A New Binary Sequence Family with Low Correlation and Large Size

Transcription

1 1 Nw inary qunc Family wit ow orrlation an arg iz Nam ul u an uang ong, mbr, bstract For o an an intgr wit, a nw amily o binary squncs o prio is construct For a givn, as maximum corrlation "#%$, amily siz )(, an maximum linar span +*,/1 2 imilarly, a nw amily o o binary squncs o prio 4 is also prsnt or vn 5 7 an an intgr wit 8, wr maximum corrlation, amily siz, an maximum linar span ar ) : /(, (, an +*;)<1 2, rspctivly nw amily (or ) contains oztas an umar s construction [1] (or Uaya s [1]) as a subst i squncs ar xclu rom bot constructions s a goo caniat wit low corrlation an larg amily siz, t amily is iscuss in tail by analyzing its istribution o corrlation valus nx rms Family o binary squncs, larg amily siz, linar span, squncs wit low corrlation NOUON n co ivision multipl accss () communication systms, psuonois squncs ar assign to istinct usrs in a common cannl at t sam tim [18] o istinguis ac usr an minimiz mutual intrrnc, w must av low crosscorrlation btwn istinct squncs Furtrmor, w must also av low autocorrlation btwn a squnc an its tim sit vrsion in orr to acquir t accurat pas inormation at t rcivr capacity o t systm can b incras by obtaining an incras numbr o squncs wic support a largr numbr o istinct usrs onsquntly, a amily o squncs wit low corrlation an larg amily siz plays important rols in communication systms construction o a amily o binary squncs o is bas on t combination o a binary squnc an its cimations suc tat t rsulting squncs av low corrlation aciving t wll known lowr bouns riv by Wlc [2], ilnikov [17], an vnstin [1] For o, ol squncs [] constitut on o t amilis wit optimal corrlation aciving t ilnikov boun amily o asami (small st) squncs [] givs squncs wit optimal corrlation aciving t Wlc s lowr boun or vn n orr to obtain binary squncs wit larg linar span as wll as low corrlation, oztas an umar construct a nw amily o binary squncs or o, so call ollik squncs [1] t as t sam prio, amily siz, an maximum corrlation as tos o t amily o ol squncs, is work was support by N rant PN 2277 autors ar wit t partmnt o lctrical an omputr nginring, Univrsity o Watrloo, Watrloo, ON N2 1, anaa (mail: nyyu@ngmailuwatrlooca, ggong@calliopuwatrlooca) but largr linar span giving bttr potntial cryptograpic proprty imilarly, Uaya construct a nw amily o binary squncs wit low corrlation but larg linar span or vn [1] n [7], im an No gnraliz ts two constructions at t pric o t cras o maximum linar span an t incras o maximum corrlation otr known approacs ar summariz as ollows n [2], ang t al sow tat a binary cyclic co bas on trtrm squncs [1] as ivvalu nonzro wigt istribution, wic is intical to t ual co o t tripl rror corrcting co From ts cyclic cos, quivalnt amilis o binary squncs wit sixvalu corrlation o maximum F %, amily siz "J, an maximum linar span ", can b construct n [12], it is sown tat N linar binary cos bcom nonlinar cyclic cos atr a propr prmutation From ts cos, anbag, umar, an llst prsnt a nw gnraliz construction o binary squnc amilis in [1], incluing rock an lsart otals squncs in [5] n tis papr, a nw amily OQP+U o binary squncs an an intgr Z For a givn, maximum corrlation o prio + is construct or o W wit squncs in O P is [ "#%$ an its amily siz is + imilarly, a nw amily OQ]@^U o binary squncs o + is also prsnt or vn W an an intgr wit a b, wr maximum corrlation an amily siz ar c rspctivly maximum an minimum linar spans o squncs in bot O P an OQ]@ /; <;/i ar Qg an Qg, rspctivly s incrass, w obtain a nw amily O P ^U (or Oc]@^U ) o xponntially incras amily siz wit maximum corrlation linarly incras rom its optimal j valu inc t amily O P (or Oc]+U ) contains O P (or Oc]@^ ) as a subst, it contains O P (or Oc]@ ) as a subst r, O P is t amily o ollik squncs construct by oztas an umar, an OQ]@ is t on construct by Uaya, wr squncs ar xclu in bot constructions For a spciic application, w can coos a propr valu an t corrsponing amily O P ^U (or OQ]@ ) For xampl, a small valu o can b cosn i low corrlation is mor crucial tan larg amily siz in t application larg amily siz is mor important, on t otr an, w can allows coos a larg valu o lxibility u to our nw squnc amily to av aaptiv amily siz an maximum corrlation or practical applications Furtrmor, our nw squnc amily as goo potntial cryptograpic proprty wit larg linar span implmntation o OkP@ (or O ] ^U ) is xtrmly asy by summing linar back sit

2 @ U r Q 2 rgistr (F) outputs amily OQP" wit maximum corrlation " an amily siz "J is a goo xampl or compromis btwn corrlation an amily siz is papr is organiz as ollows n ction, w giv som prliminaris on concpts an initions o cos an squncs lso, w rviw a wigt istribution o a linar cyclic subco o t scon orr ullr co [11], wic will b us to invstigat a corrlation istribution o squncs in our nw amily n ction, w prsnt nw amilis OQP@ an O ] ^U o binary squncs o or o an vn, rspctivly, an analyz corrlation an linar span o ac amily n ction, t istribution o corrlation valus o squncs in OFP" is riv in trms o t wigt istribution o a linar cyclic subco o t scon orr ullr co n ction, an xampl o squncs in O P is givn an implmntation bas on Fs is iscuss oncluing rmarks an som obsrvations ar givn in ction PN ollowing notations will b us trougout tis papr is t init il wit lmnts an, t multiplicativ group o i is a vctor spac ovr wit a st o all binary tupls t b positiv intgrs an trac unction rom to is not by, i, ") # N $U%$ ( *) + is simply not as + i t contxt is clar asic concpts (a) oolan unction " t, + b a vctor in i wit / an,c a unction rom to n, t unction %,c taking on valus or 1 is call a oolan unction [11] oolan unction consists o a sum o all possibl proucts o 21 s wit coicints or [4], i, %,c " N ;:<:<: 21 "" 21 :<:<: 1 wr maximum valu o wit nonzro :<:<: 1 is call t gr o t oolan unction %,c (1) is call an algbraic normal orm o a oolan unction [4] (b) ullr cos For, t rt orr ullr () is in by a st o all vctors "7i o lngt %@ o lngt givn by [11] %@ [ :<:<: 1 " wr 1 o " 5@ vctors o lngt or 7i, "" " 7 (1) (2) ar basis, is t vctor wos t lmnt is t prouct o t t lmnts rspctivly n, +J i is all zro or all on vctor o lngt, an i is always t ual o t xtn amming co wic is also obtain rom a ylvstrtyp aamar matrix [11] n (2), t t lmnt o a cowor in + i is givn by a oolan unction + " ; N wit gr o at most, wr " ; N ar t t lmnts rspctivly w rmov t t componnts corrsponing to ; " O P, tn w obtain a punctur co + i or, wr ac cowor as lngt + 4 (c) rac rprsntation o a binary prioic squnc t O b a st o all binary squncs o prio Q " an b a st o all unctions rom to For any unction, can b rprsnt as [4] * +[ %W NW wr Q" is a cost lar o a cyclotomic cost moulo, an ZN is t siz o t cyclotomic cost containing Q For any squnc []J WO, tr xists ^ suc tat ] %_ N`c " wr _ is a primitiv lmnt o n, is call a trac rprsntation o [ linar span o t squnc [ is qual to a %c, or quivalntly t gr o t sortst b linar back sit rgistrs tat can gnrat [ () Wigt an xponntial sum o a binary cowor or squnc t [5] " b a binary cowor o lngt 5] 5] Q, or a binary squnc o prio Q numbr o s in t cowor is call a (amming) wigt larly, t total sum o aitiv caractrs o a cowor or squnc [ wit wigt is givn by Q Qa an [ is rprsnt by a trac rprsntation +, i, ] _ or a primitiv lmnt _ o, tn t xponntial sum o + ovr is givn by l [ kj g g"i wit + nc, t xponntial sum o + as ontoon corrsponnc wit t wigt o a cowor or squnc givn by () orrlation o binary squncs t [ an m b binary squncs o prio Q corrlation o [ an m is in by npo +r/[ q Zs Jt ` wr r is a pas sit o t squnc m an t inics ar ruc moulo Q m is cyclically quivalnt to [, i,

3 U U g 8 m ];5]Q q " 5]7 n n o, +r< is t autocorrlation o [, o +r/ n o or sort Otrwis, +r/ is t crosscorrlation o [ q an m W may writ ]J _ or + wit a givn r an a primitiv lmnt _ o n, npo +r/[ q wit xponntial sum o () inary signal st nc, Zs t no q [ g"i kj +r< can b prsnt by t For binary cyclically istinct squncs o prio Q, " )g )g )g wit b, lt O g " g an n # $ # 1 $ r/ or any rab QJ k; b n i W larly, is maximum o all i, n wr r nontrivial auto an crosscorrlations o t squncs in O n st O will b call a +Q" signal st or amily o squncs, wr is t st siz or amily siz, an n is t maximum corrlation magnitu o O Wigt istribution o a linar subco o n tis papr, w consir a componntwis sum o a pair o binary squncs, wr t sum is quivalnt to a cowor o a linar cyclic subco o t punctur scon orr ullr co us, w can apply t wigt istribution o t subco or t istribution o corrlation valus o t squncs (a) Wigt istribution o a cowor st wit rank + For o a, w consir a cowor givn by +[ 4 1 N`( 7i () wr ac wit is an lmnt in For on t otr an, w consir a cowor givn by 1 ( wr ac i or, an Wit " rspct to a basis o, a * is an xpansion o wit or all pplying tis xpansion to () or (4), w s tat + a U ** is quivalnt to a oolan unction o gr lss tan or qual to 2, an it may b writtn as ollows: %,c " N,",$# %^,$#p, N )( wr is an +* binary uppr triangular matrix, % is a binary vctor in, an ( " o lngt Obviously, is trmin by s or nonzro, an % by Wil, + " N runs troug ac nonzro binary tupl in i, %,c proucs ac lmnt o a cowor o lngt +i in 7i quivalntly, + orms a cowor in 7i or ( (4) (5) For a givn nonzro, it is wll known tat t wigt istribution o a st o cowors o a quaratic oolan unction %,c or all % is trmin by a rank o a symplctic matrix, # [11] quivalntly, w can consir a symplctic orm j + /k + + / / associat [5] W list t wit + or givn s wit ollowing act rgaring t istribution o xponntial sums o + Fact 1 (orm 2 in [5]): t b givn suc tat at last on s o + in () or (4) is nonzro or, wr 21 For an intgr wit `, i j + / as a rank, or quivalntly j +/" as +/i54 solutions in or all /, tn t xponntial sum /74 o taks on valus o an or all, an its istribution is givn by g"i j wit + g : +/ 4 _+/ tims tims tims (b) Wigt istribution o a linar subco wit multipl ranks For a st o istinct nonzro s, w can consir a st o cowors givn by,c in (5) or all % quivalntly, w can consir a st o cowors givn by or istinct sts o s suc tat at last on is nonzro in ac st o s or 4 n, may av istinct multipl ranks ac o wic corrspons to ac st o givn onsquntly, t xponntial sum o can tak on valus +/ 4 o an or ac possibl rom Fact 1 urtr constituts a linar subco or t sts o s s, w may us t wigt istribution o t subco in orr to invstigat t istribution o xponntial sums o Nxt, w spciy t known wigt istributions o two linar cyclic subcos o 7i or o, wic will b us in a latr sction or trmining t corrlation istribution o a nw amily o squncs For ]Z in an wit ;< +Z7ik or o, a linar cyclic subco n@ givn by +[]* Q or as t istribution o wigts an corrsponing xponntial sums in abl, wr n / For ] in or o, a linar cyclic subco givn by # + %]* 2@ or ( as t istribution o wigts an corrsponing xponntial sums in abl n bot abls, t xponntial sum mans a g"i j g wit +"[ r ar svral n irnt ways to stablis t valiity o wigt istribution o, s [11], or xampl, s [] an [11] For tos o n # NW F OF N QUN W Z n tis sction, w prsnt nw amilis o binary squncs wit larg amily sizs as wll as larg linar spans or o an vn

4 @ 8 J ) g 4 W UON OF N Wigt xponntial um istribution / 2 / / / " 2 / 2 / / 2 < onstruction o OQP@ or o W UON OF # N $% () Wigt xponntial um istribution /+*, " $ 7$/ % 7$, 1 / % $ /+* 7$, 7 $ 8 7$ $U, %;: / onstruction 1: For o W an an intgr wit 4, a amily O P U o binary squncs is in by "" O P [ ) % 5@ %@ N@ " wr is a binary squnc o prio +: wit _ or a primitiv lmnt _ o, wr +, t trac rprsntation o, is givn by or ( +[ :<:<: #%$ F@ + U paramtrs o a nw signal st OFP"U ar trmin by t ollowing torm orm 1: For o 4 squncs o an an intgr () wit, t amily OQP@^U cyclically istinct binary corrlation o squncs is valu an maximum corrlation is 5 ^#%$ ror, O P constituts a ^#%$ signal st n orr to sow orm 1, i, to trmin amily siz an corrlation o O P ^U, w n t ollowing lmmas mma 1: ll squncs in O P U ar cyclically istinct us, t amily siz o O P ^U Proo: onsir a tim sit vrsion o a squnc in OcP^U rprsnt by ;J ;J ; " or J J 7 5J", an t is intical to anotr squnc o (), i, or all i an only i J "@ NJ or : kb4* (7) an or From ;< + 7$ [ or o, is a uniqu solution aciving in (7), it only givs a trivial solution or b us, squncs in O P ^U rprsnt by or in wit b, ar cyclically istinct crosscorrlation o two squncs an in O P is givn by n P Q r/k j g"i wr (8) + wr " J [ )J wit kb4 an a primitiv lmnt o n otr wors, t sum o an a r sit vrsion o can b consir as a cowor givn c b () 4 _ wr _ is by in (8) us, w n to invstigat t xponntial sum o + or t corrlation o a pair o squncs n t ollowing, w classiy t xponntial sum in trms o valus o 8 s as 1 or 4 n tis cas, + is corrspons to a trivial xponntial sum corrsponing to inpas autocorrlation o a squnc us, its xponntial sum as 2 an or ` n tis cas, + nc, w av t xponntial sum or any rom t ortogonality o t trac unction [4] as t last on 8 or : 4 n tis cas, + is quivalnt to a quaratic oolan unction us, w n to invstigat t numbr o roots o its symplctic orm j + /

5 ) W W ] 5 in orr to apply Fact 1 or trmining t istribution o xponntial sums o + + mma 2: For o an an intgr wit, lt s o + in (8) b givn suc tat at last on is nonzro or : 4 n, t symplctic orm j +/" associat wit as at most U%7 roots in or all / Proo: For givn s, t symplctic orm j +/" associat wit + is givn by j +/"F + / / / / (1) / $ $ + / + wr Na $ $ j /k or all / i an only i N From (), _+[ $ $ 8+ U [ $ ; $ [ ; $ $ 8+ U [ $ ; $ [ ; [ $ ; $ [ ; J $ [ $ $ [ ; U / [ ; Not tat < [ t n, w av wr +[ :<:<: #%$U $ N < < ; (11) + /%; [ ; + ) [ ) (12) $ i $ ogtr wit (11), + + ; (1) :<:<: #%$ [ ; 7 $ $ ) (14) For _+, w av to count t numbr o solutions in t quation + ; k (15) :<:<: #%$ s in an in or givn Nxt, w vriy tat :<:<: #%$ + is not a constant polynomial o (i, a trivial polynomial), or rom (12), an or rom () n, at last on is nonzro or bcaus / at last on is nonzro, on t otr an, cannot b zro altoug may b zro or all : ror, :<:<: #%$ + is a polynomial o wit at last on nonzro coicint o or b kb4 For t nontrivial polynomial :<:<: #%$ +, t quation in (15) can b ivi into our classs i) [ an N :<:<: #%$U N, ii) [ an ; [ :<:<: #%$ +, W iii) [ an ; k :<:<: #%$ iv) +[ an ; [ :<:<: #%$ us, t ltan si o (15) can b prsnt by W [ :<:<: #%$U 7 [ ] $ $ F or ] J )" n, #%$ kl :<:<: #%$U #%$U ] ] #%$U [ #%$ #%$ #%$ $ #%$1$ #%$ #%$ an #%$U +[ W N ) us, solutions or W #%$U +[ ar all solutions or W N W is qual to tat or W #%$ maximum gr o W #%$U + is " g, W #%$ N most " g solutions, t solutions o W #%$ N ar isjoint or irnt ] J )" numbr o solutions o (15) is at most U % g, on t otr an, t solutions o W #%$ + ar isjoint or irnt ] J 8 solutions o (15) is at most " 7 " g manwil, a possibl numbr o roots o j +/" is +/i54, an vic vrsa ror, t numbr o solutions or inc t as at wr, so t total, so t total numbr o From Fact 1, is a positiv o intgr or o For any valu o in, tror, t maximum numbr o solutions o j /k is "%7 From mma 2, w s tat j + /4 as +/i54 solutions or an intgr wr is a positiv o intgr lss tan or qual to W From Fact 1, or all, t "<74 xponntial sum o + can tak on valus o an or an intgr wr is a positiv o intgr lss tan or qual to t tis point, w n to sow tat it +<74 can tak on valus o an or all s suc tat

6 g n orr to o so, w n t ollowing act rom [11] (Not w sligtly cang t rprsntation o t rsult in [11]) is vry positiv o intgr lss tan or qual to Fact 2 ( [11], pag 454): t b a st o symplctic orms o (1) For som ix intgr wit 1, assum tat t rank o vry nonzro orm in is at last n, t maximum siz o is givn by +/ % or o ;/ g g or vn ) n t ollowing, w consir a spciic linariz polynomial an invstigat a rank o t symplctic orm corrsponing to t polynomial mma : For an intgr, consir a symplctic orm +/"[ / + wr a linariz polynomial + is givn by 7 k $ $ wr can b any lmnt in For o, t rank o +/" is at last an vry possibl rank or 5W occurs at last onc wn runs troug For vn, on t otr an, t rank is at last an vry possibl rank or a occurs at last onc wn runs troug Proo: Not tat t o cas in mma is implicitly known rom orm 1 an orollary 17 o aptr 15 in [11] r, w rprouc it or compltnss t b an intgr wit y w av /W / + wr a U $ $ n, t maximum # gr o + $ $ is g an tus t rank o / is at last ( is o, t rank is at last bcaus it soul b vn) For o, assum tat t rank nvr occurs or all s wit : À, an tus t rank o / is at last n, rom Fact 2, t maximum siz o a st +/ o suc + / s is g owvr, its actual siz g wn s wit : run troug, j wic is gratr tan ror, t rank o occurs at last onc wn s run troug For vn, 4 i t rank nvr occurs, tn t minimum rank is imilarly, t maximum siz o a st o suc ;/ / s is g g, wic is smallr tan its actual siz +/ g ror, t rank o occurs at last onc wn s run troug inc + contains vry + s or wn s wit 5 run troug, t assrtion o mma ollows Using mma, w av t ollowing rsult + mma 4: For o an an intgr wit, lt 8 s o + in (8) b givn suc tat at last on is nonzro or 4, an n, t xponntial / 4 sum o + can tak on valus o an or an intgr wr to is vry positiv o intgr lss tan or qual, tn t linariz polynomial givn in (14) (i, it is intical to (1)) is o t orm o + in mma From mma, t xponntial / 4 sum o or is qual to or or all s " suc tat us, w can say tat Proo: For, by J t xponntial sum o taks on valus o an or vry intgr suc tat J "" +/74 <74 Now, it suics to sow tat t xponntial sum taks on valus o at last onc wn Or quivalntly, w n to sow t symplctic orm j + / in (1) as a rank o at last onc or som s ssum tat tis rank nvr occurs or all s n, t rank o j / is at last 5 From Fact 2, t maximum siz o a st o suc j / s is +/ g owvr, rom () an (1), its actual siz is +U, wic is gratr tan ror, t rank o occurs at last onc wn s run troug is complts t proo o mma 4 ombining t cass 1, 2, an, w av t ollowing rsult on t corrlation o squncs in OFP" mma 5: corrlation o binary squncs in O P is n valu an maximum corrlation 5 "#%$ Proo: For a trac rprsntation + o ac squnc in O P, w can consir t xponntial sum o + in (8) n cass 1 an 2, t xponntial sum as " an valus For all s in cas, rom mma 4, t xponntial /74 sum taks on an or ac intgr suc tat 5 7 " us, t xponntial sum taks on nonzro istinct valus ncluing an, tror, t 4 ovrall xponntial sum is valu quivalntly, t 5 corrlation o squncs in OQP" is valu inc maximum valu or is trmin by rom mma 4, n N@/ 4 ^#%$ Proo o orm 1 rsults ollow irctly rom mmas 1 an 5 mark 1: ++F F@, + 5` (1) wic rprsnts t ollik quncs introuc by oztas an umar [1] From mma 4, a positiv o intgr lss tan is only <, so nc, Finally, ollik squncs givn by (1) av ourvalu + corrlation, or or xampl 1: +[ F@ or an o / an U (17) From mma 4, is a positiv o intgr lss tan, so an us, From mma 5, squncs givn by (17) av

7 8 J 7 +N % + ;, t corrsponing + in (8) constituts a sixvalu corrlation, or For squncs rprsnt by an linar cyclic subco wit iv nonzro istinct wigts i w assum tat can b any lmnt in n act, t corrlation istribution o O P can b riv rom abl tails will b iscuss in ction xampl 2:, or / +F@ + / / (18) an o imilarly, an From mma 5, squncs givn by (18) av igtvalu corrlation, or + " / or onstruction o O ] or onstruction 2: For vn an an intgr wit b, a amily O ] ^U o binary squncs is in by Oc]@[ ) %@ 5@ " wr is a binary squnc o prio +: wit _ or a primitiv lmnt _ o, wr +, t trac rprsntation o, is givn by +F@ F@ (1) + or ( + orm 2: For vn a an an intgr wit : b, t amily O ] ^U cyclically istinct binary squncs o prio +a corrlation o squncs is valu an maximum corrlation " ror, + [ O ] ^U constituts a signal st n orr to prov orm 2, w n t ollowing lmmas inc tir proos ar similar to tos or O P U, w omit t tails mma : ll squncs in OQ]@^U ar cyclically istinct nc, t amily siz o OQ]+U + Proo: imilar to t proo o mma 1, ; or all in i an only i (7) is aciv ;< +, tn is not a actor o ;< 5 +5 sinc ; < 5 or any intgr nc, i is a solution o, tn it cannot b a solution o $U anwil, w av at last two quations o, or in (7) bcaus b or vn us, (7) as a uniqu solution givn by nc, all squncs in O ] U ar cyclically istinct o invstigat t corrlation o squncs in O ] ^U, w n to consir + givn by N (2) wr [ " J kb4 an c b 4 (21) )J wit or all 5, t xponntial sum o + as a valu o Otrwis, w av to consir t numbr o solutions o j /k in orr to riv t istribution o xponntial sums o + in an an intgr wit b, lt s o in (2) b givn suc tat at last on is n, t symplctic orm j + / associat wit + as at most "% roots in ( or all / mma 7: For vn nonzro or a Proo: W av j /k / $ $ 8+ / J ) us, j +/"[ or all / i an only i Using t sam approac as w i in t proo o mma 2, w obtain intical to (1) an t quation (15) Following t sam stps as in t proo o mma 2, w can riv tat t numbr o solutions o j /k is at most mma 8: For vn an an intgr wit b, lt 8 s o in (2) b givn suc tat at last on is nonzro or 5, an n, t xponntial <74 sum o + can tak on valus o an or an intgr j wr is zro or vry positiv vn intgr lss tan or qual to nc, t corrlation o binary squncs in Oc]@^U is n valu an maximum corrlation is [ Proo: For vn, w s tat wil ac wit kb4 runs troug, ac o in (1) runs troug From mma, tror, w obtain tat t xponntial /74 sum o + taks on valus o an or vry intgr " suc tat Nxt, w will sow tat t cass an also occur all s o in (1) ar zro, tn [ as at most our solutions, i,, an wr t last tr solutions ar vali only i us, w av a uniqu solution or suc tat nc, t cas occurs at last onc pplying Fact 2, w also obtain tat t xponntial / 4 sum o + can tak on valus o at last onc suc tat in t similar way to t proo o mma 4

8 8 ror, t xponntial sum o + taks on valus o +/74 an or an intgr wr J " ncluing n an +, t corrlation is valu Furtrmor, [4 or Proo o orm 2 rsults ollow irctly rom mma an mma 8 mark [ F@, + ( (22) wic rprsnts t squncs construct by Uaya [1] From mma 8, an nc, an Finally, t squncs givn by (22) as sixvalu corrlation, + or or Q xampl :, F@ + + or ( an vn From mma 8, an tus, Q squncs givn by (2) blongs to Q 8 or %@ (2) nc, t corrlation o mark : ontrary to o, in (2) os not constitut a linar cyclic subco bcaus wit kb may not run troug all lmnts in ror, w soul us irnt mtos rom o to invstigat t corrlation istribution in O ], wic is a problm w ar working on inar spans o O P ^U an Oc]@^U linar spans o binary squncs in OFP" an O ] ^U ar trmin by t numbr o s wit Qb4 orm : n t amily O P ^U + s in ar qual to t squnc rprsnt by F@ o t squnc n, an tr ar aving linar span (or Oc]@ ), consir a U b t linar span k [ 7 squncs in O P (or Oc]@^U ) U From tis rsult, t linar span o squncs in OcP" (or O ] ^U ) an its istribution ar sown in abl Proo: Firstly, consir t linar span o squncs in O P ^U n onstruction 1, a squnc rprsnt by + as a total Q trac trms an ac trac trm as t linar span s o t squnc ar qual to, it as Q + nonzro trac trms an t corrsponing linar span o t squnc is givn by [ k ` ) N PN N NU OF OPONN QUN N (O 1 ) Numbr o squncs +*,/1 2 ( +*,/1 2 ( % 7 ( +*, 2 $ 2( % 7 ( 2 +*; 2 2 (/1 s ar s ar nonzro, t numbr o + corrsponing squncs givn by + is pplying tis rsult to ac wit, w obtain t linar span ^U o binary squncs in OFP@ wit t istribution sown in abl Using t similar approac to o cas, w s tat t linar span o squncs in O ] ^U is t sam as OQP^U orollary 1: maximum an minimum linar spans o squncs in O P an OQ]@ / ar Qg /;<% an Qg, rspctivly omparison o amilis o binary squncs n abl, w giv som comparisons o t amilis O P ^U an Oc]@^U wit wll known amilis o binary squncs wit low corrlation n abl, w inclu a amily o binary squncs construct rom ang t al s invstigation o a binary cyclic co bas on trtrm squncs [2] o, ac squnc in t amily is rprsnt + N Q or ( in On t otr an, a amily o squncs in by F@ +$7 rom t ual o t tripl n rror corrcting co, as t sam prio, amily siz,, an maximum linar span as tos o ang t al s or %@ in or o [11] n trms o prios an maximum linar spans, nw amilis O P ^U an Oc]+U ar intical to t amilis o ollik squncs an Uaya s, rspctivly t t xpns o maximum corrlation n 5, owvr, w av largr amily sizs in O P ^U an OQ])^U tan in tos amilis asically, w obtain O P an OQ]@ wit xponntially incras amily sizs by t linar incras o n 5 rom optimal corrlation Furtrmor, w can coos an t corrsponing amily O P ^U (or Oc]@^U ) or its spciic application For xampl, i low corrlation is mor crucial tan a larg amily siz in t application, tn a small valu o is cosn Otrwis, w coos a larg valu o in orr to gt a larg amily siz onsiring t lxibility u to, OFP@^U an O ] av largr amily sizs tan any otr amilis o binary squncs in abl n OcP"U an O ] ^U, squncs ar not inclu or incrasing tir minimum linar spans us, ty av largr minimum linar spans tan any otr binary squnc amilis in abl xcpt or bnt squncs, wic will b goo proprty or potntial cryptograpic applications

9 J J 8 g J OPON OF F OF N QUN W OW OON Family o quncs Prio Family iz inar pan (aximum, inimum) ol [] o asami (mall t) [] $ 2 vn asami (arg t) [] / ( 2 vn or nt [14] 1 vn oztas an umar 2 o Uaya [1] 2 vn ang t al (triplt) [2] [11] 2 5 o otaus [15] 2 5 o rock [1] / 2 o lsartotals [1] 2 / 2 o Nw Family 1 2 +*; 2 $ o Nw Family *; 2 $ vn Nw Family 1 / 2 ( 2 +*; 2 2 (<1 o Nw Family 1 2 ( 2 +*; 2 2 (<1 vn $ " # "$ 2 % / ( < [] linar spans o rock an lsartotals squncs ar not givn in [1] UON OF OON U OF O P n many applications, a amily o binary squncs wit bot low corrlation an larg amily siz is rquir us, w n to consir a amily o squncs wic can b a goo compromis btwn corrlation an amily siz s a goo caniat o suc a amily, w xtnsivly iscuss a nw squnc amily OQP" by analyzing its corrlation istribution For o a +, a amily OQP o binary squncs is givn by O P N ) F@ wr prio + wit _ o, " _ is givn by is a binary squnc o or a primitiv lmnt k F@ F@ + or ( From orm 1, w know tat t amily O P as J cyclically istinct binary squncs o prio " corrlation o squncs in O P is sixvalu an its maximum is onsquntly, OQP+ constituts J c % signal st is ata o OQP is list in abl Nxt, w will invstigat t istribution o corrlation valus o squncs in OQP" corrlation o a pair o squncs in OQP@ is riv rom t xponntial sum o [JF@ ; JF@ [ J o %@ )J J ), an an Wit Q ] O@ or 4 ", as a orm + ]* (24) or ] an pning on s, t xponntial sum o can b classii into two xclusiv cass o acilitat t analysis, assum tat can b any lmnt in as 1 c or n tis cas, + ]* ) constituts a linar cyclic subco o i or ]Z us, t istribution o t xponntial sums o + is intical to abl wit < as 2 t last on or 4 n tis cas, t istribution o t xponntial sums o + ollows rom t ollowing lmma mma : For givn s wit at last on nonzro, + in (24) as ivvalu xponntial sum or all ]Z in, an t istribution is g"i wr "4 : "4 < + 4 "4 4 /, an "54 tims tims (25) ar intgrs suc tat Proo: is ix to an lmnt in, t xponntial sum o ollows t istribution in Fact 1 pning on

10 g _ 8 : 1 t rank o its symplctic orm inc, it is obvious tat t symplctic orm o + wit at last on nonzro may av a pair o istinct ranks an pning on 4 4, wr an, rspctivly For a subst o, assum tat i, + as t rank an otrwis, as t rank, wr Q an runs troug all lmnts in, t ovrall istribution o t xponntial sums bcoms (25) atr summing up t istributions or an y combining bot cass 1 an 2, w s tat as sixvalu xponntial sum or any in n, + quivalntly constituts a linar cyclic subco o 7i wit iv nonzro istinct wigts or ], an s in wit t assumption tat ac trmin by can b any lmnt in nc, ac cowor in t subco as an its imnsion is u to ]Z, an in inc t subco contains t ual o t oubl rror corrcting co, t co contains t ual o our subco ror, it is clar tat t minimum istanc o t ual o our subco is at last anwil, w also know tat i t numbr o nonzro istinct wigts o a co is lss tan or qual to a minimum istanc o its ual co, tn its wigt istribution is an xplicit unction o its cowor lngt, imnsion, an istinct wigts (aptr, orm 2 in [11]) nc, i w apply tis to our subco, w s tat its wigt istribution is trmin by t cowor lngt, imnsion, an istinct wigts bcaus it as at most nonzro istinct wigts onsquntly, t subco as t sam wigt istribution as in abl o ction bcaus it as n t sam cowor lngt, imnsion, an wigts as tos o Now, w riv # valus o an in mma mma 1: For an in mma, / / N Proo: For givn s wit at last on nonzro, t xponntial sum o as t istribution o (25) F or 4, on t otr an, + %]* n tis cas, t xponntial sum o as t istribution / y summing up t istributions o bot cass, t ovrall istribution o xponntial sums o in abl or is givn by tims 4 : + "54 4@ 4 + "54 4 tims g"i or all sts o s in < / ) (2) orovr, tis istribution is intical to t wigt istribution in abl Not tat ompar wit abl, t valus o an ollow immiatly o ar, w assum tat can b any lmnt in n squncs aspct, owvr, s in (24) cannot b wit nonzro us, w must rmov tis cas rom t istribution in (2) in orr to obtain t istribution o corrlation valus o squncs in O P orm 4: complt istribution o corrlation valus is as ollows o any pair o binary squncs in O J tims 7$ 7$ n P P / "Ï < )@<N : 7$ tims tims (27) Proo: (2) sows t istribution o xponntial sums o + or all ]Z an s in o corrlation valus o squncs in OFP@ o invstigat t istribution, w n to consir t istribution o xponntial sums o witout ;, wic mans rmoving t istribution o (25) rom (2) orovr, not tat wit rspct to t corrlation o squncs, ] an run troug all lmnts in by J an J as wll rspctivly y aitionally multiplying ac istribution o xponntial sums by "J, tror, w gt t istribution o corrlation valus o (27) From orm, t linar span o squncs in O P t ollowing istribution [ <; + 4 <;/ Qg <;/ g g ) tim tims tims N P N PNON as n tis sction, w giv an xampl o squncs in OkP or an prsnt t implmntation o squncs in OcP"U (or O ] ) bas on linar back sit rgistrs (Fs) n xampl o O P onsir a init il gnrat by a primitiv lmnt _ satisying _ n O P, t t usr s squnc ) is givn by an valuation o (17) at _ 5), i, $ ( #" %$ " 7 $ $ ( %$ " 7 $ ( #" %$ 7 $ ( 47 %$ )( wr a or cyclically istinct binary squncs in OFP@ (28) W av 184 t )

11 n ) 2 11 wr %_, an w not ) )g cimation squnc rom ) n, ) is givn by "@ " "@" " " " " "@" is obtain rom a linar com g, or tir sits n or From orm 4, t corrlation istribution o squncs From (28), w s tat bination o squncs ) )) g ) g ) tail, or N, ) itr or, ;, in O P or is givn by ) tims 8 < " +r/[ ( : ), t (2) tims tims tims tims tims wic is also vrii rom computr xprimnts F mplmntation From t prvious xampl, w s tat t procss or ob Ẍ W taining a squnc in O P is unrstoo by aing N squncs wit irnt back polynomials an irnt initial stats nc, O P is asy to implmnt by summing F outputs just lik ol squncs + For, stag Fs ar rquir to implmnt squncs in OQP" wr t Fs av irnt caractristic polynomials or gnrating cyclically istinct squncs pciically, lt b a primitiv polynomial ovr o gr an _ b a root o in W comput wic is a minimal polynomial o _ ovr wit (or mor tails about computation o minimal polynomials, s [4]) Using 7%+ as 4 t caractristic polynomial o t t F or (st 4 +[ + ), t initial stats o F or 5 can b arbitrary incluing zro For, on t otr an, t initial stat o F is givn by "%_ g< wit, wic is ix or all usrs gnric scription o an F implmntation o OFP^U is sown in Fig For vn, t squnc amily OQ]+U can b implmnt in t similar way to OQP", wic is omitt r n, t implmntation o O ] ^U is similar to Fig 1 xcpt tat F as a siz o wit a ix initial stat Fig 2 sows t F implmntation o O P or in t prvious xampl n Fig 2, ac stag F gnrats an squnc initial stats o t uppr two Fs ar irntly loa accoring to an ) wit a usr inx On t otr an, t initial stats o t lowr two Fs ar as sown in t igur initial stats o F 2 an F ar givn by NJNJNJNJ " arying initial stats Fix initial stats F : F 1: F (( ): #%$U F ( : # F : Fig 1 F implmntation o ( 7, tn t initial stat o F is st as zro arying initial stat * * * * * * * zro initial stat or * * * * * * * zro initial stat or or, Fig 2 F implmntation o wit caractristic polynomials ( $, 2 $, an $ ( 4 $ an [, rspctivly, wic can b obtain rom (2) ccoring to irnt initial stats o uppr two Fs, 184 cyclically istinct binary squncs ar gnrat to support as many irnt usrs ONUON N O OON Nw amilis o binary squncs o prio W5 bn prsnt in tis papr For o an an intgr wit :, a nw amily O P as amily siz o +U an maximum corrlation o 5 "#%$ For vn + an an intgr wit b, on t otr an, a nw amily OQ]@^U as amily siz o +U an maximum corrlation o [4 maximum an minimum linar spans o bot OQP" an O ] / ar Qg /;<% an Qg, rspctivly From t lxibility u to, our nw squnc amilis av aaptiv amily siz an maximum corrlation larg linar spans o OQP" an O ] ^U imply tat ty av goo potntial cryptograpic proprty

12 12 For ac wit b OcP^ (or O ], OQP@^U (or O ] U ) contains ) as a subst us, t nw amily contains oztas an umar s cas corrsponing to OFP" (or Uaya s cas corrsponing to O ] ) as a subst i squncs ar xclu rom bot cass amily O P is consir as a goo caniat wit larg amily siz as wll as low corrlation For O P, w urtr riv t corrlation istribution o squncs in O P t t n, w prsnt t F implmntation o O P ^U (or OQ]@ ) an sow tat it is xtrmly asy to implmnt t squnc amilis by mans o F structurs Finally, w woul lik to point out on intrsting rsmblanc btwn our nw binary squnc amily O P (or Oc]@ ) an t quatrnary squnc amily O a in [8], an awar work by umar, llst, alrbank, an ammons Jr, in 1 From O 1, i w rplac t trac unction rom alois ring i to by t trac unction rom to, an rplac t scalar actor o t sum o tos monomial trac trms by a scalar actor, an a t sum o t rmaining quaratic monomial trac trms wit coicint, tn O a bcoms O P (or Oc]+U ) in our construction wr NOWN autors woul lik to tank t anonymous rviwrs or tir lpul commnts autors rsarc is support by N rant PN 2277 FN [1] oztas an P umar, inary squncs wit ollik corrlation but largr linar span, rans norm ory, vol 4, no 2, pp 5257, ar 14 [2] ang, P aal, W olomb, ong, llst, an P umar, On a conjctur ial autocorrlation squnc an a rlat triplrror corrcting cyclic co, rans norm ory, vol 4, no 2, pp 887, ar 2 [] ol, aximal rcursiv squncs wit valu rcursiv crosscorrlation unctions, rans norm ory, vol 14, pp 15415, Jan 18 [4] W olomb an ong, ignal sign or oo orrlation or Wirlss ommunication, ryptograpy an aar ambrig Univrsity Prss, 25 [5] llst an P umar, quncs wit ow orrlation captr in anbook o oing ory it by Plss an uman lsvir cinc Publisrs, 18 [] asami, Wigt numrators or svral classs o subcos o t 2n orr ullr cos, normation an ontrol, vol 18, pp 4, 171 [7] im an J No, Nw amilis o binary squncs wit low corrlation, rans norm ory, vol 4, no 11, pp 55, Nov 2 [8] P umar, llst, alrbank, an ammons Jr, arg amilis o quatrnary squncs wit low corrlation, rans norm ory, vol 42, no 2, pp 5752, ar 1 [] P umar an coltz, ouns on t linar span o bnt squncs, rans norm ory, vol 2, no, pp 85482, Nov 18 [1] vnstin, ouns or cos as solutions o xtrmum problms or systm o ortogonal polynomials,, ctur Nots in omputr cinc 7 rlin: pringrrlag, pp 2542, 1 [11] F J acwilliams an N J loan, ory o rrororrcting os mstram: Nortollan, 177 [12] Ncav, rock co in a cyclic orm, iscr at ppl vol 1, pp 584, 11 [1] J No, W olomb, ong,, an P aal, inary psuoranom squncs o prio wit ial autocorrlation, rans norm ory, vol44, no 2, pp , ar 18 [14] J Olsn, coltz, an Wlc, ntunction squncs, rans norm ory, vol 28, no, pp 85884, Nov 182 [15] O otaus, oii ol os, rans norm ory, vol, no 2, pp 545, ar 1 [1] anbag, P umar, an llst, mprov binary cos an squnc amilis rom 4 linar cos, rans norm ory, vol 42, no 5, pp , pt 1 [17] ilnikov, On mutual corrlation o squncs, ovit at okl,, vo12, pp 1721, 171 [18] imon, J Omura, coltz, an vitt, pra pctrum ommunications, vols omputr cinc Prss, ockvill, 185 [1] P Uaya, Polypas an rquncy opping squncs obtain rom init rings, P issrtation, pt lc ng, nian nst cnol, anpur, 12 [2] Wlc, owr bouns on t maximum cross corrlation o t signals, rans norm ory, 2, pp 7, ay 174