088/ /944//0003 choice of m is where we ca get a polyomial with t 5 o-zero coefficiets. We ca take, b 5 to get the followig irreducible polyomial as example: b f ( x ) ( 5 )( ) We ca select all the irreducible polyomials of degree ad satisfy the coditios defied by () i this case ( b, b b, b b ). Table cotais all possible variats that fit these requiremets: Table. List of all 5-wights (t=5) irreducible polyomials of the degree =. b b f(x) 5 5 ( x )( x ) 6 6 ( )( ) 7 7 ( )( ) 8 8 ( )( ) 3 3 ( )( ) 6 6 ( )( ) 7 7 ( )( ) ( )( ) ( )( ) ( )( ) I the previous example (=, m=) the complete list of irreducible polyomials is ot too log, which is ot the geeral case. The umber of polyomials will grow very fast with icreasig the parameters ad m. For example if m=5 ad the desired degree is 67 we will get a list of 47 irreducible polyomials with t=33 o-zero coefficiets. I most practical applicatios we eed a polyomials with high quality. Therefore there is a eed to fid primitive polyomials with the degree as high as possible. Table cotais some values of m, the correspodig umber of o-zero coefficiets t, ad the rage of degrees which should be used with them. Table. Values of m with correspodig umber of o-zero coefficiets t ad the suitable rages of degrees of polyomials. m t We used Mathematica software to write a program which fid all irreducible polyomials of type () with m + o-zero coefficiets for desired degree. This program eumerates all the possible values of the parameters (b, b,...,b m ) which satisfy all the previous metioed coditios (). The result is a list of umbers (b, b,...,b m ) edig with the degree of the polyomials. For example if m=5, =67, oe of the resultig elemet will be (, 3, 7, 6, 33, 67) that meas: 3

5 IO ublishig IO Cof. Series: Joural of hysics: Cof. Series (08) 0003 doi :0.088/ /944//0003 f ( x ) ( )( x )( x )( )( ) We ca expad the expressio ad get the irreducible polyomial with t=33 terms: f x x Fially, at the ed of this step we obtai the full list of irreducible polyomials which satisfy the coditios (). The we eed to select oly the primitive oes from them, ad that will be doe i the ext step. 3. Applyig primitivity tests ad fidig primitive polyomials from the list After geeratig the lists of irreducible polyomials over GF() we should test them to get oly the primitive oes []. The primitivity test of a give polyomial f is effectively performed with usig the followig set of coditios []: f 0 f mi k : f x k for all primes p, x / p mod p. k The first coditio elimiates polyomials which is divisible by x ad x+. I our case the polyomial f has the type defied by () ad this coditio is fulfilled automatically. To check if the polyomial f satisfies the secod coditio, it is ecessary to calculate residues x k mod f. The total umber of elemetary operatios over GF() which should be doe to test the coditio () is bouded by O(t ), which is big, takig ito accout the umber of polyomials that is eeded to be checked. The problem is solved i [4] where coditio () is modified by checkig two subcoditios: gcd f, f gcd f,x k, k. Here gcd f, g is the greatest commo divisor of the polyomials f ad g [3]. The computig complexity of factorizatio of, which is eeded i the third coditio, is very high. This makes it iefficiet ad ureasoable to iclude this checkig i the primitivity test ad it makes the testig process extremely slow. Fortuately we ca avoid it accordig to the famous Cuigham project [4]. We build a much faster method by storig the factors i database depedig o [4]. Furthermore, if we choose the degree to be a prime, the the umber of factors of will be very small, so the speed of the primitivity test will be very high. For example, whe =63 the factors of 63 is {{7,},{73,},{7,},{337,}, {9737,},{649657,}} we have to check six cases, but with =67 the factors of 67 will be {937077,},{ ,}. So we have to check oly two polyomials. The complexity of this primitivity check is O( F t ), where F is the umber of distict prime factors of. I our algorithm we listed the prime factors of ( ) for all primes betwee 35 ad 06 i a table accordig to [5]. The we created the program i Mathematica software [6] to geerate the lists of high efficiet irreducible polyomials of the desired degree with the suitable m (see Table II). Fially we wrote the algorithm ad program for primitivity test to pick up oly primitive polyomials from the created list of irreducible polyomials. 4. Experimetal results

6 IO ublishig IO Cof. Series: Joural of hysics: Cof. Series (08) 0003 doi :0.088/ /944//0003 Applyig the suggested method we foud all primitive polyomials with prime degree betwee 35 ad 06. Some examples are represeted i the Table 3. Table 3. Some primitive polyomials which had bee gotte by the suggested method. t f(x) 37 7 (3, 4, 9, 7, 37) 6 7 (, 3, 5, 39, 6) (,, 4, 9, 4, 67) (, 5, 0, 7, 39, 89) (3, 5, 9, 9, 37, 75, 49) 5 65 (,, 4, 4, 3, 4, 5) 33 9 (, 3, 6,, 4, 65, 69, 33) (3, 6, 3, 7, 54, 9, 66, 509) (, 5,, 8, 58, 8, 43, 5, 983) I order to icrease the efficiecy of the suggested i [,3] FRNG, we replaced two primitive characteristic polyomials (3) curretly used i FRNG by two primitive oes (4) selected from the obtaied lists with usig the proposed method x x 9 97 x x 86 x ( )( )( )( )( ) x ( )( )( x )( x )( x ) Fially we evaluated the quality of our FRNG (with two selected primitive polyomials (4)) with usig NIST [7] ad Diehard [8] budles. The ew FRNG successfully passed all radomess tests. Furthermore the ew versio of FRNG became more secure agaist algebraic attacks due to usig special type of polyomials which have high diffusio capacity ad high liear complexity, which icrease the immuity of the proposed geerator agaist correlatio attacks 5. Coclusio Fially, we ca coclude that the proposed method of selectio primitive polyomials is very useful for the field of pseudo-radom umbers geerators. Selected primitive polyomials icrease the efficiecy of pseudoradom umbers geerators. As a result the sequeces geerated by ew versio of FRNG will have low correlatio magitude, high liear complexity, less power cosumptio, will be more balaced, they will have good statistical properties. It makes the resultig FRNG more suitable for various applicatios such as modelig, telecommuicatio, cryptography, autheticatio, etc. 6. Refereces [] Lewis T G ad aye W H 973 Geeralized Feedback Shift Register seudoradom Number Algoritm Joural of the ACM (JACM) [] Aiki I V ad Alajjar K 05 Fuzzy stream cipher system roc. It. Siberia Cof. o Cotrol ad Commuicatios (Omsk). [3] Aiki I V ad Alajjar K 06 pseudo-radom umber geerator based o fuzzy logic roc. It. Siberia Cof. o Cotrol ad Commuicatios (Moscow). 89 (3) (4) 5

7 IO ublishig IO Cof. Series: Joural of hysics: Cof. Series (08) 0003 doi :0.088/ /944//0003 [4] Tausworthe R C 965 Radom umbers geerated by liear recurrece modulo two Math. Comp [5] Watso J 96 rimitive polyomials (mod ) Math. Comp [6] Stahke W 973 rimitive biary polyomials Math. Comp [7] Kurita Y ad Matsumoto M 99 rimitive t-omials (t = 3; 5) over GF() whose degree is a Mersee expoet Math. Comp [8] Wag Lag-Terg, Touba Nur A, Bret Richard, ad Wag Hui 0 O Desigig Trasformed Liear Feedback Shift Registers with Miimum Hardware Cost UT-CERC [9] Wag D ad Compager A 993 O the use of reducible polyomials as radom umber geerators Mathematics of Computatio [0] Lidholm J H 968 A aalysis of the pseudo-radomess properties of subsequeces of log m-sequece IEEE Trasactios o Iformatio Theory IT-4 (4) [] Watso E J rimitive polyomials (mod ) Mathematics of Computatio 6(79) [] Lidl R ad Niederreiter H 983 Fiite Fields, Ecyclopedia Math. Appl 0. [3] Cohe E Arithmetical fuctios of a greatest commo divisor roceedigs of the America Mathematical Society () [4] Heriga J R, Blote H W ad Compager 99 A New rimitive triomials of Merseeexpoet degrees for radom umber geeratio Iteratioal Joural of Moder hysics [5] Brillhart J, Lehmer D H, Selfridge J L, Tuckerma B, ad Wagstaf S S 988 Factorizatio of b ±, b =, 3, 5, 6, 7, 0,, up to high powers Amer.Math. Soc. [6] [7] Rukhi A, Soto J, Nechvatal J, Smid M, Barker E, Leigh S, Leveso M, Vagel M, Baks D, Heckert A, Dray J ad Vo S 00 A statistical test suite for radom ad pseudoradom umber geerators for cryptographic applicatios NIST Special ublicatio 800-, Revisio a. 6