Deterministic Random Number Generator Algorithm for Cryptosystem Keys

Transcription

1 Iteratoal Joural of Computer ad Iformato Egeerg Determstc Radom Number Geerator Algorthm for Cryptosystem Keys Ad A. Maata, Hamza A. A. Al_Sewad Abstract Oe of the crucal parameters of dgtal cryptographc systems s the selecto of the keys used ad ther dstrbuto. The radomess of the keys has a strog mpact o the system s securty stregth beg dffcult to be predcted, guessed, reproduced, or dscovered by a cryptaalyst. Therefore, adequate key radomess geerato s stll sought for the beeft of stroger cryptosystems. Ths paper suggests a algorthm desged to geerate ad test pseudo radom umber sequeces teded for cryptographc applcatos. Ths algorthm s based o mathematcally mapulatg a publcally agreed upo formato betwee seder ad recever over a publc chael. Ths formato s used as a seed for performg some mathematcal fuctos order to geerate a sequece of pseudoradom umbers that wll be used for ecrypto/decrypto purposes. Ths mapulato volves permutatos ad substtutos that fulfll Shao s prcple of cofuso ad dffuso. ASCII code characters were utlzed the geerato process stead of usg bt strgs tally, whch adds more flexblty testg dfferet seed values. Fally, the obtaed results would dcate soud dffculty of guessg keys by attackers. Keywords Cryptosystems, Iformato Securty agreemet, Key dstrbuto, Radom umbers. I. INTRODUCTION NDEPENDENT, upredctable ad uformly dstrbuted I umbers that caot be relably reproduced are referred to as radom umbers [1]. They play a major part practcal mplemetato ad stregth of most cryptographc systems. They may be used as keys for symmetrc crypto-systems, publc key parameters, sesso keys, etc. []. Falure of obtag strog keys deftely wll ed up wth data securty compromse. Therefore, strog radom umber geerators that exhbt hgh statstcal qualty ad ca wthstad cryptaalyss efforts are keely sought. Such strog radom umber geerators costtute a mportat buldg block the desg ad testg of hgh qualty crypto-systems [3]. Geerally radom umbers ca be truly radom TRN, pseudo-radom PRN or quas-radom QRN. Truly radom umbers are upredctable. Ther geerato stems from radom physcal or atural pheomea, such as radoactve decay, amplfed ose geerated by a resstor or a semcoductor dode, fed to a comparator or Schmtt trgger ad the the output s sampled to get a seres of bts whch are statstcally depedet or radom [4]. Ad A. Maata s wth the Faculty of Iformato Techology, Isra Uversty, Amma, Jorda (phoe: ; e-mal: ad.maata@u.edu.jo, Hamza A. A. Al_Sewad s wth the Faculty of Iformato Techology, Isra Uversty, Amma, Jorda (phoe: , e-mal: alsewad@hotmal.com). Pseudo-radom umbers geerators, also kow as determstc radom bt geerators, are computer programs that geerate a sequece of umbers whose propertes approxmate the propertes of sequeces of radom umbers [5], [6]. These sequeces are ot truly radom as they are completely determed by a relatvely small set of tal values called seeds; however, they are mportat practce for ther speed ad reproducblty umber geerato. Quas-radom umbers are sequeces arbtrary dmesos whch progressvely cover a d-dmesoal space wth a set of pots that are uformly dstrbuted. They are also kow as low-dscrepacy sequeces [7]. The quasradom sequece geerators use a terface that s smlar to the terface for radom umber geerators, except that seedg s ot requred as each geerator produces a sgle sequece. Recetly, a ew type of geerators whch are called Lagged Fboacc pseudo-radom umber geerators [8], [9] have become creasgly popular geerators for seral as well as scalable parallel maches. They are proved to be easy to mplemet, cheap to compute ad they are performg reasoably well o stadard statstcal tests especally whe the lag s suffcetly hgh. After the bref deftos Secto I, related works wll be summarzed Secto II. Secto III defes the mportat radomess tests that wll be executed to exame the radomess of the geerated keys. Secto IV explas the proposed Pseudo-radom geerator scheme; Secto V cludes the mplemetato of the proposed scheme ad the results of the radomess tests. Fally Secto VI cocludes the paper. II. RELATED WORK Radom umber geerators may be classfed to Iteger geerators, sequece geerators, teger set geerators, Gaussa geerators, decmal fracto geerators or row radom byte geerators depedg f they geerate tegers, teger sequece, set of radom tegers, tegers that fts ormal dstrbuto or umbers the 0 ad 1 rage wth cofgurable decmal places, respectvely. Each of the metoed types s useful for may cryptographc purposes [10]. Spltable pseudoradom umber geerators (PRNGs) were very useful for structurg purely fuctoal programs that deal wth radomess, because they allow dfferet parts of the program to depedetly geerate radom values, thus avodg radom seed threadg through the whole program [11]. The avalable pseudoradom umber geerators (PRNGs) Iteratoal Scholarly ad Scetfc Research & Iovato 9(4)

2 Iteratoal Joural of Computer ad Iformato Egeerg are ether secure but slow, or fast but secure [1]. Besdes, they are ether ot effcet eough, have heret flaws, or lack formal argumets for ther radomess. Claesse ad Palka [1] provded proofs order to show strog guaratees of radomess uder assumptos commoly made cryptography. Mxg secure ad fast PRNGs order to beeft from ther respectve qualtes was sought recetly, for example [14] proposed chaotc dyamcal systems whch appear to be good caddates to acheve ths mxture for optmzato by topologcal chaos ad chaotc teratos for hash fuctos, [13]. PRNGs based o chaotc teratos were suggested also for watermarkg applcato [14] ad [15]. Chaotc systems have may advatages as upredctablty or dsorder-lke, whch are requred buldg complex sequeces, [16], [17]. Ths s why chaos has bee appled to secure optcal commucatos as suggested by [18]. However, chaotc systems of real-umber or fte bt represetato realzed fte computg precso have the problems of o-deal dstrbuto ad short cycle legth. Hece fte space of tegers was cosdered lately leadg to the proposto of usg chaotc teratos (CIs) techques. Such proposal has to a ew famly of statstcally perfect ad fast PRNGs, [19], where a ew verso of ths famly has bee proposed. It uses decmato strateges that lead to mprovemets both radom umber geerato speed ad statstcal qualtes. Other terestg PRNGs used a ew Iterato Fucto System (IFS) that measures the sestvty of the IFS to certa tal values order to geerate chaotc radom umbers, [0]. PRNGs were also suggested based o teratve mplemetato of oe-way fuctos utlzg a radomly selected start value wth a key, [1]. Both the start value ad the key for subsequet teratos were selected from the already geerated radom umber the prevous terato. Aother terestg PRNG method was suggested by []. It troduced a dyamc system to produce a terestg herarchy of radom umbers based o the revew of radom umbers characterstcs ad chaotc fuctos theory. The authors had carred out certa statstcal tests o a seres of umbers obtaed from the troduced herarchy. Orue et al. [3] suggested cryptographcally secure PRNGs that are based o the combato of the sequeces geerated by three coupled Lagged Fboacc geerators, mutually perturbed. The mutual perturbato method cossts of the btwse XOR cross-addto of the output of each geerator wth the rght-shfted output of the earby geerator. The proposed geerator has better etropy ad much loger repetto perod tha the covetoal Lagged Fboacc Geerator. Ths paper proposes a pseudo-radom umber geerato scheme that s based o Shao s cocept of cofuso ad dffuso. The geerated radom umbers are to be used for cryptographc applcato. It suggests a house scearo process that mplemets a oe-tme-pad key for secure commucato. The scheme geerates cotuous strgs of radom bt sequeces to be used as oe tme keys progressvely for the subsequet messages. III. KEY RANDOMNESS TEST The geerated pseudoradom bary sequeces ca be tested for radomess by some of the statstcal tests outled by NIST [7]. These tests wll focus o a varety of dfferet types of o-radomess. The selected tests here clude Frequecy (Moobt) test, Frequecy test wth a Block, the Rus test, ad the test for the Logest-Ru-of-Oes a Block ad wll be summarzed below. For ay of these tests the P- value s calculated ad compared agast the level of sgfcace α (whose value s commoly set to about 0.01 for cryptographc applcatos). α s defed as the probablty that the geerated umber s ot radom whe t s really radom ad P-value s the probablty that a perfect radom umber geerator would have produced a sequece less radom tha the sequece that was tested, gve the kd of oradomess assessed by the test [7]. The crtera s f P- value>=α, the sequece appears to be radom but f P- value<α, the the sequece appears to be o-radom. A. Frequecy (Moobt) Test: It tests the proporto of zeroes ad oes for the etre sequece. The purpose of ths test s to determe whether the umber of oes ad zeros a sequece are approxmately the same as would be expected for a truly radom sequece. For the purpose of testg the radomess of a umber wth a bt strg legth of bts, such that = b 1, b,b, a observed value S obs s used as a test statstc whch s defed by (1). S obs S where S s the sum of all strg bts after covertg zeros ad oes to -1 ad +1, respectvely. The P-value for ths test s calculated by () (1) S obs () P value erfc where erfc s the complemetary error fucto [7] B. Frequecy Test wth a Block test: It tests the proporto of 1 s wth M-bt blocks. The purpose of ths test s to determe whether the frequecy of 1 s a M-bt blocks s approxmately M/, as would be expected uder a assumpto of radomess. The P-value s calculated by (3): P /, value gamc N ( obs ) / (3) where gamc s the complete gamma fucto, N s the umber of M-bt blocks to be tested, ad χ (obs) s the ch fucto of the observed proporto of 1 s wth a gve M- bt block gve by (4). Iteratoal Scholarly ad Scetfc Research & Iovato 9(4)

3 Iteratoal Joural of Computer ad Iformato Egeerg ( 1) 1 N M j1 1 ( obs ) 4M ( ) (4) 1 M C. The Rus Test: It tests s the total umber uterrupted sequece of detcal bts,.e. whether the umber of rus of 1 s ad 0 s of varous legths s as expected for a radom sequece. Ths test dcates the speed of 1 s ad 0 s whether t s too fast or too slow. The P-value s calculated by (5): M for example, the 64 bts key for DES eeds 8 characters ad the 18 bts key for AES eeds 16 characters, ad so o. The work flow dagram of suggested PRNG scheme s llustrated Fg. 1. P value V ( obs ) (1 ) (5) erfc 1 (1 ) Where V (obs) s the total umber of ru across ad s the pre-test proporto the put sequece gve by (6): j j D. Logest-Ru-of-Oes a Block Test It tests the logest ru of 1 s wth M-bt blocks, ad show whether t s cosst wth the legth of the logest ru of 1 s that would be expected a radom sequece. The P-value s calculated by (7) ad χ (obs) s a measure of matchg betwee observed logest ru legth wth M-bt blocks wth the expected logest legth wth M-bt blocks, gve by (8): (6) K ( obs ) P value, (7) K ( N ) (8) ( obs ) N 0 Where ν s the frequeces of the logest rus of 1 s each block categorzed for =0 to K whle the values of K ad N are determed by the value of M accordace wth the pre-set Table I. TABLE I PRE-SET VALUES FOR M, K AND N WITHIN THE NUMBER Mmum key legth M K N IV. PSEUDORANDOM NUMBER GENERATION METHODOLOGY Ths work proposes a scheme for pseudoradom umber geerator (PRNG) that combes btwse logcal operato ad bts mapulato order to fulfll the cofuso ad dffuso prcple. It starts wth a radomly selected put key (seed) that cossts of ay combato of letters (lower or upper case) ad umbers (.e. a b c z, A B C Z, _, 0 1 9). Ths seed ca be exchaged betwee seder ad recever publcly but would eve better f t s exchaged secretly. They are replaced by ther ASCII code bary represetato as they eter to the PRNG. The legth of ths key s decded by the cryptographc system that s gog to be mplemeted, Fg. 1 Work flow dagram for the proposed PRNG Pror to the operato of the proposed PRNG, the characters of the suppled key are coverted to ASCII codes represetato, ad the the followg steps are performed accordg to the work flow of Fg. 1. Step 1. The selected key characters are frst replaced by ther bary represetatos. Let the etered key cossts of -characters, the the legth of ths key wll be 8* (.e. -bytes). Step. Abtwse XOR operatos are performed o the bt blocks of each two successve bytes,.e. (1 st XOR d ) replaces the 1stbyte, ( d XOR 3 rd ) replaces d character, etc., utl the last byte where t s XOR ed wth the frst oe, or ( th XOR1 st ) replaces the th -byte, as gve by (9):.e. th th XOR ( 1) mod, for =1 to (9) Step 3. Successve bytes are exchaged wth each other pars. However, f s odd umber, the the last byte s left ualtered. Step 4. The resultg bt strg of the prevous step s dvded to halve, left ad rght each of 4* bts legth. The resultg bt sequece of 8* ca be take as the frst pseudoradom radom key K 1. Step 5. The geerated key step 4 ca be fed back as a put to step order to geerate ext radom key. Step 6. I order to geerate more keys, steps -5 ca be repeated as may as requred. A computer program s wrtte to perform these steps wrtte C# laguage ad tabulate the results excel sheet together wth ther radomess tests order to be ready for use ay cryptographc system. Securty of the key agreemet: Mxg of btwse Boolea operatos (XOR), btwse ad Iteratoal Scholarly ad Scetfc Research & Iovato 9(4)

4 Iteratoal Joural of Computer ad Iformato Egeerg operatos oreted (mapulato) serves to avod purely algebrac attacks ad the purely bt oreted attacks ad prevets the mathematcal behavor of the scheme from beg shaped easly. They both cotrbute to a great mathematcal complexty together wth hgh computatoal effcecy. Moreover, oly very effcet operatos are used, btwse Boolea operatos, ad bts dsplacemets, ad they are both of easy mplemetato ether by hardware or by software. Ths method geerates pseudo radom umbers by selectg a certa pece of secure formato. For the users volved ths system to geerate these radom umbers, they must agree upo ths formato advace to eable them to geerate ther ow radom umbers whch are the used cryptographc systems. The users also agree upo the predefed structure of ths secret formato or the key. The structure ths method cossts of a certa umber of dgts costtutg certa dgtal data that ca ether be represeted as characters or as a bt sequece. The agreemet upo the sze () of ths key, whch depeds practcally o the cryptographc system that s teded to be used wth, represets oe of the costrats of ths structure. Ad aother costrat s the method used for splttg ths pece of formato for the varous operatos. The secure formato s dealt wth at byte level segmets. Obvously each byte cotas dfferet dgtal cotets. The algorthm geerates the frst pseudoradom key at the ed of the frst ru, however to geerate more radom umbers, further rus ca be doe. Each of the successve rus takes the prevously geerated radom key as the put. Therefore, the dffculty ad complexty of the geerated keys wll crease. The acceptace stregth of the geerated radom umbers wll be decded by the tests that wll be performed o the results the followg secto. V. IMPLEMENTATION AND RESULTS A Computer program s wrtte C# laguage for the proposed PRNG algorthm llustrated Fg. 1. It s desged to accept a seed of ay umber of characters ad geerate as may radom umbers as practcally requred wth ay legth of bt sequece. It has bee expermeted for the geerato ad testg of radom keys of 64, 18, ad 51 bts legths. A example demostratg the mplemetato steps for the algorthm executo ad elemetary radomess tests are cluded the ext subsecto, the four elaborate tests were carred out ad ther results were lsted ext subsecto. A. PRNG Implemetato The followg example ca llustrate the mplemetato of the scheme for a 64 bts key sequece,.e. the seed cossts of 8 characters. Let us assume that step 1 produce the bary represetato of the seed characters to be: Step produces: , The step 3 produces: Ad fally step 4 wll produce: Ths s the geerated pseudoradom umber of the frst roud. The more radom umbers ca be geerated by repeatg the PRNG algorthm usg the geerated umber of a roud as put to the algorthm for the ext roud. To test the radomess of the geerated radom umber, the followg smple frequecy test ca be coducted. Ths test uses the ch-fucto formula gve (10): ( 0 1) (10) where 0 ad 1 are the umbers of 0 s ad 1 s the geerated key sequece, respectvely. Good sequece the geerated radom umber should have χ values the rage 0<χ <3.84[4]. Sce χ = (9-35)/64 = whch s < 3.84 for the above example, therefore t s cosdered as acceptable radom umber. Coutg the 0 s ad 1 s the geerated key the above example resulted to 0 = 34 ad 1 = 30, therefore χ = (34-30) /64 = 0.5, ad sce ths value s < 3.84, hece t s cosdered as acceptable radom umber. Other test lke frequecy (moobt) ca also be coducted, for the above example above. The strg legth = 64 ad the sum of all umber strg bts sequece after covertg 0 s ad 1 s to -1 ad +1, respectvely, S s determed as S = = - 4. The from (1), the test statstc observed value S obs 6 / 64 = 0.5. Now applyg (), the P-value s calculated ad t s here P-value=erfc (0.5/8) = The sce 0.996s > 0.01, the umber s radom otherwse t s ot. Aother seres test for the frequecy of occurrece of sequeces of two bts,.e. 00, 01, 10, ad 11 ca be coducted. For the above example are 00 =16, 01 =16, 10 =17 ad 11 =10 ad the umber of 1 s ad 0 s the key sequece 1 =30 ad 0 =34. Applyg the ch-fucto for ths test whch s gve by (11) [4]: 4 ( ) ( ) 1 (11) 1 The crtero for good radomess sequece s χ Applyg (11) results to 4 ( ) (30 ( 34 ) ) Therefore, ths key passes the radomess test. Iteratoal Scholarly ad Scetfc Research & Iovato 9(4)

5 Iteratoal Joural of Computer ad Iformato Egeerg B. PRNG Testg The potetal problems wth determstc geerators are ther falure statstcal patter-detecto tests. These problems clude lack of dstrbuto uformty, correlato of successve values; output sequece has poor dmesoal dstrbuto, ad shorter seed state. The radom sequeces geerated by the proposed PRNG are tested for radomess by the four tests processes outled Secto III. Although, the tests were doe for 64, 18, 1d 51 bts key sequeces, however, t ca be practcally doe for ay key legth. Table II lsts the calculated percetages of successfully geerated radom key sequeces whe the used seed for geeratg radom umbers s oly umerc s,.e. dgts 0, 1,,, 9. The tests cluded are explaed four tests secto 3 amely; frequecy (moobt) test, Frequecy wth a Block test, Rus test, ad Logest-Ru-of-Oes a Block test. Ths table lsts the results for three commoly requred radom key sequeces; 64, 18, ad 51 bts. However, Table III lsts the percetages of successfully radom keys geerated usg alphaumerc seed (.e. both upper ad lower case letters addto to the umbers). It s also coducted for the same legths ad tests selected. TABLE II SUCCESSFUL RANDOMNESS FOR ONLY NUMERIC KEYS PERCENTAGE OF SUCCESSFUL Test RANDOMNESS (NUMERIC SEED ONLY) =64=18=51 Frequecy (moobt) 73% 40% 13% Frequecy wth a block 84% 80% 3% The Rus test 91% 6% 3% Logest ru of 1 s a block N/A 99% 99% TABLE III SUCCESSFUL RANDOMNESS FOR ALPHANUMERIC KEYS PERCENTAGE OF SUCCESSFUL RANDOMNESS Test (ALPHANUMERIC SEEDS KEYS) =64=18=51 Frequecy (moobt) 96% 89% 8% Frequecy wth a block 98% 99% 98% The Rus test 100% 99% 89% Logest ru of 1 s a block N/A 98% 98% The results lsted ths secto have show that the proposed PRNG algorthm effectvely geerates key sequeces wth a cosderably acceptable radomess. The radomess tests showed that creasg the umber of rouds to geerate more keys ehaces the effcecy of radom key geerato for cryptographc system applcatos. The scheme s characterzed by ts smple yet practcal desg as t does ot have complcated ad legthy expoetato processes. Ths leads to more effcet software ad hardware mplemetatos. VI. CONCLUSION A algorthm for computatoally fast, cryptographcally secure pseudoradom key geerator has bee proposed ad descrbed ths paper. It s based o mxg btwse Boolea operatos wth bts mapulatos ad dsplacemets for secret splttg. The mplemeted radomess tests have show that the geerated sequeces were upredctable ad passed successfully strget test sutes. It obvously reles o the seder/recever agreemet protocol regardg the cryptosystem they are usg, the key legth, umber of keys ad way of ther geerato. Therefore the requred radom keys wll be securely geerated accordgly. The algorthm was programmed C# laguage o a 64 bts word legth computer usg oly btwse XOR ad blocks exchages. Hece excellet performace was acheved. Besdes, the program produces as may umber of radom sequece keys as requred, whch each radom sequece s geerated from a radom oe. REFERENCES [1] B. Scheer, Appled cryptography: protocols, algorthms, ad source code C, Secod Edto, Joh Wley & Sos, [] D. Dll, Madhu S., Desg of a New Cryptography Algorthm usg Reseedg -Mxg Pseudo Radom Number Geerator, IJITEE, vol. 5, No. 5, 013 [3] K. Marto, A. Sucu, C. Sacarea, ad Octava Cret, Geerato ad Testg of Radom Numbers for Cryptographc Applcatos, Proceedgs of the Ramaa Academy, Seres A, Vol. 13, No. 4, 01, PP [4] S. Marta, Testg of True Radom Number Geerator Used Cryptography, Iteratoal Joural of Computer Applcatos IJCA, Vol., No. 4, 01. [5] Wkpeda, Pseudoradom umber geerator, Last vsted December 014. [6] D. Dll, ad S. 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6 Iteratoal Joural of Computer ad Iformato Egeerg [0] Q. Wag, J. Bah, C. Guyeux, ad X. Fag, Radomess qualty of CI chaotc geerators; applcato to teret securty, INTERNET 010. The d Iteratoal Coferece o Evolvg Iteret, Valeca, Spa, September 010. IEEE Computer Socety Press. Best Paper award, PP [1] H. B. Neuma, S. Scholze, ad M. Voegeler, Method of geeratg pseudo-radom umbers, US A1, Ju 11, 009. [] M. N. Elsherbey, ad M. Raha, Pseudo Radom Number Geerator Usg Determstc Chaotc System, Iteratoal Joural of Scetfc ad Techology Research, vol. 1, o. 9, Oct. 01. [3] S. Beha, A. Akhava, A. Akhsha, ad A.Samsud, A ovel dyamc model of pseudo radom umber geerator, Joural of Computatoal ad Appled Mathematcs J COMPUT APPL MATH, vol. 35, o. 1, 011, PP [4] W. Bhaya ad W. Mahd, Fgerprt Securty Approach for Iformato Exchage o Networks, Europea Joural of Scetfc Research, vol. 13, o, 014, PP Hamza A. A. Al_Sewad s curretly a professor at the Faculty of Iformato Techology, Isra Uversty (Jorda). He got hs B.Sc. degree 1968 from Basrah Uversty, Iraq, the M.Sc. ad Ph.D. degrees 1973 ad 1977 respectvely, from Uversty of Lodo(UK). He worked as assocate professor at varous uverstes such as Basrah Uversty (Iraq), Zarqa Uversty ad Isra Uversty (Jorda), vstg professor at Uversty of Azu (Japa). Hs research terests clude Cryptography, Stegaography, Iformato ad Computer Network Securty, Artfcal Itellgece ad Neural Networks. Ad A. Maata s curretly a assstat professor at the Faculty of Iformato Techology, Isra Uversty. He receved hs B.Sc. degree 00 from the Uversty of Jorda (Jorda), hs M.Sc. 003 from the New York Isttute of Techology (Jorda).The receved hs Ph.D. from the Uversty of Lecester (UK) 008. Hs research terests clude Cryptography, Stegaography, Iformato ad Computer Network Securty, Geetc Algorthms, Neural Networks, ad Software Modelg. Iteratoal Scholarly ad Scetfc Research & Iovato 9(4)