2-Adic Complexity of a Sequence Obtained from a Periodic Binary Sequence by Either Inserting or Deleting k Symbols within One Period

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1 -Adc Comlexty of a Seuence Obtaned from a Perodc Bnary Seuence by Ether Insertng or Deletng Symbols wthn One Perod ZHAO Lu, WEN Qao-yan (State Key Laboratory of Networng and Swtchng echnology, Bejng Unversty of Posts and elecommuncatons, Bejng 876, Chna) Abstract. In ths aer, we roose a method to get the lower bounds of the -adc comlexty of a seuence obtaned from a erodc seuence over GF() by ether nsertng or deletng symbols wthn one erod. he results show the varaton of the dstrbuton of the -adc comlexty becomes as ncreases. Partcularly, we dscuss the lower bounds when resectvely. Key words: Perodc Bnary Seuence; -Adc Comlexty; -symbol deleton; -symbol nserton Introducton Many modern stream chers are desgned by combnng the outut of several LFSRs n varous nonlnear ways. Snce 955, large amount of efforts have been devoted nto the study of other ( nonlnear ) feedbac archtectures. In [], Klaer and Goresy ntroduced a new feedbac archtecture for shft regster generaton of seudorandom bnary seuences called feedbac wth carry shft regster (FCSR) and dscussed some basc roertes of FCSR seuences, such as the erods, the exonental reresentatons and so on. Based on FCSR they resented the concet of the -adc comlexty (denoted φ() ) as an mortant ndex of the securty of seudorandom seuences and summarzed the ratonal aroxmaton algorthm. For every erodc seuence S, only nowledge of φ ( s) + bts s suffcent to reroduce t usng the algorthm. In another word, the hgher -adc comlexty a seuence has, the more secure t s. In recent years, more and more researchers show ther nterest n the -adc comlexty of seuences. It s well nown that the lnear comlexty of a erodc seuence s unstable under small erturbatons, so -error lnear comlexty s roosed to study of the stablty of stream chers. Smlar to -error lnear comlexty, Honggang roosed the concet of -error -adc comlexty and gave the exected value and varance of a erodc bnary seuence s -adc comlexty []. In [3], Chen Lanfang rovded an analog of the extended Games-Chan algorthm whch can yeld uer and lower bounds for the -adc comlexty of erodc bnary seuences

2 wth erod m n. Jang Shaouan and Da Zongduo dscussed lnear comlexty of a seuence by substtutng, nsertng or deletng symbols n a erodc seuence [4]. Motvated by ther wor, we study the lower bounds of the -adc comlexty of a seuence obtaned from a erodc seuence over GF() by ether nsertng or deletng symbols wthn one erod. he method of ths aer s smle, can deal both the two dfferent cases wth the same manner and the results show the varaton of the dstrbuton of the -adc comlexty becomes as gve out the lower bounds when. ncreases. Fnally, we also Defntons and Lemmas An FCSR s determned by r coeffcents,,, r, where {,},,,, r, and an ntal memory. he contents of the regster at any gven tme consst of r bts, mr denoted ( an, an,, an r +, an r) and the memory s m n. he oeraton of the shft regster s defned as followng:. Form the nteger sumσ n a n + mn ; r. Shft the contents one ste to the rght, whle oututtng the rghtmost bt an r; 3. Put a σ mod nto the leftmost cell of the shft regster; n n 4. Relace the memory nteger m wth m ( σ a ). n n n n he nteger r r s called the connecton nteger of the FCSR. mn an n a an r + an r dv mod r r

3 Any nfnte bnary seuence S { s } Fg. Feedbac wth carry shft regster can be resented by a formal ower seres α s called -adc number. Such ower seres forms the rng of -adc number, denoted Z. Ss eventually erodc f and only f the -adc numberα s ratonal,.e. there exst nteger such thatα Z. If S s strctly erodc wth mnmal erod, then, s α s where. f and only f S s the all- seuence. If gcd(, ), called the reduced ratonal exresson of S. s Defnton. Let S be a erodc bnary seuence wth reduced ratonal exresson, then the -adc comlexty φ( S) s the real number log. Obvously, f S s the all- seuence or the all- seuence, thenφ ( S). Lemma. [] Let, be two ostve ntegers, where < <. Let h be a nonzero nteger and ( h) ' ' mod, where the notaton ( h) mod means the reduced resdue of h modulo, and < ' < '. hen ' gcd( ', ') gcd(, ) he eualty holds f and only f gcd( h, ) gcd(, ) Proof. Put d gcd(, ). hen d d, and gcd(, ). We have, hus ( h) mod ( d h) d ( h) d mod( ) mod( )

4 gcd(, ) gcd(, ) he eualty holds f and only f gcd( h, ) gcd( h, ) gcd(, ) Defnton. Let S ( s, s, s, s, ) be a erodc bnary seuence, then S ( s, s,, s, s,) s the comlementary seuence of S, f s + s, <. Lemma. Let S ( s, s, s, s, ) be a erodc bnary seuence, -adc comlexty of S sφ ( S), f S s the comlementary seuence of S, and -adc comlexty of S s ( S), then φ( S) φ( S ). Proof. Let the reduced ratonal exresson of S and S be and resectvely, that s, φ s s and. hus φ ( S) log and φ ( S ) log. Hence s ( ) s hen we can get gcd(, ) gcd(, ) s s φ( S) φ( S ) 3 -Symbol Inserton heorem. Let S ( s, s,, s, s, ) be a erodc bnary seuence wth erod. Suose that the ratonal exresson of S s α, where < <, gcd(, ). Let S ( s, s,, s, s, ) be a seuence obtaned from S by -symbol a (,,, ) +

5 nserton wthn one erod, where t < t < < t <, a {,},,,,, then we have Proof. Let m max, n log ( ) φ( S) ( t + ) < φ( S ) log ( ) s, < t + + max a, + S () ' Let be the ratonal exresson of S, where g cd( ', '),, and + ' < ' < ' + + S () s t t tj j t j s s s s s a t tj j+ t + t If n< m+, let A() S + () S () t t tj j t t j s s s s t ( ) tj j+ t ( ) + + ( ) + + ( ) + ( ) + a t hen we have + ' S () S () + A() ( ) + A() ' ( ) So + + { [ ( ) + A()]}mod[ ( )] [ ( ) + A()]mod( ) + + ( ) ( ) A() + ( ) From lemma we have: ' () + gcd(, A() + ( ) ) A() + ( ) In the followng, we dscuss the value of A() + ( ). For each j,,, we have tj j tj j j j tj tj j+ + tj + tj j+ + s + tj j+ tj j+, ( ) ( ) hen

6 tj j j tj tj j+ + tj + tj j+ + tj t t+ t+ [( ) s ] ( + ) + j tj j+ j j Whle t t t t+ ( ) s ( ) + So If tj t a j j j A() + ( ) + + ( ) n< m+, then A () < t So tj tj t + A() + ( ) A() + ( ) ( )( ) < < j j From (), + + ' > t + A() + ( ) + So φ( S ) > log ( ) φ( S) ( t + ) If n> m+, let S and S be comlementary seuence of S and S resectvely. By lemma, we have φ( S) φ( S ) andφ( S ) φ( S ). Let and be the reduced ratonal exresson of S and S resectvely, that s, S + () S () and, where gcd( ', '), < ' < ' and () s S t t tj j t j s s s s s t tj j+ t + t a where a + a Let A + () S () S () t

7 t tj j t t j s s s s t ( ) tj j+ t ( ) + + ( ) + + ( ) + ( ) + a + S () S () + A () ( ) + A () Hence ( ) + + { [ ( ) + A ()]}mod[ ( )] [ ( ) + A ()]mod( ) + + ( ) ( ) A () + ( ) + By lemma, we have t + + ' + gcd(, A () + ( ) ) A () + ( ) ( ) In the followng, we dscuss the value of A () + ( ). For each j,,, we have tj j tj j j j tj tj j+ + tj + tj j+ + s + tj j+ tj j+ ( ) ( ) hen tj j j tj tj j+ + tj + tj j+ + tj t t+ t+ [( ) s ] ( + ) + j tj j+ j j t t t t+ Whle ( ) s ( ) + tj t t j So A () + ( ) + a + ( ) j j If n> m+, then A () < So tj tj t + A () + ( ) A () + ( ) ( )( ) < < + + From ( ), ' > t + A () + ( ) j j

8 So φ( S + ) > log ( ) φ( S + ) ( t + ), that s φ( S ) > log ( ) φ( S) ( t + ). If n m+, we can get the concluson wth the same method above. 4 -Symbol Deleton heorem. Let S ( s, s,, s, s, ) be a erodc bnary seuence wth erod. Suose that the ratonal exresson of S s α, where < <, gcd(, ). Let S ( s, s,, s, s, ) be a seuence obtaned from S by -symbol s (,,, ) t deleton wthn one erod, where t < t < < t <, a {,},,,,, then we have log ( ) + φ( S) ( t + ) < φ( S ) log ( ) Proof. Let m max, n max s, < t, tj, j {,,, } st, S () ' Let ' be the ratonal exresson of S, where gcd( ', '), < ' < ', and () s S t t t j+ t + j ( ) s s s s s t + tj+ t + t Hence φ( S ) log log ( ) If n< m+, let () S S () B() t t t j+ t j s s s s s t t + tj+ t + t [( ) + ( ) + + ( ) + + ( ) ] ( ) + B() ' S () [ S () + B()] ( ) + B() hen ' ( )

9 So { [ ( ) + B()]}mod[ ( )] [ ( ) + B()]mod( ) ( ) + B() ( ) ( ) From lemma we have: ' () gcd(, ( ) + B()) ( ) + B() For each j,,, we have tj+ tj+ j j s tj+ tj+ tj+ + j tj+ j+ tj+ tj+ ( ) ( ) + hen t j+ j tj+ + j tj+ j+ tj+ tj+ t j [( ) ] ( ) t + t t + s j tj + j j t t t+ t Whle ( ) s ( ) + tj st j j j So B() ( ) + ( ) ( ) + If n< m+, then B()> tj tj tj t + So ( ) + B() ( )( ) < < j j From () ' > t gcd(, ( ) B()) ( ) B() So log ( ) ( ) ( ) ( ) log ( + φ S t + < φ S ) If n> m+, let S and S be comlementary seuence of S and S resectvely. By lemma, we have φ( S) φ( S ) andφ( S ) φ( S ). Let and be the reduced ratonal exresson of S and S resectvely, that s, S () S () and, where g cd( ', '), < ' < ' and

10 () s S t t t j+ t + j ( ) s s s s t + tj+ t + t s So φ( S ) log log ( ) Let S () S () B () t t t j+ t j s s s s s t t + tj+ t + t [( ) + ( ) + + ( ) + + ( ) ] ( ) + B () ' S () [ S () + B ()] ( ) + B () So we have ' ( ) herefore, { [ ( ) + B ()]}mod[ ( )] [ ( ) + B ()]mod( ) ( ) + B () ( ) ( ) By lemma, we have ' gcd(, ( ) + B ()) ( ) + B () ( ) For each j,,, we have tj+ tj+ j j s tj+ tj+ tj+ + j tj+ j+ tj+ tj+ ( ) ( ) + hen t j+ j tj+ + j tj+ j+ tj+ tj+ t j [( ) ] ( ) t + t t + s j tj + j j t t t+ t Whle ( ) s ( ) + tj s t j j j t So B () ( ) + ( ) ( ) + If n> m+, then B ()> j

11 tj tj t + So ( ) + B () ( )( ) < < j j From () ' > gcd(, ( ) + B ()) ( ) + B () t + So log ( ) ( ) ( ) ( ) log ( + S t + < S ) φ hat s log ( ) ( ) ( ) ( ) log ( + φ S t + < φ S ). φ If n m+, we can get the concluson wth the same method above. 5 One Symbol Inserton or Deleton In ths secton, we loo nto a secal case. We get the lower bounds of the -adc comlexty for a seuence obtaned from a erodc bnary seuence by ether nsertng or deletng bt. heorem 3. Let S ( s, s,, s, s, ) be a erodc bnary seuence wth erod. Suose that the ratonal exresson of S s α, where < <, gcd(, ). Let S ( s, s,, s, s, ) be a seuence obtaned from S by symbol a nserton wthn one erod, then we have log ( ) φ( S) < φ( S ) log ( ) + + Proof. It s well nown that -adc comlexty of a seuence obtaned from a seuence by shftng s not changed, wthout losng generalty we nsert symbol behnd the last bt wthn one erod. + () + S If a, let s the ratonal exresson of S, where + S () s + s + + s + a s + a s S () S () So Snce gcd(, ), so > + gcd(, )

12 + hereforeφ( S ) log > log ( ) φ( S ) If a, let S and S be comlementary seuence of S and S resectvely. By lemma, we haveφ( S) φ( S ) andφ( S ) φ( S ). Because a, we have φ > φ + ( S ) log ( ) ( S) + hat sφ( S ) > log ( ) φ ( S) heorem 4. Let S ( s, s,, s, s, ) be a erodc bnary seuence wth erod. Suose that the ratonal exresson of S s α, where < <, gcd(, ). Let S ( s, s,, s, s, ) be a seuence obtaned from S by symbol a deleton wthn one erod, then we have Proof. he method s smlar to theorem 3. log ( ) φ( S) < φ( S ) log ( ) 6 Concluson In communcatons, nose can easly cause modfcaton to the orgnal ey stream, even comromse the securty. In ths aer, we dscuss the lower bounds of the -adc comlexty of a seuence obtaned from a erodc seuence over GF() by ether nsertng or deletng symbols wthn one erod. Interestngly, two tyes of changes can be treated wth the same method. he results show the stablty of the stream chers whch generated by FCSR. Partcularly, we gve out the lower bounds of -adc comlexty when resectvely. References. A. laer, M. Goresy. Feedbac shft regsters, -adc san, and combners wth memory, J. Crytogoly, Vol.,.-47, Honggang Hu, Dengguo Feng. On the -Adc Comlexty and the -Error -Adc Comlexty of Perodc Bnary Seuences. IEEE rans. Inform. heory, Vol.54, No., , Lanfang Chen, Wenfeng Q. -Adc Comlexty of Bnary Seuences wth Perod m n.

13 Journal on Communcatons, Vol.6, No.6, Shaouan Jang, Zongduo Da et al. Lnear Comlexty of a Seuence Obtaned from a Perodc Seuence by Ether Substtutng, Insertng, or Deletng Symbols Whtn One Perod. IEEE rans. Inform. heory, Vol.46, No.3,.74-77,. 5. Le Wang, Man Ca et al. On Stablty of -Adc Comlexty of Perodc Seuence. Journal of Xdan Unversty, Vol.7, No.3, ,. 6. Lhua Dong, Yuu Hu et al. -Error -Adc Comlexty Algorthm for the Bnary Seuences wth Perod n. Chnese Journal of Comuters, Vol.9, No.9, , Yaodong Zhao, Wenfeng Q. he Lnear Comlexty of the Perod Seuences Obtaned from the Bnary Seuences by One-Symbol Delete or Insert. Journal of Informaton Engneerng Unversty, Vol.5, No.,.-3, 4.