Formation of Pseudo-Random Sequences of Maximum Period of Transformation of Elliptic Curves

Transcription

1 Internatonal Journal of Computatonal Engneerng Research Vol 03 Issue 5 Formaton of Pseudo-Random Sequences of Maxmum Perod of Transformaton of Ellptc Curves Alexandr Kuznetsov 1 Dmtry Prokopovych-Tkachenko 2 Alexey Smrnov 3 1 Professor n the Department of Informaton Technology Securty of Kharkv Natonal Unversty of Radoelectronc Kharkv Ukrane 2 Ukranan Academy of Customs Dnpropetrovsk Ukrane 3 Professor n the Department of Software Krovohrad Natonal Techncal Unversty Krovohrad Ukrane ABSTRACT It s consdered methods of formng pseudo-random sequences for cryptographc of applcatons n partcular for domestc mechansms to ensure securty of nformaton systems and technologes We nvestgate the propertes of pseudo-random sequences generator usng the changes n the group of ponts of ellptc curves [accordng to the standard NIST SP ] there are some dsadvantages of test generator wth respect to the propertes of the formed sequences and ther statstcal vsblty wth unformly dstrbuted sequences It s developed an mproved method by ntroducng addtonal recurrent converson whch allows you to create sequences of pseudo-random numbers maxmum perod Keywords: Pseudo-Random Sequences Ellptc Curves I INTRODUCTION The analyss and comparatve studes have shown that the most effectve n terms of ndvsblty of molded sequences wth realzaton of stochastc process s a method of formng pseudo-random numbers whch are based on the use of modular transformatons or changes n the group of ponts of an ellptc curve [1-3] The most promsng are generators of pseudo-random sequences (PRS) whch are bult usng transformatons under ponts of an ellptc curve At the same tme as shown n the work [4-5] carred out studes the man drawback of the known method of formng the PRS usng transformatons on ellptc curves (accordng to the standard NIST SP ) s that t does not allow to form a sequence of pseudo-random numbers maxmum perod that sgnfcantly reduces ts effcency and lmts the possbltes for practcal use Indeed appled scalar pont multplcaton operaton of an ellptc curve and coordnates reflecton of obtanng pont for formaton of pseudo-random numbers does not provde the maxmum amount of molded sequences [4-5] In ths paper tasked to develop a method of formng of sequences of pseudo-random numbers s due to the addtonal drvng recursve transformaton n conjuncton wth the use of transforms under ponts of ellptc curve wll generate a maxmum perod of PRS ncreasng ts effcency and expand opportuntes for use n practce II A KNOWN METHOD OF FORMATION OF PSEUDO-RANDOM SEQUENCES (PRS) ON THE ELLIPTIC CURVES A method of formaton of PRS usng transformatons on the ellptc curves that suggested n the recommendatons NIST SP s based on the use of two scalar products of ponts of an ellptc curve and mappng of correspondng x-coordnates of receved results nto non-zero nteger values [3] The frst scalar product on a fxed pont P s performed n order to form the ntermedate phase s and t s cyclcally changed at each teraton durng the functonng of the correspondng generator So the value of state s depends on the value of the prevous state s -1 (at the prevous teraton) and the value of fxed pont P: (x(s P)) (1) where x(a) s the x-coordnate of the pont A ) numbers s 1 (x feld elements mappng functon nto non-zero nteger wwwjceronlnecom May 2013 Page 26

2 An ntal value of the parameter s0 s formed wth the use of ntalzaton procedure that ncludes nserton of a secret key (Key) whch sets the ntal entropy and nserted key hashng wth the formng of receved results to the specfed length of the bts The receved value Seed ntates the ntal value of the parameter: s 0 Seed The second scalar product on a fxed pont Q s performed n order to form an ntermedate state r Ths scalar product sets the value of generated pseudo-random bts after the correspondng converson The value of parameter r depends on the frst scalar product of parameter s and the value of fxed pont Q : r (x(sq)) (2) The value r s ntal for formng of pseudo-random bts These bts are formed by readng of the block wth the least sgnfcant bts of number r PRS s formed by a concatenaton of read-n bts of generated numbers r The values of fxed ponts are set as constants They are not changed durng the formng of PRS The structure chart of PRS generator wth the use of conversons on the ellptc curves n accordance wth the recommendatons of the standard NIST SP s shown n a Fg 1 Fg 1 The structure chart of PRS generator wth the use of conversons on the ellptc curves (accordng to the recommendatons of the standard NIST SP ) Ths method of PRS formng apples converson n a cluster of ponts of an ellptc curve n order to form ntermedate states s and r The back acton or n the other words formng of s -1 by the known s and/or formng of s by the known r s connected wth a soluton of a dffcult theoretcal task of dscrete logarthm n a cluster of ponts of an ellptc curve Generator s ntermedate states formaton chart s represented n a Fg 2 s 0 s 1 s 2 s 3 r 1 r 2 r 3 Fg 2 Generator s ntermedate states formaton chart wwwjceronlnecom May 2013 Page 27

3 As t s seen from the Fg 2 the sequence of states s -1 s s 1 forms from the ntal value s 0 Seed whch n turn forms from secret key (Key) data Each followng value s depends on prevous value of s -1 and forms by the nstrumentalty of ellptc curve s basc pont scalar multplcaton accordng to formula (1) Some bts of PRS s formed by readng bt of sequence of numbers r -1 r r 1 e by readng data from the result of another scalar multplcaton of the base pont on the value of states s -1 s s 1 accordng to the formula 2 Snce the secret Key whch sets the results for formng sequences after certan transformatons determnes the ntal value of the parameter s 0 relevant sustanablty of consdered generator s based on the reducton of the problem of secrete key data recovery solutons to well-known and hghly complcated mathematcal task of dscrete logarthm n the group of ponts of an ellptc curve Besdes fragments PRS also lnked by scalar multplcaton of ellptc curve ponts e to recover any pece of PRS by any other known fragment to solve the problem of dscrete logarthm n an ellptc curve group The paper [4-5] studed the propertes of perodc reportng generator PRS ncludng a comparson of the obtaned lengths of the perods of sequences wth maxmum perod that can be obtaned for gven length of keys and groups of ponts of an ellptc curve For maxmum perod of molded PRS take the meanng [4-5]: where:: Lmax mn(lmax (K)Lmax (S)k) (3) L (K) 2 K max l 1 l K the length of secret key (bts); L (S) 2 S max l 1 ls log2(seed) bt length of meanng Seed ; k order of pont P of ellptc curve The generated sequences wll reach the maxmal perod when the elements of the sequence: r 0 r 1 r -1 r r 1 r L-1 r 0 r 1 (4) wll possess each: l mn( 2 K l 1 2 S 1 k) non-zero value Practcally t means that the feld elements mappng functon (x) nto non-zero nteger numbers at each teraton for each generated pont s 1 P s to generate unque nteger number But t s mpossble The order m of the group H EC of the ellptc curves ponts whch are used n cryptographc addtons n cases provded by the recommendatons of standard NIST SP s lmted by the form: p 1 2 p m p 1 2 p where p s the order of smple Galos feld GF (p) over whch the ellptc curve s consdered So the cases when the order of cluster of ponts can be hgher than the order of Galos feld can emerge Practcally t means that for some elements of the group H EC for example the functon (x) wll return dentcal value for ponts P і P j P Pj In ths case the use of ellptc curves arthmetc n the generator of pseudorandom numbers wll mean the equalty of states values s s for some j where: and and j Lmax j s (x(s 1P)) (x(p )) s j (x(s j 1P)) (x(pj)) wwwjceronlnecom May 2013 Page 28

4 So the value of real perods L of generated consequences of states (4) wll be lower than maxmal perod (3) But n ths context we should consder the exstence of negatons for each element of the group H EC It means that for each pont P (xy) HEC exsts the pont P (x y) HEC and ts x- coordnate concdes wth the pont P (xy) and y-coordnate s nverse to correspondng y-coordnate of the pont P (xy) relatve to the operaton of the addton n the Galos feld GF p arthmetc The ponts of zero y-coordnate (the ponts of type P (x0) P (x0) HEC) are the exceptons In ths case the mappng functon (x) of x-coordnate of the pont P nto non-zero nteger numbers wll return dentcal values lke n cases P P (xy) and P P (x y) so we wll have the next case: ( x(p (xy))) (x( P (x y))) And equalty correspondngly: s (x(p (x y ))) (x( P (x y ))) s Practcally n means that accordng to the rule of PRC generatng wth the use of arthmetc of ellptc curves that s descrbed n the recommendatons of the standard NIST SP the maxmal perods of sequences wll not be reached Furthermore the expermental studes [4 5] show that real perods wll be lower than maxmal III DEVELOPMENT OF IMPROVED METHOD OF PRS GENERATING OF THE MAXIMAL PERIOD WITH THE USE OF TRANSFORMATIONS ON ELLIPTIC CURVES The task of provdng maxmal perod of generated PRS s solved by the addtonal recurrence transformatons nto the generator A structure chart of an mproved PRS generator wth the use of transformatons on the ellptc curves s shown n a Fg 3 The frst scalar multplcaton on a fxed pont P lke n the generator that meets the recommendatons of NIST SP s performed n order to form an ntermedate state s And t s cyclcally updated at each teraton durng the functonng of the correspondng generator But there s a fundamental dfference It s a formng process of ths ntermedate state The mproved method proposes to use a recurrence transformaton whch ntates by a secret key nserton (Key) n order to provde a maxmal perod of sequences s -1 s s 1 So every next state value s depends not only on the prevous state value s -1 (at prevous teraton) and on the value of a fxed pont P but also t depends on the result of recurrence transformaton ( LRR (y) ): where x(a) s x-coordnate of the pont A ) nteger numbers s (x((s 1 LRR(y))P)) (x s a mappng functon of the feld elements nto non-zero j Fg 3 A structure chart of the mproved PRS generator wth the use of transformatons on the ellptc curve wwwjceronlnecom May 2013 Page 29

5 A recurrence transformaton can be bult n dfferent ways The smplest way s the use of a crcut of lnear recurrence regsters (LRR) wth a feedback (Fg 4) The taps of the chan are set by coeffcents of polynomal wth bnary coeffcents g(x) 2 m g0 g1x g2x bmx If the polynomal g (x) s prmtve over Galos feld GF (2 m ) so the sequence whch s formed by LRR wth the correspondng logc of a feedback has a maxmal perod 2 m 1 The secret key value (Key) that ntates the work of LRR (y) s wrtten down nto LRR as an ntal value of the regster а) b) Fg 4 A structure chart of a block of recurrence transformatons wth the use of LRR n Fbonacc confguraton (a) and Galos confguraton (b) Devces represented at the chart 4 have the followng prncple of operaton Frst of all ntalzaton procedure s held Durng ths procedure the key of the devces s n a lower poston; the ntal value y 0 whch s equal to the value of a secret key y 0 Key s put down nto a shft regster Then the key swtches on an upper poston that s mean that nothng goes to a shft regster Durng the work of the block the nformaton kept n a shft regster shfts to the rght for one cell And n the feedback crcut a value of feedback functon goes to the frst cell So at tme nterval the value of y s kept n a shft regster and ths value of y s read as a value of the functon L(y ) The feedback functon provdes the generatng of PRS of the maxmal perod; t also sets a concrete look of a commutaton of a feedback crcut Accordng to the charts of the devces at the Fg 4 at each step only the value of the last left cell s changed n the lnear regster under the Fbonacc confguraton In the lnear regster under the Galos confguraton at each step the values of all cells whch take part n a generatng of the value of feedback functon are changed The man pont of the mproved method of PRS generatng s that the key sequence s represented as a vector x 0 whch ntalzes an ntal value of an argument of a scalar product functon of a pont of ellptc curve f x x P where P s a pont of the ellptc curve that belongs to a cluster of ponts EC n of multple of N and the ntal value of y 0 of recurrence transformaton L y whch s mplemented for example wth the help of lnear recurrence regsters wth a feedback wwwjceronlnecom May 2013 Page 30

6 The next value x of an argument of a functon f x s calculated wth the help of recurrence transformaton (whch s mplemented for example wth the help of lnear recurrence regsters wth a feedback) wth the help of devces of the scalar multplcaton x 1 on a fxed pont P: P' x 1 L y 1 P and transformaton P' coordnates of the receved pont P' ECn wth the help of the correspondng devces (for example x can be equal to the value of the coordnates of the pont P' ) so x P' f x 1 L y 1 x 1 L y 1 P The receved values of x are represented as an argument of a functon of scalar product of an ellptc curve pont f ' x x Q where Q s a pont of an ellptc pont whch belongs to a cluster of ponts EC n of multple of N Parent elements of sequence pseudorandom numbers forms by readng the meanng of scalar product functon wth the ad of correspondng devces that s requred sequence of length bt m wll be the sequence: where b s a lower bt of number z functon f x The problem calculus of functon b 0 m 1 0 b1 b2 b bm z 1 f' x x Q 1 f (x) whch s nverse to the ellptc curve s pont scalar product x P that s calculaton of some sense of x 1 when x s known s naggng theoretcalcomplcated problem of takng the dscrete logarthm n cluster of ponts of ellptc curve Conformably the 1 problem of calculaton the functon f'(x) whch s nverse to the ellptc curve s pont scalar product functon f ' x x Q x when z s known s naggng theoretcal- that s calculaton of some sense of complcated problem of takng the dscrete logarthm n cluster of ponts of ellptc curve The effectve logarthms of calculaton dscrete logarthms for bass ponts of large-scale order for resoluton of ths problem are rested unknown for today That s why ths way of formaton of sequence of pseudorandom numbers s cryptographcally resstant Formally the formaton of PRS (ndcated as LRR) could be shown n such a way Secret key: Key; Constants: P Q ponts ЕC of the multple of n; Intal condton: x0 = Key y0 = Key ; Cycle functon: (f(x LRR(y))) ((x LRR(y))P) LRR(y {u1 u2 um}) :u - a ju- j u j when the lnear recurrent regsters are used where: {u1 u2 um} s the condton of LRR {a1 a2 am} are the coeffcents whch specfy the functon of nverse lason of LRR ( P' ) transformaton of coordnates of pont P' ECn (fe readng of sense of one of the ponts s P ' coordntaes) Formed PRS: (b0 b1 b ) where b the less meanngful bt (the bt of twoness) number z z (f'( ((x-1 LRR(y-1))P))) ( ((x-1 LRR(y-1))P)Q) y LRR(y -1) To analyze perodcal propertes of enhanced generator PRS let us consder the structure chart whch s shown at the Fg 3 The ntal value of parameter s 0 as t s n the generator whch corresponds to the recommendatons of NIST SP forms wth the use of ntalzaton procedure whch ncludes the enter of secret key (Key) whch defnes the ntal entropy (ambguty) and hashng of the key entered wth formattng of receved result to concrete length of bts Receved sense Seed ntate the startng value of the parameter: s 0 Seed wwwjceronlnecom May 2013 Page 31

7 The second scalar multplcaton on a fxed pont Q s performed n order to form an ntermedate state r whch after the correspondng transformaton set the value of the generated pseudorandom bts Every next value of the state s depends on the results of an mplementaton of a recurrence transformaton LRR (y) whch provdes a maxmal perod of generated sequences so the value of parameter r that depends on the parameter s and the value of the fxed pont Q : r (x(sq)) wll depend on the results of results of an mplementaton of a recurrence transformaton LRR (y) so the generated sequence of states r -1 r r 1 wll have a maxmal perod The receved value r s an ntal for generatng of pseudorandom bts whch are generated by readng of block from the least sgnfcant bts of generated numbers r PRS s generated by concatenaton of read-n bts of generated numbers r The values of fxed ponts are defned as constants and durng PRS generatng they don t change So the perodc features of states of the mproved generator are defned by the perodc features of an addtonal recurrence transformaton LRR (y) On the Fg 5 we see an orgnal sequence LRR (y) ndcated by y -1 y y 1 We sketch the nfluence of perodcty of ths sequence on perodcty of sequences s -1 s s 1 and r -1 r r 1 Fg 5 The chart of formng of state sequences of the generator wth maxmal perod The perodc features of sequences s -1 s s 1 and r -1 r r 1 depends on features of sequences y -1 y y 1 so the use of recurrence transformaton LRR (y) wth the maxmal perod of the orgnal sequences provdes the maxmal perod of the orgnal sequence The addtonal recurrence transformatons whch are mplemented wth the help of lnear recurrence regsters wth feedbacks help to form sequences of pseudorandom numbers of maxmal perod It helps to ncrease the effcency and wden the practcal use IV CONCLUSIONS Durng the research of PRS generator on the ellptc curves whch s descrbed n a standard NIST SP we found out further shortcomngs: a cyclc functon of the generator that provdes the maxmal perod of the generated PRS of the nner states and the correspondng ponts of the ellptc curves s not defned; the formng of PRS bts from the sequence of the ellptc curve ponts by a sample of a block wth the least sgnfcant bts and ts concatenaton doesn t meet the requrements of the statstcal dscrepancy wth unformly separated sequence So the PRS generator on the ellptc curves (NIST SP ) doesn t meet the requrements suffcently The mproved method of formng PRS of the maxmal perod wth the use of transformatons on ellptc curves was elaborated durng the research Ths method unlke the others helps to generate sequences of pseudorandom numbers of the maxmal perod and t ncreases the effcency and wdens the practcal use wwwjceronlnecom May 2013 Page 32

8 The elaborated method s based on the brngng of a task of a secret key fndng to the solvng of a dffcult theoretcal task of a dscrete logarthm n a cluster of ponts; t also helps to form PRS wth a maxmal perod An ntroduced method of formng of PRS wth maxmal perod wth the use of transformatons on ellptc curves removes the uncovered drawbacks of the generator descrbed n the standard NIST SP : t was proposed to use lnear recurrence transformatons whch help to form a maxmal perod of sequences of nner states and correspondng ponts of an ellptc curve; formng of an orgnal PRS by readng of one the least sgnfcant bt from coordnates of ellptc curve ponts meets the requrements of a statstcal safety So the ntroduced method of PRS formng meets modern requrements and can be used n order to mprove dfferent safety mechansms for an nformaton securty of telecommuncatons networks and systems Prospectve course of a further research are the argumentaton of practcal recommendatons concernng a realzaton of the ntroduced PRS and the ways of ts use n dfferent mechansms of an nformaton securty of telecommuncatons networks and systems REFERENCES [1] Alfred J Menezes Paul C van Oorschot Scott A Vanstone Handbook of Appled Cryptography CRC Press р [2] Fnal report of European project number IST named New European Schemes for Sgnatures Integrty and Encrypton Aprl Verson 015 (beta) Sprnger-Verlag 829 p [3] Elane Barker and John Kelsey Recommendaton for random number generaton usng determnstc random bt generators Natonal Insttute of Standards and Technology January p [4] LSSoroka OOKuznetsov DIProkopovych-Tkachenko The features of the generators of pseudorandom sequences on the ellptc curves // the bulletn of Ukranan Academy of Customs Edton Techncs 1 (47) 2012 p 5-15 [5] LSSoroka OOKuznetsov DIProkopovych-Tkachenko The research of the generators of pseudorandom sequences on the ellptc curves // An mplementaton of nformaton technologes for tranng and actvtes of law-enforcement agences: theses of reports from theoretcal and practcal conference of Academy of nternal mltary forces of Mnstry of nternal affars of Ukrane March H: Academy of nternal mltary forces of Mnstry of nternal affars of Ukrane 2012 p AUTHORS NAMES Alexandr Kuznetsov Graduated from Kharkv Mltary Unversty wth a degree n Automated Control Systems n 1996 Doctor of Techncal Scences Professor Kharkov Natonal Unversty of Rado Electroncs Kharkov Ukrane Feld of nterest: nformaton securty and routng ssues Dmtry Prokopovych-Tkachenko Graduated from Kharkov Hgher Mltary Command School of Engneerngwth a degree n Automated Control Systems Graduate student Ukranan Academy of Customs Dnpropetrovsk Ukrane Feld of nterest: nformaton securty and routng ssues Alexey Smrnov Graduated from Kharkv Mltary Unversty wth a degree n Automated Control Systems n 1999 Canddate of Techncal Scences (PhD) Professor of Department of Software of Krovohrad Natonal Techncal Unversty Ukrane Feld of nterest: nformaton securty and routng ssues wwwjceronlnecom May 2013 Page 33