Linear Complexity and Correlation of some Cyclotomic Sequences

Transcription

1 Lnear Complexty and Correlaton of some Cyclotomc Sequences Young-Joon Km The Graduate School Yonse Unversty Department of Electrcal and Electronc Engneerng

2 Lnear Complexty and Correlaton of some Cyclotomc Sequences by Young-Joon Km A Dssertaton Submtted to the Graduate School of Yonse Unversty n partal fulfllment of the requrements for the degree of Doctor of Phlosophy Supervsed by Professor Hong-Yeop Song, Ph.D. Department of Electrcal and Electronc Engneerng The Graduate School YONSEI Unversty December 8

3 Ths certfes that the dssertaton of Young-Joon Km s approved. Thess Supervsor: Hong-Yeop Song Seong-Lyun Km Jeung-Yoon Cho Jang-Won Lee Habong Chung The Graduate School Yonse Unversty December 8

4 감사의글 먼저본논문의주제를정해주시고논문이완성되기까지철저한가르침을주신송홍엽교수님께큰감사드립니다. 보다좋은논문이되도록많은가르침을주신정하봉교수님, 김성륜교수님, 최정윤교수님과이장원교수님께도감사드립니다. 아울러학부및대학원재학기간동안엄하고애정어린지도로학문의길을인도해주셨던모든교수님들께감사드립니다. 교수님들의학문에대한대단한열정과제자들에대한사랑이제가느즈막이사회에내딪는첫걸음에등불이되어저를인도해줄것이라믿습니다. 연세대학교에입학해서학교에다닌기간만무려 년이흘렀습니다. 년겨울부호및정보이론연구실의신입생으로서교수님과 Abstract Algebra 세미나를매일함께하면서하루하루새로운것을배워가는기쁨을느꼈던그기억은평생잊지못할추억이될것입니다. 매해주옥같은논문들을쓰시고졸업하시는선배님들을보면서나도열심히연구하여좋은논문을많이써야지하며느슨해진마음을다잡고열의에불탔던그많은날들이주마등처럼지나갑니다. 부족하지만제가졸업하는이날까지희노애락을같이했던모든선후배님들께깊은감사를드립니다. 먼저동아리 년후배이자연구실한학기선배인 년지기로연구실생활뿐만아니라많은면에서도움을준진석용학형에게감사를드립니다. 오랫동안대학원생활을함께하면서친형들처럼정이든홍윤표, 김대선, 김준성선배님과만학도이지만재치와열정으로많은가르침을준주영이형에게도각별한우정과감사를드립니다. 이제박사과정으로서연구실의기둥역할을맡게될천재성과순수성을겸비한박기현학형과연구실의석사과정최고참으로써연구실을이끌어갈똑똑하고성실한남미영학형, 신입생박진수, 임상훈에게도감사를드리며앞으로남은연구실생활이보람되고, 학문에더욱정진하시어부디훌륭한논문으로대학원생활을마무리하기를기원합니다.

5 평생을같이할죽마고우승구 정아부부, 순고친구들에게도고마운마음을전하며우리의우정이끝까지계속되길기원합니다. 공대 YCV 선 후배들과동기들, 기계전자공학부 96 반친구들모두에게감사를드립니다. 모두들자신의분야에서최고의성취를거두길바랍니다. 저를낳아주시고언제나사랑으로길러주신부모님과아들처럼사랑으로감싸주신장인, 장모님께가장큰감사를드립니다. 그리고항상의지할수있고, 든든한후원자가되어준동생영훈, 영민이, 처남동훈이에게도고마운마음을전합니다. 언제나조카를믿어주시고물심양면으로도움을주신작은아버지들, 외삼촌들과이모님들, 처외할머니를비롯한처가식구들께도이자리를빌어감사드리고, 하시는모든일들잘되시길빕니다. 위에서언급하지는못했지만제가사랑하고존경하는모든분들과저를아껴주고믿어주는모든분들께감사드립니다. 마지막으로이세상누구보다도사랑하는아내성희와내년에태어날우리복덩이아가에게본논문을바칩니다. 8 년 월 김영준드림

6 Contents Lst of Fgures Lst of Tables v Abstract v Introducton. Motvaton Overvew Generalzed Cyclotomc Sequences of length p n 8. Generalzed Cyclotomc Sequences of length p n Lnear Complexty of Generalzed Cyclotomc Sequences Another Proof of Lnear Complexty of Generalzed Cyclotomc Sequences 9.4 Autocorrelaton of Generalzed Cyclotomc Sequences Example : Prme Cube Sequences (n = 3) Hardware Implementaton Remarks

7 3 Quaternary Cyclotomc Sequences of length p 3 3. Quaternary Cyclotomc Sequences of Length p Autocorrelaton of Quaternary Cyclotomc Sequences of Length p Remarks Non-bnary Power Resdue Sequences of length p Propertes of k-ary Power Resdue Sequences (PRS) Characterzaton of k-ary PRS s Crosscorrelaton of k-ary PRS s Non-bnary Sdel nkov Sequences of length p m and Ther Constant Multples Propertes of k-ary Sdel nkov Sequences Crosscorrelaton of k-ary Sdel nkov Sequence and Ther Constant Multples Comparson wth Welch Bound Concludng Remarks Summary Future Drectons Bblography 83 Abstract (n Korean) 88

8 Lst of Fgures. Implementaton of Generalzed Cyclotomc Sequences of Length p n Autocorrelaton and ts nverse of new Quaternary Cyclotomc Sequences of Length 38 when p = 9 ( 3 mod 8)

9 Lst of Tables 5. Comparson of crosscorrelaton and ts bound of Sdel nkov sequences 77 v

10 ABSTRACT Lnear Complexty and Correlaton of some Cyclotomc Sequences Young-Joon Km Department of Electrcal and Electronc Eng. The Graduate School Yonse Unversty In such systems as mltary communcaton system and stream cpher, t s desrable to use sequences havng large lnear complexty n order to present a hgh degree of protecton. Here s another and most mportant desgn crtera of sequences, called, correlaton. In code dvson multple access (CDMA) communcaton systems, the sequences used as sgnature codes are requred to have a good correlaton property. The correlatons are classfed nto two categores. One s the autocorrelaton property and an mpulse lke autocorrelaton s essental for synchronzaton. The other category s a crosscorrelaton of a par of or a number of sequences. For the purpose of usng sequences as sgnature codes n the multple access system, we need a set of sequences wth a low crosscorrelaton. In ths dssertaton, we frst revew the new generalzed cyclotomc sequences, whch s defned by Dng and Helleseth. Then we present the lnear complexty of these sev

11 quences of perod p n for any n >, where p s an odd prme number, by usng the degree of the mnmal polynomal of the sequences. We show that these sequences have lnear complexty that s of the order of the perod p n. We also compute the autocorrelaton of the sequences. Next, we defne new quaternary cyclotomc sequences of length p, where p s an odd prme. We compute the autocorrelaton of these sequences. In terms of magntude, the autocorrelaton of these sequences s not good. Nevertheless, t s an nterestng observaton that these sequences have the autocorrelatons wth at most 4 values. Furthermore, for the specfc cases of p ±3 (mod 8), we ntroduce how to utlze these NOT-GOOD sequences n applcatons. In 969, V. M. Sdel nkov ntroduced two knds of non-bnary sequences. One type sequences are known as power resdue sequences (PRS) and the other type sequences are known as Sdel nkov sequences. k-ary PRS are k-ary classcal cyclotomc sequences of length p. Frst we descrbe dstnct k-ary PRS s and show that two dstnct k-ary PRS s are related each other as constant multples. We prove that the perodc crosscorrelaton of k-ary PRS of perod p and ts constant multple s upper bounded by p +. Unlke k-ary PRS, k-ary Sdel nkov sequences are not cyclotomc sequences but these two have lots of smlarty. In the last, we prove that the crosscorrelaton of a k-ary Sdel nkov sequence of perod q and ts constant multple sequence s upper bounded by q+3, whch acheves approxmately Welch s crosscorrelaton bound, where q = p m and here p s an odd prme and m s a postve nteger. Key words : Cyclotomc sequences, lnear complexty, autocorrelaton, crosscorrelaton, Sdel nkov sequences v

12 Chapter Introducton. Motvaton Sequences or dscrete tme sgnals have been wdely used n radar, sonar, drect-sequence code dvson multple access (DS-CDMA) system, frequency-hoppng CDMA systems, synchronzaton, channel estmaton, cryptography, test, measurements and etc. The requred propertes of sequences are dependent upon the applcatons for those sequences. Dfferent applcatons requres dfferent sequence characterstcs []. But, some basc propertes are very mportant n many applcatons so that they should be treated carefully when desgnng of sequences. In general, the smplfcaton of system mplementaton requres the use of perodc sgnals. So, we restrct our attenton to perodc sequences n ths dssertaton. Here we summarze some useful propertes of perodc sequences.. Perod (or length) N : the number of sequence elements n a perod.. Famly sze M : the number of sequences n a sequence famly S. 3. Maxmum mbalance I max : the maxmum mbalance between numbers of df-

13 ferent symbols n a k-ary sequence per perod. It s desrable that the numbers of each symbol per perod are the same, or almost the same. 4. Lnear complexty L(s) s the length of the shortest lnear feedback shft regster (LFSR) that can reproduce the sequence {s(n)}. In other words, t s defned to be the least postve nteger l such that there are constants c =, c,, c l over a feld F satsfyng followng lnear recurson s() = c s( ) + c s( ) + + c l s( l) for all l < N. The polynomal c(x) = c + c x + + c l x l s called a mnmal polynomal of {s(n)}. The lnear complexty and mnmal polynomal of perodc sequences can be expressed smply as follows. Let {s(n)} be a sequence of perod N over a feld F, and S(x) = s() + s()x + + s(n )x N. It s well known that [] (a) the mmmal polynomal of {s(n)} s gven by c(x) = (x N )/ gcd(x N, S(x)) (b) the lnear complexty of {s(n)} s gven by L(s) = N deg(gcd(x N, S(x))) 5. Autocorrelaton functon C s (τ) of k-ary sequence {s(n)}: C s (τ) = N n= w s(n+τ) s(n) (.) where w s a complex prmtve k-th root of unty,.e., w = exp(j π k ).

14 6. Crosscorrelaton functon C u,v (τ) between two cyclcally dstnct k-ary sequences {u(n)} and {v(n)} of the same perod N : C u,v (τ) = N n= w u(n+τ) v(n) (.) where w s a complex prmtve k-th root of unty,.e., w = exp(j π k ). 7. Maxmum nontrval correlaton value C max : C max = max{c u amax, C u,v cmax u, v( u) S}} (.3) where C u amax s the maxmum out-of-phase autocorrelaton of a gven sequence {u(n)}, and C u,v cmax s the maxmum crosscorrelaton of two sequences u = {u(n)} and v = {v(n)},.e. C u amax = max τ<n {C u (τ)} (.4) C u,v cmax = max τ<n {C u,v (τ)} (.5) In usual, the requred length of sequences and famly sze are gven or fxed and balance condton should be satsfed at the most of applcatons. Hence, lnear complexty and (auto- and/or cross-) correlatons are man concern when we desgn a sequence and/or a set of sequences for specfc applcatons. In such systems as mltary communcaton system and stream cpher, t s desrable to use sequences havng large lnear complexty n order to reduce the mpact of jammng and ntercepton by an unfrendly recever [3 5]. Ths s mportant because t s possble to generate a replca of a perodc sequence {s(n)} usng the Berlekamp and Massey 3

15 algorthm from the observaton of L(s) consecutve symbols of the sequence, where L(s) s the lnear complexty of the sequence {s(n)} [6]. Autocorrelaton s used to dstngush a sequence from the tme shfted verson of tself. An mpulse lke autocorrelaton s essental for synchronzaton, rangng system, radar systems, and spread-spectrum communcatons. On the other hand, crosscorrelaton s used to dstngush a sequence from every other sequences n the set and tmeshfted verson of those. For the purpose of usng sequences as sgnature codes n the multple access system, we need a set of sequences wth a low crosscorrelaton. In an envronment n whch multple users share the resources (e.g., tme and frequency), t s more desrable to use the set of sequences havng as low crosscorrelaton values as possble wth each other [7 9]. In ths dssertaton, we focus on the study of lnear complexty and correlatons of some cyclotomc sequences whch we can analyze usng number theory and some algebra. Lterally, the word cyclotomy means crcle-dvson and refers to the problem of dvdng the crcumference of the unt crcle nto a gven number n of arcs of equal lengths. Next, we clarfy some notatons commonly used to refer cyclotomy. Defnton. Let n be a postve nteger and Zn be the multplcatve group of the nteger rng Z n. For a partton {D =,,, d } of Zn, f there exst elements g,, g d of Zn satsfyng D = g D for all where D s a multplcatve subgroup of Zn, the D are called generalzed cyclotomc classes of order d when n s composte, and classcal cyclotomc classes of order d when n s prme [ ]. 4

16 Defnton. Let (, j) d denote the number of solutons (x, y) of the equaton = y x, (x, y) D D j or equvalently (, j) d = (D + ) D j. These constants (, j) d are called (generalzed) cyclotomc numbers of order d. There have been lots of studes about cyclotomy wth respect to p or p or pq where p and q are dstnct odd prmes [3 7]. Before [8] was publshed, the generalzed cyclotomy had been known wth respect to p and pq, where p and q = p + are dstnct two prmes. In 998, Dng and Helleseth [8] ntroduced the new generalzed cyclotomy wth respect to p e pe t t, where p,, p t are dstnct odd prmes and e,, e t are postve ntegers. These new generalzed cyclotomy ncludes classcal cyclotomy as a specal case. In that paper, they also defned a balanced bnary sequence based on ther own generalzed cyclotomy wth respect to p e pe t t. For notatonal convenence, we wll refer these sequences as generalzed cyclotomc sequences of length p e pet t. These sequences ncludes the bnary quadratc resdue sequences also known as Legendre Sequences because these sequences can be understood as the generalzed cyclotomc sequences of length p [9]. In 998, C. Dng [3] presented some cyclotomc sequences of perod p whch are not balanced. They are slghtly dfferent from the generalzed cyclotomc sequences of length p. In [], t s shown that the lnear complexty of the sequence s not so good. In general, the lnear complexty of a sequence s consdered as good when t s of order of perod. Recently, n [5], Yan et al. computed the lnear complexty and 5

17 autocorrelaton of generalzed cyclotomc sequences of length p and n [], Km et al. computed the lnear complexty and autocorrelaton of generalzed cyclotomc sequences of length p 3. The results of [4, 5, ] show that the lnear complextes of generalzed cyclotomc sequences of length p, p and p 3 are qute good. In ths dssertaton we wll compute completely the lnear complexty and autocorrelaton of generalzed cyclotomc sequences of length p n for any postve nteger n. We do not lmt our nterest only to bnary sequences. We extend our researches to non-bnary sequences. In ths frst place, we wll construct a new quaternary cyclotomc sequences of length p and compute ther autocorrelaton. Up to now, most of studes on the quaternary sequences have been carred out wth respect to optmal or good correlaton [ 4]. Unlke those researches, we wll show that the autocorrelaton of proposed sequences s not good. Nevertheless, we wll exhbt how to utlze those sequences n applcatons. The next thng we do s the study of k-ary sequences. In 969, V. M. Sdelnkov showed that two knds of k-ary sequences have a good autocorrelaton. One sequences are well known as k-ary power resdue sequences (PRS) whch are exactly equvalent to k-ary Legengre sequences or generalzed cyclotomc sequences of order k of length p. The other knds of sequences are known as Sdel nkov sequences, whch are not cyclotomc sequneces, strctly speakng, but have lots of smlarty wth PRS. In ths dssertaton, we wll study crosscorrelaton of these two knds of sequences and ther constant multples as a another topc. 6

18 . Overvew In chapter, we revew the defnton of generalzed cyclotomc sequences wth respect to p n and then we compute the lnear complexty and autocorrelaton of these sequences. In addton, t s descrbed how the generalzed cyclotomc sequences w.r.t p n can be mplemented. In chapter 3, we construct a new quaternary cyclotomc sequences of perod p and compute the autocorrelaton of these quaternary sequences. In chapter 4, we descrbe dstnct k-ary PRS s and show that two dstnct k-ary PRS s are related each other as constant multples. We compute the crosscorrelaton of k-ary PRS of perod p and ts constant multple. In chapter 5 we compute the crosscorrelaton of a k-ary Sdel nkov sequence of perod q and ts constant multple sequence, where q = p m, here p s an odd prme and m s a postve nteger. Fnally all those results of ths dssertaton are summarzed and some dscussons follow. 7

19 Chapter Generalzed Cyclotomc Sequences of length p n In ths chapter, we compute the lnear complexty and autocorrelaton of generalzed cyclotomc sequences of length p n for any n. The rest of ths chapter s organzed as followng. In Secton., we revew the generalzed cyclotomc sequences of length p n. In Secton. and.3, we present the proof of the lnear complexty by two dfferent manners. In Secton.4, we compute the autocorrelaton of the sequences. In Secton.5, we present an example of prme cube sequences as a partal result. In Secton.6, we show how the generaton of these sequences can be easly mplemented n hardware. Fnally, n Secton.7, some remarks wll be followed.. Generalzed Cyclotomc Sequences of length p n In ths secton, we go over the generalzed cyclotomc sequences of length p n. We frst clarfy terms and notatons that are convenently used for the remander of ths chapter. Gven a prme p, let g be a prmtve root of p. Then t follows that g s also a prmtve root of p k, k (see [5]). By defnton, the order of g modulo p k s 8

20 p k p k for k n. Let D (pk ) = (g ) (mod p k ) the cyclc group generated by g modulo p k. Analogously, let D (pk ) = gd (pk ) (mod p k ) the coset of D (pk ) by g. It then follows that D (pk ) D (pk ) s the multplcatve group Z p k. In fact, Z p k = Z p k pz p k, for whch pz p k = {, p, p,, p k p}. It can be dentfed that Let C = Z p n = ( n k= ( n k= pn k D (pk ) p n k D (pk ) ) ) and C = ( n k= p n k D (pk ) ( n k= pn k D (pk ) ) {}. ) {}. In [8], Dng and Helleseth defned the generalzed cyclotomc sequences s(n) of perod p n as shown below: s() = {, f ( mod p n ) C, f ( mod p n ) C.. Lnear Complexty of Generalzed Cyclotomc Sequences Lemma. For a Zp n, =,, and k =,,, n ad (pk ) = D (pk ), f a D (pk ) D (pk ) + (mod ), f a ) D(pk. where all the computatons are done n modulo p k. Proof: It can be proved n the smlar way as [3]. Lemma. Let b be any nteger. Then D (pk ) +bp = D (pk ) and =,. (mod p k ) for k =,,, n Proof: See ( [], Lemma ). The proof can be easly extended for p k, k =,,, n. Lemma.3 For k =,,, n, we have 9

21 . (mod p k ) D (pk ) f and only f p (mod 4).. D (pk ) f and only f p ± (mod 8). Proof: Snce the proof of ) and ) are smlar, we wll show the proof of ) only. It s well known that (mod p) D (p) f and only f p (mod 4) [5]. Usng Lemma., we can show (mod p) D (p) mples (mod p ) D (p ) and (mod p 3 ) D (p3 ), and so on. The converse s obvous. Let m be the order of modulo p n and θ a prmtve p n -th root of unty n GF ( m ). Note that for j =,, n, = θ pn = (θ pn j )( + θ pn j + θ pn j + + θ (pj )p n j ). It follows that + θ pn j + θ pn j + + θ (pj )p n j = + By smple calculaton, we obtan p n k D (pk ) θ + p n k D (pk ) j k= p n k D (pk ) θ =. θ = for k =,, n. (.) and + p n D (p) θ + p n D (p) θ =. (.) Lemma.4 For k =,, n, p n k D (pk ) θ = p n k D (pk ) θ =. (.3)

22 Proof: The equalty of (.3) drectly comes from (.). We only have to show that t s zero. Let u k (θ) = p n k D (pk ) u k (θ) = p n k D (pk ) = θ for k =, 3,, n. Then, θ = p n k D (pk ) p n k D (pk ) ( from Lemma.3) p n k D (pk ) θ θ = u k (θ), f D (pn ) θ = u k (θ), f D (pn ) Thus, u k (θ)(u k (θ) ) =. That s u k (θ) {, }. Furthermore, we can show that u k (θ) =. When p (mod 4), D (pk ) = D (pk ) (mod p k ) for =,, and so (j, ) k D (pk ) (D (pk ) j + ) (mod p k ) = (D (pk ) ) D (pk ) j = D (pk ) j (D (pk ) ) = ( D (pk ) j ) ( D (pk ) + ) = (D (pk ) j ) (D (pk ) + ) = (, j) k ( from Lemma.3)

23 Note that θ pn k θ k s a prmtve p k -th root of unty n GF ( m ). u k (θ) = u k (θ) = p n k D (pk ) θ j p n k D (pk ) θ j (Here the ndces, j s are n Z p n) = θ = = D (pk ) k j D (pk ) θ j k (Hereafter the ndces, j s are n Z p k) D (pk ) p k b= + a D (pk ) + a D (pk ) j D (pk ) p k θ j k = D (pk ) (D (pk ) + a) θk a a= D (pk ) (D (pk ) + bp) θ bp k D (pk ) (D (pk ) + a) θ a k D (pk ) (D (pk ) + a) θ a k

24 = ( D (pk ) (D (pk ) + bp) = D (pk ) D (pk ) = ) + a D (pk ) (a D (pk ) + ) θk a a D (pk ) + = a D (pk ) a D (pk ) + a D (pk ) = (, ) k a D (pk ) (a D (pk ) + ) θ a k D (pk ) (D (pk ) + ) θ a k D (pk ) (D (pk ) + ) θ a k a D (pk ) θ a k + (, ) k = [(, ) k + (, ) k ]u k (θ) = a D (pk ) θ a k Assume now that p 3 (mod 4), then D pk (mod p k ). Note that (, ) k = D (pk ) (D (pk ) + ) = D (pk ) ( D (pk ) ) = D (pk ) (D (pk ) ) = (D (pk ) + ) D (pk ) = (, ) k (.4) 3

25 Then, u k (θ) = u k (θ) = p n k D (pk ) θ j p n k D (pk ) (Here the ndces s are n Z p n) = = D (pk ) θ k j D (pk ) θ j k θ j (Hereafter the ndces s are n Z p k) D (pk ) j D (pk ) θ j k p k = D (pk ) + D (pk ) (D (pk ) + a) θk a a= p k = D (pk ) + D (pk ) (D (pk ) + bp) θ bp k + a D (pk ) b= D (pk ) (D (pk ) + a) θ a k + a D (pk ) D (pk ) (D (pk ) + a) θ a k = D (pk ) D (pk ) ( D (pk ) (D (pk ) + bp) = D (pk ) D (pk ) = D (pk ) + (, ) k a D (pk ) by Lemma. and p k b= θ bp k = ) θk a + (, ) k a D (pk ) = [(, ) k + (, ) k ]u k (θ) = ( from eq.(.4)) θ a k 4

26 Defnton. Let the symbols be the same as before. Let t(θ) be defned as follows: t(θ) = p n D (p) Lemma.5 [] t(θ) {, } f and only f p ± (mod 8). θ. (.5) Defnton. Let d (pk ) (x) be defned as follows: for =, and k =,,, n. d (pk ) (x) = a p n k D (pk ) (x θ a ) (.6) Note that x pn can be factored as follows. x pn = (x ) n k= d (pk ) (x) n k= d (pk ) (x). Lemma.6 For k =,,, n and =,, d (pk ) (x) GF ()[x] f and only f p ± mod 8. Proof: Almost the same proof n [3] can be appled. Lemma.7 Let S(x) be the polynomal correspondng to one perod of the generalzed cyclotomc sequences of perod p n,.e., Then, S(x) = x = + ( + C S(θ a ) = where t(θ) = p n D (p) D (pn ) pd (pn ) + + p n + (mod ), f a = p n k + + t(θ), f a p n k D (pk ) p n D (p) )x GF ()[x]. and k =,,..., n p n k + t(θ), f a p n k D (pk ) and k =,,..., n θ. 5

27 Proof: For the case a =, we have S(θ a ) = S() = + n k= p n k D (pk ) = pn + (mod ). Note that for k =,,, n, and any postve nteger satsfyng n k + n, we have p p n k D (pk ) (mod p n ) = p n k+ D (pk ) (mod p n ) and p p n k D (pk ) (mod p n ) = p p n k+ D (pk ) (mod p n ). (.7) Obvously, for the case of such that n k + n, p p n k D (pk ) (mod p n ) = {} (mod p n ). (.8) When a p n k D (pk ) p n k D (pk ) for k =,,, n, we can let a = p n k b for some b Z p k = D (pk ) D (pk ). In ths case, we have S(θ a ) = + ( p n k D (pn ) = + p n k ( + ( p n D (pn k ) + p n k D (pk ) p n k+ D (pn ) p n k+ D (pk ) p n k+n D (p) + + )θ b p n D (p) }{{} + + k summatons ( from eq. (.7)) p n k+n D (p) }{{} ) θ b n k summatons (.9) ) θ b Note that from the equaton (.8) the latter n k summatons n the equaton (.9) becomes pn k. Now, we only have to consder the former k summatons of the equaton (.9). When b D (pk ), bd (pj ) (mod p j ) = D (pj ) (mod p j ) for j =,,, k. 6

28 Hence bp n j D (pj ) = p n j D (pj ) for =,. Lkewse, when b D (pk ), we have bp n j D (pj ) = p n j D (pj ) + (mod ) for =,. Hence, n the former k summatons of the equaton (.9), frst k summatons vansh from Lemma.4. The last summaton term whch only survves becomes t(θ) or t(θ) + from defnton of t(θ) and relaton of equaton (.), whch depends on whether b s n D (pk ) or D (pk ), respectvely. It completes the proof. Theorem. Let p be an odd prme. Then the lnear complexty of generalzed cyclotomc sequences of length p n s as follows: When n s even, When n s odd, L(s) = { p n +, f p ± mod 8 p n, f p ±3 mod 8. p n +, f p mod 8 p n, f p 3 mod 8 L(s) = p n, f p 5 mod 8 p n, f p 7 mod 8. Proof : From Lemma.7, whether the equaton S(x) = has a soluton depends on the values t(θ) and pl +, where l s a non negatve nteger value. Note that p l = { p k = (z + ) k = k ( k ) = (z) + 4kz + (mod 4) p k+ = (z + ) k+ = k+ = ( k+ ) (z) + 4kz + z + p (mod 4) (.) Consder the case of n even frst. Wth Lemma.5 and the equaton (.), Lemma.7 can be computed as followng: 7

29 If p mod 8, t(θ) {, }. So,, f a = S(θ a ) = + t(θ), f a p n k D (pk ) and k =,,, n t(θ), f a p n k D (pk ) and k =,,, n Therefore, m(x) = { x pn (x ) n ) gcd(x pn, S(x)) = k= d(pk (x), f t(θ) = (x ) n ) k= d(pk (x), f t(θ) = If p 3 mod 8, t(θ) {, }, and, f a = Therefore, S(θ a ) = t(θ), f a p n k D (pk ) and k = odd + t(θ), f a p n k D (pk ) and k = even + t(θ), f a p n k D (pk ) and k = odd t(θ), f a p n k D (pk ) and k = even. m(x) = x pn. If p 5 mod 8, t(θ) {, }, and, f a = S(θ a ) = + t(θ), f a p n k D (pk ) and k =,,, n t(θ), f a p n k D (pk ) and k =,,, n. Therefore, m(x) = x pn. If p 7 mod 8, from Lemmas.5 and.7, t(θ) {, } and, f a = S(θ a ) = t(θ), f a p n k D (pk ) and k = odd + t(θ), f a p n k D (pk ) and k = even + t(θ), f a p n k D (pk ) and k = odd t(θ), f a p n k D (pk ) and k = even. 8

30 Therefore, m(x) = { (x ) n k= d (pk ) (x) n k= d (pk ) (x), f t(θ) = (x ) n k= d (pk ) (x) n k= d (pk ) (x), f t(θ) =. Now consder the case of n odd. Smlarly, we have the followng: [ (x ) n ) k= d(pk (x), f t(θ) = (x ) f p mod 8 n ) k= d(pk (x), f t(θ) =. (x pn )/(x ) f p 3 mod 8 m(x) = x pn f p 5 mod 8 n+ d(pk ) k= (x) n ) k= d(pk (x), f t(θ) = (x) f p 7 mod 8 n+ (x), f t(θ) =. n k= d(pk ) k= d(pk ) Fnally, we can obtan the lnear complexty of generalzed cyclotomc sequences of perod p n by computng the degree of mnmal polynomal..3 Another Proof of Lnear Complexty of Generalzed Cyclotomc Sequences In ths secton, we present another proof of lnear complexty of generalzed cyclotomc sequences of perod p n by computng Hammng weght of dscrete Fourer transform of the sequences. For ths, we ntroduce the well known Blahut s theorem. Theorem. [6] Let {s(n)} be a sequence over F q wth perod N where gcd(n, q) =. Then the lnear complexty of the sequence {s(n)} s equal to the Hammng weght of the dscrete Fourer transform (DFT) {A t } of {s(n)} defned by A t N = s()α t, t =,,..., N, where Hammng weght of {A t } s defned as {t A t, t < N} and α s a prmtve N-th root of unty n some extenson feld of F q. 9

31 Now, we are gong to apply ths Blahut s theorem to generalzed cyclotomc sequences to get lnear complexty. In contrast to the noton n the prevous secton, let C be the set wthout. Then we have C C = Z p n \ {}. The lnear complexty s just the number of DFT values A t = + C α t (We have to add a here snce we deleted from C ). Now we have to dstngush some cases. Here we only consder the case n odd. If p, 5 (mod 8), then A = p n = s() = + C = pn + (mod ) s odd and conversely ths value s even f p 3, 7 (mod 8). So from now on, we just look at A t for t < p n. Note that C α t + C α t =. If C, equvalently p ± (mod 8), then C and hence C are fxed by the automorphsm g g (by Lemma.), whch mples that the DFT values are fxed by the Frobenus automorphsm ζ ζ, where ζ s n F m [7]. Ths mples A t {, } for t < p n f p ± (mod 8). Ths s because A t = + C α t and A t = + C α t = + C α t = + C α t = A t. Furthermore, we obtan that exactly one of C α t and C α t s, the other value s. Snce the lnear complextes of the sequences correspondng to C and C are the same, ths mples that there are (p n )/ nontrval DFT values dfferent from, and we may conclude as above. On the other hand, f p ±3 (mod 8), A t = C α t = C α t = + A t. So, n ths case, A t can be nether nor. Ths gves the lnear complextes p n for p 5 (mod 8) and p n for p 3 (mod 8) (ths dfference comes from the DFT value A ). Remark. In fact, computng the degree of mnmal polynomal and countng the

32 Hammng weght of DFT of gven sequences are equvalent way to compute the lnear complexty [8]. Nevertheless, the proof of ths secton does not need mnmal polynomal so that all the complcated computatons n Lemma.4 and.7 are not needed..4 Autocorrelaton of Generalzed Cyclotomc Sequences The perodc autocorrelaton of a bnary sequence {s(n)} of perod N s defned by C s (τ) = L n= ( ) s(n+τ) s(n), where τ < L To compute the autocorrelaton of generalzed cyclotomc sequences of perod p n, we use the generalzed cyclotomc numbers of order wth respect to p k for k defned n [] (, j) p k = (D (pk ) + ) D (pk ) j, j =,, and k =,,, n. Lemma.8 [8] If p 3 (mod 4), then (, ) p k = (, ) p k = (, ) p k = pk (p 3) 4 If p (mod 4), then (, ) p k = (, ) p k = (, ) p k = pk (p ) 4, and (, ) p k = pk (p + ). 4, and (, ) p k = pk (p 5). 4 To prove the autocorrelaton, we need the followng Lemma.9 and.. Lemma.9 Let the symbols be the same as before. Then, ( ) {} p n k D (pk ) + τ =, f τ p n k D (pk ) and p (mod 4), f τ p n k D (pk ) and p 3 (mod 4), othewse.

33 Proof: Only when τ p n k D (pk ), zero s an element of p n k D (pk ) + τ. If p (mod 4), from Lemma.3, τ p n k D (pk ) f and only f τ p n k D (pk ). Lkewse, f p 3 (mod 4), from Lemma.3, τ p n k D (pk ) p n k D (pk ) + (mod ). It completes the proof. f and only f τ Lemma. Let l,k (τ) be defned as follows. l,k (τ) = ( p n l D (pl ) p n k D (pk ) + τ), for l, k {,,, n}. Then,. When l = k, (, ) p k, f τ p n k D (pk ) k,k (τ) = (, ) p k, f τ p n k D (pk ), othewse.. When l < k, p l p l, f τ p n k D (pk ) and p (mod 4) l,k (τ) = p l p l, f τ p n k D (pk ) and p 3 (mod 4), othewse. 3. When l > k, Proof: l,k (τ) = { p k p k, f τ p n l D (pl ), othewse.. When l = k,

34 (a) If τ p n k D (pk ) p n k D (pk ), we can let τ = p n k a for some a Z p k. Then, k,k (τ) = p n k D (pk ) ( = = D (pk ) ( ) p n k D (pk ) + p n k a ) + a (mod p k ) D (pk ) { (, )p k, f τ p n k D (pk ) (, ) p k, f τ p n k D (pk ). (mod p n ) ( from Lemma.) (b) For τ p n u D (pu ) p n u D (pu ) such that u k, let τ = p n u b for some b Zp u. Then, If u > k, t mples p n u < p n k, so k,k (τ) = p n k D (pk ) = ( p n k D (pk ) p n k D (pk ) p n u ) + p n u b (mod p n ) ( ) p u k D (pk ) + b (mod p u ) (mod p n ) (Denote Λ p u k D (pk ) + b (mod p u ), from Lemma., Λ Z p u) = p n k D (pk ) p n u Λ = =. If u < k, t mples p n u > p n k, so ( ) k,k (τ) = p n k D (pk ) p n k D (pk ) + p n u b (mod p n ) ( ) = p n k D (pk ) p n k D (pk ) + p k u b (mod p n ) = p n k D (pk ) p n k D (pk ) = =. ( from Lemma.). When l < k, (a) If τ p n k D (pk ) p n k D (pk ), we can let τ = p n k a for some a Z p k. 3

35 Then, ( ) l,k (τ) = p n l D (pl ) p n k D (pk ) + p n k a (mod p n ) ( ) = p n l D (pl ) p n k a p n k D (pk ) (mod p n ) ( ) = p k l D (pl ) a D (pk ) (mod p k ). When p (mod 4), from Lemma.3, a D (pk ) D (pk ) f and only f a. From Lemma., f a D (pk ), for any element x of p k l D pl a (mod p k ), x becomes an element of D (pk ). Hence, p k l D pl a D (pk ). It follows that l,k (τ) = p k l D (pl ) a = pl p l, f a D (pk ), f a D (pk ). When p 3 (mod 4), f a D (pk ), any element x of p k l D pl a (mod p k ) becomes an element of D (pk ) + (mod ). Hence, pk l D pl a D (pk ) + (mod ). It follows that l,k (τ) = { (p, f a D k ) p l p l, f a D (pk ). (b) For τ p n u D (pu ) p n u D (pu ) such that u k, let τ = p n u b for some b Zp u. Then, ( ) When u > k, note that p n k D (pk ) + τ = p n u p u k D (pk ) + b p n u Zp u. Hence, l,k(τ) = =. ( ) When u < k, note that p n k D (pk ) + τ = p n k D (pk ) + p k u b p n k D (pk ). Hence, l,k (τ) = p n l D (pl ) p n k D (pk ) = =. = 4

36 3. When l > k, the proof can be easly done wth smlar way to the case of l < k. Theorem.3 Let p be an odd prme. Then the autocorrelaton profle of the generalzed cyclotomc sequences of length p n s as follows:. p (mod 4) p n, τ = (mod p n ) C s (τ) = p n p u p u, τ p n u D (pu ) for u =,,, n p n p u p u +, τ p n u D (pu ) for u =,,, n. p 3 (mod 4) C s (τ) = Proof: Defne { p n, τ = (mod p n ) p n p u p u, τ p n u D (pu ) p n u D (pu ) for u =,,, n. d s (, j; τ) = C (C j + τ), τ < L,, j =,. Snce C s (τ) = p n 4d s (, ; τ), we need to calculate d s (, ; τ). Note that d s (, ; τ) = C (C + τ) = = n k= n + + n k= {} (p n k D (pk ) + τ) + n l= k=l+ n l l= k= C (p n k D (pk ) + τ) n k= p n l D (pl ) (p n k D (pk ) + τ) p n k D (pk ) (p n k D (pk ) + τ) p n l D (pl ) (p n k D (pk ) + τ). (.) 5

37 Note that the values of the frst, the second, the thrd, and the fourth summatons n (.) are determned from Lemma.9 and.. In other words, d s (, ; τ) = n k= n + {} (p n k D (pk ) + τ) + n l= k=l+ Consder the case p (mod 4) frst. l,k (τ) + n k,k (τ) k= n l l,k (τ). l= k= If τ p n u D (pu ), then from Lemma.9 and., the equaton (.) becomes u d s (, ; τ) = + (, ) p u + l,u (τ) + = + pu + p u, for u =,,, n. 4 l= If τ p n u D (pu ), then the equaton (.) becomes d s (, ; τ) = + pu + p u, for u =,,, n. 4 In case of p 3 (mod 4), the computaton of d s (, ; τ) can be done n a smlar way. Fnally, we have that d s (, ; τ) = pu +p u 4, f τ p n u Zp u for u =,,..., n. Snce C s (τ) = p n 4d s (, ; τ), t completes the proof..5 Example : Prme Cube Sequences (n = 3) An example of bnary prme cube sequence of length 7 (when p=3, g=) s shown below: D (3) = ( ) (mod 3) = {} D (3) = D (3) (mod 3) = {} D (9) = ( ) (mod 9) = {, 4, 7} D (9) = D (9) (mod 9) = {, 5, 8} D (7) = ( ) (mod 7) = {, 4, 7,, 3, 6, 9,, 5} D (7) = D (7) (mod 7) = {, 5, 8,, 4, 7,, 3, 6} 6

38 C = D (7) 3D (9) 9D (3) = {, 3, 4, 7, 9,,, 3, 6, 9,,, 5} C = D (7) 3D (9) 9D (3) {} = {,, 5, 6, 8,, 4, 5, 7, 8,, 3, 4, 6} Therefore, a bnary prme cube sequence {s(n)} of length 7 are gven as follows: n s(n) n s(n) Corollary. [] Let p be an odd prme and {s(n)} be a prme cube sequence of perod p 3. Then the lnear complexty L(s) of {s(n)} s as follows: p 3 +, f p mod 8 p 3, f p 3 mod 8 L(s) = p 3, f p 5 mod 8 p 3, f p 7 mod 8. Corollary. [] Let p be an odd prme. Then the autocorrelaton profle of the bnary prme cube sequence of perod p 3 s as follows:. p (mod 4) p 3, τ = (mod p 3 ) p 3 p 3, p 3 p +, C s (τ) = τ p D (p) τ p D (p) p 3 p p, τ pd (p ) p 3 p p +, τ pd (p ) p, τ D (p3 ) p +, τ D (p3 ) 7

39 . p 3 (mod 4) C s (τ) = p 3, τ = (mod p 3 ) p 3 p, τ p D (p) p D (p) p 3 p p, τ pd (p ) pd (p ) p, τ D (p3 ) D (p3 )..6 Hardware Implementaton Generalzed cyclotomc sequences of order two of perod p n s an output sequence of the generator descrbed n Fg., where cyclc counter counts the numbers {,,, p n } cyclcally and Modular Exponentaton Computaton s a chp for fast modular exponentaton wth respect to p. The chp computes mod p (mod [ ] p p). a gcd(a,p n ) It s clear that f a D (pk ), a mod p D (p) for =,, k {,, n}. For each a < p n, let us consder the followng : [ { } p ] a gcd(a, p n ) mod p mod p. (.) When a =, the equaton (.) release. If a p n k D (pk ) for k =,,, n, { } p then gcd(a, p n ) = p n k a. So mod p = {D (pk ) gcd(a,p n ) (mod p)} p = { } {D (p) } p p = ( ) (mod p). Therefore, the output s(a) = mod p a gcd(a,p n ) of ths mplementaton s just generalzed cyclotomc sequences of perod p n, where mples modulo operaton..7 Remarks Legendre sequences (n=), prme square sequences (n=), and prme cube sequences (n=3) are subclasses of generalzed cyclotomc sequences of length p n [4, 5,, 9]. 8

40 n {,,, P } A GCD( A, P n ) mod P P mod P Fgure.: Implementaton of Generalzed Cyclotomc Sequences of Length p n. We get the lnear complexty by two dfferent manners. One s computng degree of mnmal polynomals and the other s countng the nonzero dscrete Fourer transform values. We can descrbe the lnear complexty completely and show that the lnear complexty of these sequences are of the order of perod p n, whch s consdered as qute good. On the other hand, the autocorrelaton s not good n terms of the absolute value. We presented the result for lnear complexty of generalzed cyclotomc sequences of length p n n [3]. Recently, Yan, L, and Xao determne the lnear complexty of generalzed cyclotomc sequences of length p m [3]. After we fnshed our presentaton, we eventually found out that Yan, L, and Xao had done the same thng. The methodologes of two ndependent works are almost same n that they are based on computng degree of mnmal polynomals. Nonetheless, there s our own contrbuton that we also compute the autocorrelaton of these sequences completely. 9

41 Chapter 3 Quaternary Cyclotomc Sequences of length p 3. Quaternary Cyclotomc Sequences of Length p Let p be an odd prme and g a prmtve root of p. Snce ether g or g + p s odd modulo p and both of them are prmtve roots of p, smply we wll assume that g s an odd nteger. Then t s well known that g s also a prmtve root of p [5]. Defne D (p) = (g ), D (p) = gd (p), D (p) = (g ), D (p) = gd (p) where (g ) denotes the subgroup of Z p and Z p generated by g, respectvely. Then Z p = {, p} D (p) D (p) D (p) D (p). Defne a quaternary sequence {s(n)} of length p as follows [3]: 3

42 s(n) =, f n = (mod p), f n = p (mod p), f n D (p), f n D (p), f n D (p) 3, f n D (p). (3.) Hereafter, we wll call ths sequence as a quaternary cyclotomc sequence of length p. Example 3. An example of a quaternary cyclotomc sequence of length (when p=5, g=3) s shown below: D (5) = {3, 3 4 } = {, 4} = {, 8}, D (5) = {3, 3 3 } = {, 3} = {4, 6}, D () = {3, 3 4 } = {, 9}, D () = {3, 3 3 } = {3, 7}. Therefore, a quaternary cyclotomc sequence {s(n)} of length are gven as follows: n s(n) Autocorrelaton of Quaternary Cyclotomc Sequences of Length p The perodc autocorrelaton of a quaternary sequence {u(n)} of perod N s defned by C u (τ) = N n= w u(n+τ) u(n) 3

43 where w = exp(j π 4 ) s a complex prmtve quadratc root of unty. Snce the quaternary cyclotomc sequence of length p s defned n the smlar way wth the bnary cyclotomc sequence of length p, t s desrable to revew how to construct the bnary cyclotomc sequence of length p and ther propertes. Defnton 3. Let p be an odd prme, then the sequence {t(n)} of length p defned as { (p), f n C D (p) t(n) =, f n C (p) {} D (p). (3.) s called the bnary cyclotomc sequence of length p, whch s well known as the bnary quardratc resdue sequence or Legendre sequence of length p. Defne d t (, j; τ) = C (p) (C (p) j + τ), τ Z p,, j =,. For the notatonal smplcty, we recall the cyclotomc numbers of order wth respect to p defned by (, j) p = (D (p) + ) D (p) j. Lemma 3. For a Z p, For b Z p, { { (p) (p) D =, f a D(p), ad (p) D, f a = D(p) ad (p) D (p) { (p) bd (p) D, f b D (p) = D (p), f b D (p), bd (p) = D (p), f a D(p), f a D(p) { (p) D, f b D (p) D (p), f b D (p) Proof: Frst part of ths lemma s ponted out n [3] and t s a specal case of Lemma.. Second part can be proved n the same way as [3]... 3

44 Lemma 3. (mod p) D (p) f and only f (mod p) D (p). Proof: Obvous. Lemma 3.3 When p (mod 4), then we have d t (, ; τ) = When p 3 (mod 4), then we have { p+3 4 p 4, f τ D(p), f τ D(p). d t (, ; τ) = p +, f τ D(p) D (p) 4. Proof: It can be easly proved from Lemma.8. For a gven quaternary cyclotomc sequence of length p, defne s () (n) = s(n), n p, and s () (n) = s(n + ), n p. That s, they are defned as follows, respectvely: s () (n) = s () (n) =, f n (mod p), f n D (p) n D (p) 3, f n D (p) n D (p),, f n + p (mod p), f n + D (p), f n + D (p). Lemma 3.4 Let p be an odd prme. Then the autocorrelaton of the quaternary cyclo- 33

45 tomc sequence {s(n)} of length p can be expressed as follows: C s (τ) = p = = w s(+τ) s() C s ()(k) + C s ()(k), f τ k (mod p) and k p C s () s ()(k) + C s () s ()(k ), f τ k (mod p) and k p where w = exp(j π 4 ) = j s a complex prmtve quadratc root of unty and C s () s (j)(t) = p z= ws() (z+t) s (j) (z) s a crosscorrelaton of s () and s (j) for (, j) {(, ), (, )}. Proof: Obvous. Wthout loss of generalty, we can let s () (n) = t(n) +, n p. Snce s () (n) s a quaternary sequence, all the plus and mnus operaton should be done n modulo 4 throughout ths chapter even f {t(n)} seems lke a bnary sequence. Lemma 3.5 The autocorrelaton of a quaternary sequence {s () (n)} of length p s as follows:. When p (mod 4),. When p 3 (mod 4), C s ()(k) = C s ()(k) = { p 7 p 3 { p 5, f k D(p), f k D(p). (3.3) + j, f k D (p) p 5 j, f k D (p). (3.4) 34

46 Proof: The autocorrelaton of {s () (n)} can be expressed as follows: C s ()(k) = p j s() (+k) s () () = = j s() (k) + j s() ( k) + j s() (+k) s () () Z p, k (Here, the computaton n the power of j s done n modulo 4) = j t(k)+ + j t( k) j t(k) t() j t() t( k) + j t(+k) t() (3.5) Z p If t( + k) = and t() =, then + k C (p) and C (p). It gves us C(p) k and C (p). Hence, there are such s as many as { (C(p) k) C (p) } = C (p) (C(p) +k) = d t(, ; k). Lkewse, there are d t (, ; k), d t (, ; k) and d t (, ; k) number of pars (t( + k), t()) = (, ), (, ) and (, ), respectvely. Therefore the equaton (3.5) becomes as follows: C s ()(k) = j t(k)+ + j t( k) j t(k) j t( k) + d t (, ; k)j +d t (, ; k)j +d t (, ; k)j +d t (, ; k)j. Snce d t (, ; k)+d t (, ; k)+d t (, ; k)+d t (, ; k) = p and d t (, ; k) = d t (, ; k), t becomes C s ()(k)=j t(k)+ +j t( k) j t(k) j t( k) +p d t (, ; k). When p (mod 4), k D (p) f and only f k D (p), =,. So k D (p) mples t( k) = t(k) =, for =,. Smlarly, when p 3 (mod 4), k D (p) f 35

47 and only f k D (p) + mod, =,. So k D(p) mples t(k) = and t( k) = + (mod ), for =,. By Lemma.3 and 3.3, we complete the proof. The autocorrelaton of {s () (n)} can be calculated n the smlar way. For smplcty, we ntroduce the followng {, } sequence of length p. v(n) = { (p), f n + C D (p), f n + C (p) {p} D (p). (3.6) Although {v(n)} s {, } sequence, the symbol or s not over Z but over Z 4. Defne d v (, j; τ) = C (p) (C (p) j + τ), τ Z p,, j =,. where all the operatons are computed n the modulo p. Lemma 3.6 The autocorrelaton of a quaternary sequence {s () (n)} of length p s as follows:. When p (mod 8),. When p 3 (mod 8), 3. When p 5 (mod 8), C s ()(k) = C s ()(k) = C s ()(k) = { p 7 p 3 { p 5, f k D(p), f k D(p). (3.7) j, f k D (p) p 5 + j, f k D (p). (3.8) { p 3 p 7, f k D(p), f k D(p). (3.9) 36

48 4. When p 7 (mod 8), C s ()(k) = { p 5 + j, f k D (p) p 5 j, f k D (p). (3.) Proof: Wthout loss of generalty, we can let s () (n) = v(n), n p, n p. Then, by smlar procedure wth the case of {s () (n)}, the autocorrelaton of {s () (n)} can be expressed as follows: p v( C s ()(k) = j +k) p v( + j k) p v( j +k) p v( j k) + p d v (, ; k). Note that d v (, ; k) = C (p) (C (p) + k) = {p} (D (p) + k) + D (p) (D (p) + k) Snce {p} (D (p) + k) = {} (D (p) + k) (mod p), by Lemma.3,. When p, 3 (mod 8), we have {p} (D (p) + k) =. When p 5, 7 (mod 8), we have {p} (D (p) + k) = {, f k D (p), f k D (p). {, f k D (p), f k D (p). Snce D (p) (D (p) + k) = D (p) (D (p) + k) (mod p) = k D (p) ( k D (p) + ) and by Lemmas 3. and.3, 37

49 . When p ± (mod 8), we have D (p) (D (p) + k) =. When p ±3 (mod 8), we have D (p) (D (p) + k) = { (, )p, f k D (p) (, ) p, f k D (p). { (, )p, f k D (p) (, ) p, f k D (p). Note that ( p + k) + = p + k D (p) means k D (p) and ( p k) + = p k D (p) we have means k D (p) for =, and the converse s also true. Therefore,. When p (mod 8), k D (p) f and only f k D (p) f and only f k D (p), =,. So k D (p) mples v( p + k) = v( p k) = for =,.. When p 3 (mod 8), k D (p) D (p), =, So k D (p) for =,. f and only f k D (p) + f and only f k mples v( p p +k) = + (mod ) and v( k) = 3. When p 5 (mod 8), k D (p) f and only f k D (p) + f and only f k D (p) +, =, So k D(p) mples v( p + k) = v( p k) = + (mod ) for =,. 4. When p 7 (mod 8), k D (p) f and only f k D (p) f and only f k D (p) +, =, So k D(p) mples v( p + k) = and v( p k) = + (mod ) for =,. 38

50 Combnng all of these computaton, we can get to the (3.7),(3.8),(3.9) and (3.). Next, we are gong to consder the autocorrelaton of {s(n)} when the tme shft τ s odd. As s mentoned n Lemma 3.4, when τ = k, k p, C s (τ) = C s () s ()(k) + C s () s ()(k ). Defne and d t,v (, j; k) = C (p) d v,t (, j; k) = C (p) (C (p) j + k ), k Z p,, j =,, (C (p) j + k ), k Z p,, j =,, where all the operatons are computed n the modulo p. Lemma 3.7 The crosscorrelaton of two quaternary sequences {s () (n)} and {s () (n)} of length p s as follows:. When p (mod 8), p, f τ =k =p (mod p) C s () s ()(k)= p+7, f τ = k D (p) p+3, f τ = k D (p).. When p 3 (mod 8),, f τ =k =p (mod p) C s () s ()(k)= p+, f τ = k D (p) p+5, f τ = k D (p). 3. When p 5 (mod 8),, f τ =k =p (mod p) C s () s ()(k)= p+3 + j, f τ =k D (p) p+3 j, f τ =k D (p). (3.) (3.) (3.3) 39

51 4. When p 7 (mod 8), p, f τ = k =p (mod p) C s () s ()(k)= p+5 j, f τ =k D (p) p+5 + j, f τ =k D (p). (3.4) Proof: To begn wth, we are gong to consder C s () s ()(k). When k = p (mod p), C s () s ()(p + p ) = = j s() (+ p+ ) s() () = j s() () s () ( p ) p t()+ v( j ) + Z p j t(+k)+ v() = Z p j t(+k)+ v() When τ = k D (p) D (p), p C s () s ()(k) = = j s() (+k) s () () = j v( k) p t( +j +k) j 3 v( k) p t( j +k)+ + Z p j t(+k)+ v() By smlar procedure wth the case of autocorrelaton of {s () (n)} and {s () (n)}, t becomes p C s () s ()(k)=j v( k) t( +j +k) j 3 v( k) p t( j +k)+ p + d t,v (, ; k). (3.5) Note that d t,v (, ; k)= C (p) (C (p) + k ) = {} (D (p) +k ) + D (p) (D(p) +k ) 4

52 If p (mod 4), by Lemmas.3 and 3., a D (p) the other hand, f p 3 (mod 4), then a D (p) Therefore, f and only f a D (p). On f and only f a D (p) + mod.. When p (mod 4), f k p (mod p) {} (D (p) + k ) =, f k D (p), f k D (p).. When p 3 (mod 4), f k p (mod p) {} (D (p) + k ) =, f k D (p), f k D (p). Snce D (p) (D(p) +k ) = D (p) (D(p) +τ)(mod p) = τ D (p) (τ D (p) +) and by Lemmas 3. and.3,. When p ± (mod 8), we have D (p) (D (p) + k ) =. When p ±3 (mod 8), we have D (p) (D (p), f τ =k = p (, ) p, f τ =k D (p) + k ) = (, ) p, f τ =k D (p). p, f τ =k = p (, ) p, f τ =k D (p) (, ) p, f τ =k D (p). Now, we are ready to compute (3.5). When p (mod 4), ( k) + = τ D (p) f and only f τ D (p). So, n ths case, v( k) = f τ D (p) for =,. 4

53 On the other hand, when p 3 (mod 4), ( k) + = τ D (p) f and only f τ D (p) + mod. So, n ths case, v( k) = + (mod ) f τ D(p) for =,. When p ±(mod 8), f τ D (p), then p + τ (mod p) s an element of ether D (p) (mod p) or D (p) + mod (mod p). Snce t s vald when we apply modulo p reducton, τ (mod p) s an element of ether D (p) = D (p) (mod p) or D (p) + mod = D (p) + mod (mod p). Because τ D(p) mples τ D (p) (mod p), p + τ (mod p) = ( p + k) D (p) (mod p). Therefore, t( p +k) =. On the other hand, f p ±3 (mod 8), t( p +k)=+(mod ) when τ D(p). Combnng all of these computaton, we can get to the (3.),(3.),(3.3) and (3.4). Lemma 3.8 The crosscorrelaton of two quaternary sequences {s () (n)} and {s () (n)} of length p s as follows:. When p (mod 8), p, f τ =k =p (mod p) p+7 C s () s ()(k )=, f τ =k D (p) p+3, f τ =k D (p).. When p 3 (mod 8),, f τ = p (mod p) p+5 C s () s ()(k )=, f τ D (p) p+, f τ D (p). 3. When p 5 (mod 8),, f τ = p (mod p) p+3 C s () s ()(k )= j,f τ D (p) p+3 + j,f τ D (p). (3.6) (3.7) (3.8) 4

54 4. When p 7 (mod 8), p, f τ = p (mod p) p+5 C s () s ()(k )= j,f τ D (p) p+5 + j,f τ D (p). Proof: When k = p (mod p), C s () s ()(p + p ) = = j s() (+ p ) s() () = j s() ( p ) s()() p v( j ) t() (3.9) + Z p j v(+k ) t() = Z p j v(+k ) t(). When τ = k D (p) D (p), by smlar procedure wth C s () s ()(k), the crosscorrelaton C s () s ()(k ) becomes Note that p C s () s ()(k )=jv(k ) t( +j k+) j v(k ) 3 p t( j k+) p + d v,t (, ; k). (3.) d v,t (, ; k)= C (p) (C (p) + k ) = {p} (D (p) +k ) + D(p) (D (p) +k ) If p (mod 4), by Lemmas.3 and 3., a D (p) the other hand, f p 3 (mod 4), then a D (p) Therefore, we have f and only f a D (p). On f and only f a D (p) + mod.. When p, 3 (mod 8),, f k p (mod p) {p} (D (p) + k ) =, f k D (p), f k D (p), 43

55 . When p 5, 7 (mod 8),, f k p (mod p) {p} (D (p) + k ) =, f k D (p), f k D (p). Snce D (p) (D (p) + k ) = D (p) (D (p) + τ) (mod p) = τ D (p) (τ D (p) + ) and by Lemmas 3. and.3, we have. When p ± (mod 8), D (p). When p ±3 (mod 8), (D (p) +k ), f τ =k =p (mod p) = (, ) p, f τ =k D (p), (, ) p, f τ =k D (p) D (p) Now, we are ready to compute (3.). (D (p) +k ) p, f τ =k =p (mod p) = (, ) p, f τ =k D (p). (, ) p, f τ =k D (p) Snce (k ) + = τ, τ D (p) mples v(k ) = for =,. When p ± (mod 8), f τ D (p), then p τ (mod p) s an element of ether D (p) (mod p) or D (p) + mod (mod p). Snce t s vald when we apply modulo p reducton, τ (mod p) s an element of ether D (p) (mod p). Because τ D (p) ( p k + ) D (p) = D (p) (mod p) or D (p) + mod = D(p) + mod mples τ D (p) (mod p), p τ (mod p) = (mod p). Therefore, t( p k + ) =. On the other hand, f p ±3 (mod 8), t( p k + ) = + (mod ) when τ D (p). 44

56 Combnng all of these computaton, we can get to the (3.6),(3.7),(3.8) and (3.9). Theorem 3. [Man Result] Let p be an odd prme. Then the autocorrelaton of the quaternary sequence of length p defned at (3.) s as follows:. If p (mod 8), C s (τ) = p, f τ = (mod p) p, f τ = p (mod p) p 7, p 3, p + 7, p + 3, f τ D (p) f τ D (p) f τ D (p) f τ D (p).. If p ±3 (mod 8), C s (τ) = p, f τ = (mod p), f τ = p (mod p) p 5, f τ D (p) D (p) p + 3, f τ D (p) D (p). 3. If p 7 (mod 8), C s (τ) = p, f τ = (mod p) p, f τ = p (mod p) p 5 + 4j, p 5 4j, p + 5 4j, p j, f τ D (p) f τ D (p) f τ D (p) f τ D (p). Proof: When τ (mod p), obvously C s (τ) = p. When τ k (mod p) and k p, we obtan what we want by combnng (3.3),(3.4),(3.7),(3.8),(3.9) 45

57 and (3.). Lkewse, when τ k (mod p) and k p, we also obtan what we want to prove by combnng (3.),(3.),(3.3),(3.4),(3.6),(3.7),(3.8) and (3.9). Next, we are gong to consder assgnng another symbol to the each sets D (p), D (p), D (p) and D (p). Let us ntroduce a vector d = (d, d, d, d 3 ), where d {,,, 3}, < 4. Each d, d, d and d 3 correspond to the symbol assgned to D (p), D (p), D (p) and D (p) respectvely. Snce t s also possble to change the symbols at and p poston, we ntroduce another vector e = (e, e ), where e {,,, 3}, =,. For the followng pars of vector (d, e), ((,,, 3), (, )), ((,,, 3), (, )), ((,, 3, ),(, )),((,, 3, ),(, )),((,,, 3),(, )),((,,, 3),(, )),((,, 3, ),(, )), and ((,, 3, ), (, )), we can verfy that the number of dstnct absolute values of autocorrelaton s up to 4 and for any τ {, p}, C s (τ) p +. Furthermore, when e {(, ), (, )} s fxed, t s easly checked that d = (,,, 3) and d = (,, 3, ) gve us the same autocorrelaton profle from the pont of vew of absolute value. Lkewse d = (,,, 3) and d = (,, 3, ) gve us the same autocorrelaton profle. 3.3 Remarks Ths chapter proposes a smple method of constructng quaternary cyclotomc sequences of perod p for any odd prme p, and calculates ther autocorrelaton functons. In general, t s preferred to have low out-of-phase correlaton [] [33], and the sequences proposed n ths chapter can be sad to be not good n ths sense. One contrbuton of ths chapter s the exact calculaton of such correlaton values, though t s not good. 46

58 38 C s C s Fgure 3.: Autocorrelaton and ts nverse of new Quaternary Cyclotomc Sequences of Length 38 when p = 9 ( 3 mod 8). One nterestng observaton s the case where p ±3 (mod 8). Observe n ths case that the correlaton value C s (τ = p) s at the phase shft p and C s (τ p) s of the order p at all other phase shfts ncludng the zero shft. Snce the perod s p, t s almost the same as havng deal autocorrelaton wth the peak at the phase shft p f we take the correlaton measure as /C s (τ). For nstance, see Fg 3.. Ths property of the proposed sequences can be equally useful when 4 ary modulaton s used and the deal peak correlaton measure must be exploted. 47

59 Chapter 4 Non-bnary Power Resdue Sequences of length p In 969, V. M. Sdelnkov showed that two knds of k-ary sequences have a good autocorrelaton. One sequences are well known as k-ary power resdue sequences (PRS) and the other knds of sequences are known as Sdel nkov sequences. The latter sequences wll be studed n the next chapter. In ths chapter we characterze k-ary PRS of perod p. k-ary PRS s are exactly k- ary verson of Legengre sequences, whch means that they are equvalnet to generalzed cyclotomc sequences of order k of length p. Then, we generate a set H wth sze φ(k) whch conssts of k-ary PRS s, and calculate the crosscorrelatons of the sequences n the set [34, 35]. Defnton 4. ( [36] [37]) Let p be an odd prme and k be a dvsor of p. Let T = (p )/k and µ be a prmtve root mod p. The nonzero ntegers mod p can be parttoned nto k cosets C, k, where C s the set of the k-th power resdues mod p, and C = µ C for >. 48