Period-Different $m$-Sequences With At Most A Four-Valued Cross Correlation

PERIOD-DIFFERENT m -SEQUEN CES WITH A T MOST A F OUR-V ALUED CR OSS CORRELA TI ON LEI HU, XIAN GYONG ZENG, NIAN LI, A ND WENFENG JIANG Abstra ct. In this pap er, we fol lo w the recent w ork of Helleseth, Kholosha, Johanssen and Ness to stu d y the cross correlation betw een an m - sequence of p erio d 2 m − 1 and t h e d -decimation of an m -sequ ence of shorter p erio d 2 n − 1 for an even number m = 2 n . Assuming that d satisfies d (2 l + 1) = 2 i (mod 2 n − 1) for some l and i , we p rove the cross correlation takes exactly either three or four v alues, d ep ending on gcd( l, n ) is equal to or larger than 1. The distribution of the correlation v alues is also completely determined. Our result confirms th e numerical phen omenon Helleseth et al found. I t is conjectured that there are n o m ore other cases of d th at give at most a four-v alued cross correlation apart from the ones prov ed here. 1. Introduction Sequences with go o d co rrelation prop erties h a ve imp ortan t applicatio ns in co m m u nication systems. The maximal p erio d sequences ( m -sequences) and their decimations are widely u sed to design sequence families w ith low-c orrelation [3 , 4, 8, 16]. Recen tly , Ness and Helleseth initiated the studies on the cr oss correlation b et w een an m - sequence { s t } of p erio d 2 m − 1 and the d -decimation { u dt } of an m -sequence { u t } with shorter p erio d 2 n − 1 for m = 2 n and d with gcd( d, 2 n − 1) = 1 [12]. Th ese tw o p erio d -differen t sequences are exactly the m -sequences used to constru ct the Kasami sequence family [9, 10, 17]. They prop osed the fi rst family of d ecimation d that the cross correlation tak es three v alues in [12], where n is o d d and d is take n as d = 2 n +1 3 . They also completed a full searc h f or all the decimation d giving at most a six-v alued correlation for m ≤ 26 [12]. Later, they f ound another family of d = 2 n +1 2 − 1 giving a three-v alued cross correlation for o d d n [13]. F u rther, in [7], Helleseth, Kholosha, and Ness show ed for a larger family of d satisfying d (2 l + 1) = 2 i (mo d 2 n − 1) , (1.1) the cross correlation is thr ee-v alued, where n is o dd and gcd( l, n ) = 1. Based on their numerical exp eriment, the authors in [7] conjectured that they had f ou n d all d giving a three-v alued cross correlation. F or the cross correlation taking f our v alues, Ness and Helleseth [14] pro ved this is the case if d = 2 3 k +1 2 k +1 and m = 6 k . This result was v ery r ecen tly generalized to th e case of d = 2 kr +1 2 k +1 and m = 2 k r with o d d r b y Helleseth, K holosha and Johanssen in [6]. As they had p oint ed out, there still existed some examples of four-v alued cross correlation that did not fit into all the kno wn families they had found . In this pap er, w e follo w ab o ve work to stud y th e cross correlation of { s t } and { u dt } with the decimation d satisfying E q u alit y (1.1) for a general case of parameters m and l , namely , without an y restrictio n on n and l . W e p r o ve the cross correlation tak es exactly ei ther three or four v alues, dep ending on gcd( l, n ) is equal to or larger th an 1. The distribution of the correlation v alues is also completely determined. Key wor ds and phr ases. m -sequence, cross correlation, qu adratic form. 1 2 LEI HU, XIANGYONG ZENG, NIAN LI, AND WENFENG JIANG Our result theoretically confi rms the numerical exp erimen ts Helleseth et al done in a unifi ed w ay , that is, except some exceptional d in small m = 8 and 10, all d found in the exp eriments and giving at most a four-v alued cross correlation are co vered b y our result. This d efinitely includes new classes of d that can n ot b e explained in p revious results, for instance, d = 181 for m = 20. T he full numerical searc h of u p to m = 26 [12] together with our result su ggests one to conjecture th at f or m > 10 th er e are n o more other d giving at most a four-v alued cross correlation apart from the ones satisfying (1.1). Our pro of is mainly based on the theory of quadratic forms and on the stud y of a class of equations o ver a finite field, which is a little differen t from the treatmen ts of Helleseth et al. This pap er is organized as follo w s . Section 2 is preliminaries on cross correlation and quadratic forms. Section 3 d etermines the ranks of quadratic p olynomials ρ a ( x ) and the enumeration of the r anks as a v aries in F 2 n . In Section 4, w e obtain the desired four-v alued cross correlation distribution. 2. Preliminaries on Cr oss Correla tion and Quadra tic F orms Let F 2 m b e the finite field with 2 m elemen ts and F ∗ 2 m = F 2 m \ { 0 } . F or any integ ers m and n with m | n , th e trace function from F 2 m to F 2 n is d efined by tr m n ( x ) = m/n − 1 P i =0 x 2 ni , where x is an elemen t in F 2 m . Let m = 2 n b e an ev en in teger, α b e a pr imitiv e element of F 2 m and T = 2 n + 1. Then β = α T is a primitiv e elemen t of F 2 n . The m -sequence { s t } of p erio d 2 m − 1 and the m -sequence { u t } of shorter p erio d 2 n − 1 are giv en by s t = tr m 1 ( α t ) , u t = tr n 1 ( β t ) , resp ectiv ely . The cross corr elation at shif t 0 ≤ τ ≤ 2 n − 2 b etw een { s t } and the d decimation of { u t } is defined by C d ( τ ) = 2 m − 2 X t =0 ( − 1) s t + u d ( t + τ ) . In this pap er, w e consider the cross correlation for the decimatio n d satisfying d (2 l + 1) = 2 i (mo d 2 n − 1) (2.1) for s ome l and i for an arbitrary n . Note th at in [7], only th e case for od d n and gcd( l, n ) = 1 is considered. Obviously , if d satisfies Equ ality (2.1), then b oth d and 2 l + 1 are prime to 2 n − 1. In fact, for eac h d satisfying (2.1 ), one can c ho ose some l < n suc h that b oth (2.1) and gcd(2 l + 1 , 2 m − 1) = 1 hold. T o exp lain this, w e need the f ollo wing standard consequence of the division algorithm. Lemma 1. L et u and u b e p ositive inte gers with w = gcd( u, v ) , then gcd(2 u − 1 , 2 v − 1) = 2 w − 1 and gcd(2 u − 1 , 2 v + 1) =  1 , if u/w is o dd , 1 + 2 w , otherwise . Assume that Equalit y (2.1) holds. W r ite n = 2 e n 1 with o dd n 1 . By Lemm a 1, 2 e divides l , and l can b e wr itten as l = 2 e l 1 . If l 1 is ev en, then by Lemma 1 again, gcd(2 l + 1 , 2 m − 1) = 1. Otherwise, if l 1 is o dd, let l ′ = n − l . Then one has d (2 l ′ + 1) = 2 n − l + i (mo d 2 n − 1) and gcd(2 l ′ + 1 , 2 m − 1) = 1. This leads w e use the follo wing notation on in tegers: • n = k r and m = 2 k r with an arbitrary integ er k and od d r ≥ 3. 3 • l = k s , w here 0 < s < r , s is ev en, and gcd( r , s ) = 1. Note that gcd(2 l + 1 , 2 m − 1) = 1 b y Lemm a 1. • d is a decimation satisfying d (2 l + 1) = 2 i (mo d 2 n − 1) for some i . Additionally , w e let • N ( f , F 2 u ) denote the num b er of ro ots in a fin ite field F 2 u of a p olynomial f . F or an y function f ( x ) from F 2 m to F 2 , the trace transform b f ( λ ) of f ( x ) is defin ed b y b f ( λ ) = X x ∈ F 2 m ( − 1) f ( x )+t r m 1 ( λx ) , λ ∈ F 2 m . f ( x ) is calle d to b e a quadratic form, if it can b e expressed as a degree t wo p olynomial of the form f ( x 1 , x 2 , · · · , x m ) = P 1 ≤ i ≤ j ≤ m a ij x i x j b y using a basis of F 2 m o ve r F 2 . Th e rank of th e symmetric matrix w ith zero diagonal en tries and a ij as the ( i, j ) and ( j, i ) en tries for i 6 = j , is defined as the rank of f ( x ). It ca n b e calculate d from the num b er N of x ∈ F 2 m suc h that B f ( x, z ) := f ( x + z ) + f ( x ) + f ( z ) = 0 holds for all z ∈ F 2 m , namely , rank ( f ) = m − log 2 ( N ). The B f ( x, z ) is called the s ymplectic form of f ( x ). It is known that the distrib ution of trace transform v alues of a quadratic form is determined by its r ank as the follo wing lemma. F or more details, the reader is referr ed to [8]. Lemma 2 ([8]) . Th e r ank of any quadr atic form is even. L et f ( x ) b e a q u adr atic f orm on F 2 m with r ank 2 h , 1 ≤ h ≤ m/ 2 . Then i ts tr ac e tr ansform values have the fol lowing distribution: b f ( λ ) =  ± 2 m − h , 2 2 h − 1 ± 2 h − 1 times , 0 , 2 m − 2 2 h times . Using the trace represent ation of the sequences, C d ( τ ) can b e written as C d ( τ ) = 2 m − 2 P t =0 ( − 1) s t + u d ( t + τ ) = 2 m − 2 P t =0 ( − 1) tr m 1 ( α t )+tr n 1 ( β dτ α tdT ) = P x ∈ F ∗ 2 m ( − 1) tr m 1 ( x )+tr n 1 ( ax dT ) , where a = β dτ ∈ F 2 n . Now making a substitution x = y 2 l +1 , w e h a ve C d ( τ ) = P y ∈ F ∗ 2 m ( − 1) tr m 1 ( y 2 l +1 )+tr n 1 ( ay 2 n +1 ) . = P y ∈ F ∗ 2 m ( − 1) ρ a ( x ) = − 1 + b ρ a (0) , (2.2) where ρ a ( x ) = tr m 1 ( x 2 l +1 ) + tr n 1 ( ax 2 n +1 ) . (2.3) The idea of the ab ov e work handling C d ( τ ) comes from Ness and Helleseth [12]. Ho w eve r , th e observ ation that gcd (2 l + 1 , 2 m − 1) = 1 and the sub stitution x = y 2 l +1 is one-to-one o v er F 2 m enables us to simplify the calculation of C d ( τ ), comparin g with the w ork of [12]. 3. Ranks of Quadra tic F orms ρ a ( x ) and Their Enume ra tion In this section, we determine the ranks of the quadratic forms ρ a ( x ) defined by Equalit y (2.3), and the en u m eration of the ranks when a ranges ov er F ∗ 2 n . 4 LEI HU, XIANGYONG ZENG, NIAN LI, AND WENFENG JIANG The symplectic form of ρ a ( x ) is giv en b y B ρ a ( x, z ) = ρ a ( x ) + ρ a ( z ) + ρ a ( x + z ) = tr m 1 ( z x 2 l + xz 2 l ) + tr n 1 ( az x 2 n + axz 2 n ) = tr m 1 ( z x 2 l + z x 2 m − l ) + tr n 1 (tr m n ( az x 2 n )) = tr m 1 ( z ( x 2 l + x 2 m − l + ax 2 n )) = tr m 1 ( z 2 l ( x 2 2 t + a 2 l x 2 t + x )) , where t = n + l . Therefore, w e consider the n u m b er of ro ots in F 2 m of the p olynomial f a ( x ) := x 2 2 t + a 2 l x 2 t + x, a ∈ F ∗ 2 n . (3.1) Note that gcd ( t, m ) = k , all these r o ots form an F 2 k -v ector space. Th us, the n u m b er of ro ots of f a ( x ) is a p ow er of 2 k . Denote y = x 2 t − 1 . W e hav e f a ( x ) = x ( y 2 t +1 + a 2 l y + 1) . (3.2) The nonzero ro ots of f a ( x ) are closely related to that of the p olynomial g a ( y ) := y 2 t +1 + a 2 l y + 1 , (3.3) or equiv alen tly , to th at of the ro ots of the p olynomial h c ( z ) := z 2 t +1 + cz + c, (3.4) whic h is obtained fr om Equation (3.3) by substituting y = a − 2 l z , then dividing by a − 2 l (2 t +1) and letting c = a 2 l (2 t +1) . Noti ce that there exists a one-to-o ne corresp ondence b et ween a ∈ F ∗ 2 n and c ∈ F ∗ 2 n since gcd(2 t + 1 , 2 n − 1) = 1. The prop erties ab out the r o ots of a p olynomial with th e form as Equ ation (3.4) were inv es- tigated tec hnically in [1] and [5]. T o f acilitat e the in tro duction of their results, we still use th e notation h c ( x ) defined by Equalit y (3.4) in th e follo w ing Lemmas 3. Bu t notice that this lemma holds for an y p ositiv e in tegers m and t , and c ∈ F 2 m . Lemma 3 (Theorem 5.4 of [1]) . Denot e Ξ m = { ξ ∈ e F 2 m | ξ 2 t − 1 = 1 η +1 , where η ∈ e F 2 m and h c ( η ) = 0 } , (3.5) wher e e F 2 m denotes the algebr aic closur e of F 2 m . L et k = gcd( t, m ) . Then h c ( x ) has either 0 , 1 , 2 or 2 k + 1 r o ots in F 2 m . Mor e over, i f h ( x ) has e xactly one r o ot in F 2 m , then Ξ m ∩ F 2 m is nonempty, and tr m k ( ξ ) 6 = 0 for any ξ ∈ Ξ m ∩ F 2 m . Prop osition 1. F or any a ∈ F ∗ 2 n , g a ( y ) has either 0 , 2 , or 2 k + 1 nonzer o r o ots in F 2 m . Pr o of. Since g a ( y ) h as the same n u m b er of ro ots as h c ( z ) f or c = a 2 t (2 t +1) , it is sufficient to pro ve h c ( z ) has either 0, 2, or 2 k + 1 nonzero ro ots in F 2 m . By gcd( t, n ) = k and Lemma 3, h c ( z ) has either 0, 1, 2, or 2 k + 1 n onzero ro ots in F 2 m . Thus, w e only n eed to prov e h c ( z ) can not ha ve exactly one ro ot in F 2 m . Assume that h c ( z ) has exactly one r o ot in F 2 m , denoted by η . T h en η 2 n also is a ro ot of h c ( z ), whic h implies that η = η 2 n and then η ∈ F 2 n . Th us, h c ( z ) has exactly one ro ot in F 2 n to o. By Lemma 3, the set Ξ n ∩ F 2 n ⊆ Ξ m ∩ F 2 m is n onempt y , and for an y ξ ∈ Ξ n ∩ F 2 n ⊆ Ξ m ∩ F 2 m , we ha ve tr m k ( ξ ) 6 = 0, which con tradicts with the fact tr m k ( ξ ) = tr n k (tr m n ( ξ ))) = tr n k ( ξ · tr m n (1)) = 0.  5 Prop osition 2. L et a ∈ F ∗ 2 n . Then (1) If y 1 and y 2 ar e two differ ent r o ots of g a ( y ) in F 2 m , then y 1 y 2 is a (2 k − 1) -th p ower in F 2 m ; and (2) If ther e exist at le ast thr e e differ ent r o ots of g a ( y ) in F 2 m , then e ach r o ot of g a ( y ) in F 2 m is (2 k − 1) -th p ower in F 2 m . Pr o of. (1) If g c ( y i ) = 0 for i = 1 and 2, then we hav e y 1 y 2 ( y 1 + y 2 ) 2 t = y 2 t +1 1 y 2 + y 1 y 2 t +1 2 = ( a 2 l y 1 + 1) y 2 + ( a 2 l y 2 + 1) y 1 = y 1 + y 2 . Since gcd(2 t − 1 , 2 m − 1) = 2 k − 1, so y 1 y 2 = ( y 1 + y 2 ) − (2 t − 1) is a (2 k − 1)-th p ow er in F 2 m . (2) Let y 1 , y 2 and y 3 b e th r ee different ro ots of g a ( y ) in F 2 m . Th en by (1), all of y 1 y 2 , y 1 y 3 and y 2 y 3 are (2 k − 1)-th p o wer in F 2 m , and hence so are y 2 1 = ( y 1 y 2 )( y 1 y 3 ) / ( y 2 y 3 ) and y 1 . This finishes the pro of.  Prop osition 3. L et a ∈ F ∗ 2 n . Th en f a ( x ) has e xactly e i ther 1 or 2 2 k r o ots in F 2 m , that is, ρ a ( x ) has r ank of m or m − 2 k . F u rthermor e, N ( f a , F 2 m ) = 1 ⇐ ⇒ N ( g a , F 2 m ) = 0 or 2 and N ( f a , F 2 m ) = 2 2 k ⇐ ⇒ N ( g a , F 2 m ) = 2 k + 1 . Pr o of. This is shown by analyzing the num b er of ro ots in F 2 m of g a ( y ). Note th at y = x 2 t − 1 and gcd(2 m − 1 , 2 t − 1) = 2 k − 1. Thus, the corresp ondence from x ∈ F ∗ 2 m to y is (2 k − 1)-to 1. If g a ( y ) has exactly 2 k + 1 ro ots y in F 2 m , then by Prop osition 2 (2), all its ro ots in F 2 m are (2 k − 1)-t h p o wer in F 2 m , and th ey corresp ond (2 k + 1)(2 k − 1) = 2 2 k − 1 non zero r o ots of f a ( x ) under the inv erse of the mapping x 7→ y = x 2 t − 1 . T hus f a ( x ) has exactly 2 2 k ro ots in F 2 m . If g a ( y ) has exactly 2 ro ots y 1 , y 2 ∈ F 2 m , then by Prop osition 2(1), y 1 , y 2 are either b oth (2 k − 1)-th p ow ers in F 2 m or b oth not (2 k − 1)-th p o wer in F 2 m . The former can not b e true. If it is that case, then y 1 and y 2 w ould corresp ond total 2(2 k − 1) = 2 k +1 − 2 nonzero ro ots of f a ( x ), and thus f a ( x ) would ha ve exactly 2 k +1 − 1 r o ots in F 2 m . This is a contradict ion with that the num b er of ro ots is a p ow er of 2 k . T h erefore, b oth t w o r o ots are not (2 k − 1)-th p o wers in F 2 m and f a ( x ) has no nonzero ro ot in F 2 m .  In the s equel, w e determine how many a ∈ F 2 n there are suc h that f a ( x ) has exact ly 1 or 2 2 k ro ots in F 2 m . Note that w e encounter h ere t wo d ifferent fields F 2 n and F 2 m . T o this end, we first ha v e Prop osition 4. N ( g a , F 2 m ) = N ( g a , F 2 n )( mo d 2) . Pr o of. Since g a ( y ) is a p olynomial ov er F 2 n , we know that for any ro ot γ ∈ F 2 m of g a ( y ), its 2 n -th p o wer γ 2 n is also a r o ot of g a ( y ) in F 2 m . If this ro ot γ / ∈ F 2 n , then γ 6 = γ 2 n , and these t wo elemen ts form a pair { γ , γ 2 n } as roots of g a ( y ) in F 2 m , By ( γ 2 n ) 2 n = γ , eve ry t w o pairs of ro ots of this form are either just the same one or disjoin ted. T h us, N ( g a , F 2 m ) = N ( g a , F 2 n )( mo d 2).  T o d etermine N ( f a , F 2 n ), w e need the follo w in g lemma. Notice that in this lemma, in tegers n and t refer to an y p ositiv e int egers, and c r efers to an y element in F 2 n . 6 LEI HU, XIANGYONG ZENG, NIAN LI, AND WENFENG JIANG Lemma 4 (Theorem 5.6 of [1]) . L e t k = gcd( t, n ) and q = 2 k , then h c ( x ) has either 0 , 1 , 2 or q + 1 r o ots in F 2 n . Mor e over, let N i denote the numb er of c ∈ F ∗ 2 n such that h c ( x ) has e xactly i r o ots in F 2 n , wher e i = 0 , 1 , 2 , q + 1 . If µ = n/k is o dd, then N 0 = q µ +1 + q 2( q +1) , N 1 = q µ − 1 − 1 , N 2 = ( q − 2)( q µ − 1) 2( q − 1) , N q +1 = q µ − 1 − 1 q 2 − 1 . Prop osition 5. F or any a ∈ F ∗ 2 n , the r ank of ρ a ( x ) is m or m − 2 k . F urthermor e, let R m ( R m − 2 k , r esp e ctively) b e the numb er of a ∈ F ∗ 2 n such that rank( f a ) = m ( rank( f a ) = m − 2 k , r esp e ctively), then R m = 2 n +2 k − 2 n + k − 2 n + 1 2 2 k − 1 , R m − 2 k = 2 n + k − 2 2 k 2 2 k − 1 . Pr o of. By Prop ositions 1, 3 and 4, we ha v e N ( f a , F 2 m ) = 1 ⇐ ⇒ N ( g a , F 2 m ) = 0 or 2 ⇐ ⇒ N ( g a , F 2 n ) = 0 or 2 (3.6) and N ( f a , F 2 m ) = 2 2 k ⇐ ⇒ N ( g a , F 2 m ) = 2 k + 1 ⇐ ⇒ N ( g a , F 2 n ) = 1 or 2 k + 1 . (3.7) Since gcd ( t, n ) = k and n gcd( t,n ) = r is o dd , by L emma 4 , the n um b er of ro ots of h c ( z ) is equal to either 0, 1, 2 or 2 k + 1. F or i = 0, 1, 2 and 2 k + 1, let N i denotes the n um b er of c ∈ F ∗ 2 n suc h that h c ( z ) has exactly i ro ots in F 2 n . T h en N 0 = 2 k ( r +1) +2 k 2(2 k +1) , N 1 = 2 k ( r − 1) − 1 , N 2 = (2 k − 2)(2 kr − 1) 2(2 k − 1) , N 2 k +1 = 2 k ( r − 1) − 1 2 2 k − 1 . Since g a ( y ) has the s ame num b er of ro ots as h c ( z ) for c = a 2 l (2 t +1) ∈ F ∗ 2 n , and the corresp onden ce from c ∈ F ∗ 2 n to a ∈ F ∗ 2 n is bijectiv e b y gcd(2 t + 1 , 2 n − 1)) = 1, s o there are N i elemen ts a ∈ F ∗ 2 n suc h g a ( y ) h as exactly i ro ots in F 2 n , wh ic h implies that ther e are N 0 + N 2 elemen ts a ∈ F ∗ 2 n suc h th at f a ( x ) has exactly one ro ot in F 2 m b y Equ alit y (3.6). T hen w e ha ve R m = N 0 + N 2 = 2 k ( r +1) +2 k 2(2 k +1) + (2 k − 2)(2 kr − 1) 2(2 k − 1) = 2 n +2 k − 2 n + k − 2 n +1 2 2 k − 1 , and R m − 2 k = N 1 + N 2 k +1 = 2 n + k − 2 2 k 2 2 k − 1 .  4. Distribution of the Cross Corre la t ion C d ( τ ) Lemma 5 ([1 2]) . L et m = 2 n , then for a ny de cimation d with gcd( d, 2 n − 1) = 1 , the cr oss c orr elation value C d ( τ ) satisfies the fol lowing r elations: (1) 2 n − 2 P τ =0 C d ( τ ) = 1 ; (2) 2 n − 2 P τ =0 ( C d ( τ ) + 1) 2 = 2 m (2 n − 1) ; (3) 2 n − 2 P τ =0 ( C d ( τ ) + 1) 3 = − 2 2 m + ( ν + 3)2 n + m , wher e ν is the numb er of solutions ( x 1 , x 2 ) ∈ ( F ∗ 2 m , F ∗ 2 m ) of the e quations  x 1 + x 2 + 1 = 0 , x d (2 n +1) 1 + x d (2 n +1) 2 + 1 = 0 . 7 Prop osition 6. L et d b e define d by Equality (2.1). Then ther e ar e 2 k − 2 solutions ( x 1 , x 2 ) ∈ ( F ∗ 2 m , F ∗ 2 m ) of the system of e quations  x 1 + x 2 + 1 = 0 , x d (2 n +1) 1 + x d (2 n +1) 2 + 1 = 0 . (4.1) Pr o of. Supp ose that x 1 , x 2 ∈ F ∗ 2 m and let x 1 = y 2 l +1 1 , x 2 = y 2 l +1 1 . Since gcd(2 m − 1 , 2 l + 1) = 1, so there exists a one-to -one corresp ond ence b et ween ( x 1 , x 2 ) ∈ ( F ∗ 2 m , F ∗ 2 m ) and ( y 1 , y 2 ) ∈ ( F ∗ 2 m , F ∗ 2 m ). Then Equ ation (4.1) equiv alen tly b ecomes ( y 2 l +1 1 + y 2 l +1 2 + 1 = 0 , y d (2 l +1)(2 n +1) 1 + y d (2 l +1)(2 n +1) 2 + 1 = 0 . Since d (2 l + 1) ≡ 2 i (mo d 2 n − 1), w e ha ve y d (2 l +1)(2 n +1) 1 + y d (2 l +1)(2 n +1) 2 + 1 = y 2 i (2 n +1) 1 + y 2 i (2 n +1) 2 + 1 = ( y 2 n +1 1 + y 2 n +1 2 + 1) 2 i = 0 . Th us, the ab o ve system of equations b ecomes ( y 2 l +1 1 + y 2 l +1 2 + 1 = 0 , y 2 n +1 1 + y 2 n +1 2 + 1 = 0 , (4.2) whic h im p lies y (2 n +1)(2 l +1) 1 = ( y 2 m +1 2 + 1) 2 n +1 = y (2 n +1)(2 l +1) 2 + y (2 l +1)2 n 2 + y 2 l +1 2 + 1 and y (2 n +1)(2 l +1) 1 = ( y 2 n +1 2 + 1) 2 l +1 = y (2 n +1)(2 l +1) 2 + y (2 n +1)2 l 2 + y 2 n +1 2 + 1 . Then w e h a ve y (2 l +1)2 n 2 + y 2 l +1 2 + y (2 n +1)2 l 2 + y 2 n +1 2 = ( y 2 n + l 2 + y 2 )( y 2 n 2 + y 2 l 2 ) = ( y 2 n + l 2 + y 2 )( y 2 n − l 2 + y 2 ) 2 l = 0 . So, y 2 b elongs to F 2 n + l T F 2 m = F 2 k or F 2 n − l T F 2 m = F 2 k . In either case, w e al w ays ha ve y 2 ∈ F 2 k . S imilarly , we ha ve y 1 ∈ F 2 k . Then, Equation (4.2) b ecomes y 2 1 + y 2 2 + 1 = 0 , (4.3) or equiv alen tly , y 1 + y 2 + 1 = 0 (4.4) with y 1 , y 2 ∈ F ∗ 2 k , whic h has exactly 2 k − 2 solutions. Therefore, Equation (4.1) has 2 k − 2 solutions ( x 1 , x 2 ) ∈ ( F ∗ 2 m , F ∗ 2 m ). This finishes the pro of.  Theorem 1. Th e cr oss-c orr elation function C d ( τ ) has the fol lowing distribution:            − 1 , o ccurs 2 n − k − 1 times , − 1 + 2 n , o ccurs (2 n +1)2 k − 1 2 k +1 times , − 1 − 2 n , o ccurs (2 n − 1)(2 k − 1 − 1) 2 k − 1 times , − 1 − 2 n + k , o ccurs 2 n − k − 1 2 2 k − 1 times . 8 LEI HU, XIANGYONG ZENG, NIAN LI, AND WENFENG JIANG Pr o of. By Equalit y (2.2), w e ha v e C d ( τ ) + 1 = b ρ a (0) . F r om Prop osition 5, the rank of ρ a is m or m − 2 k . By th is together w ith Lemma 2, one can conclude th at b ρ a (0) tak es v alues from { 0 , ± 2 n , ± 2 n + k } , and C d ( τ ) p ossibly tak es v alues − 1, − 1 ± 2 n , − 1 ± 2 n + k . Supp ose that C d ( τ ) tak es v alues − 1, − 1 + 2 n , − 1 − 2 n , − 1 + 2 n + k and − 1 − 2 n + k exactly M 1 , M 2 , M 3 , M 4 and M 5 times, resp ectiv ely . Ob viously , M 1 + M 2 + M 3 + M 4 + M 5 = 2 n − 1 . (4.5) Since b ρ a (0) tak es v alues ± 2 n if and only if the rank of ρ a ( x ) is m , by Prop osition 5, w e hav e M 2 + M 3 = R m = 2 n +2 k − 2 n + k − 2 n + 1 2 2 k − 1 . (4.6) By Lemma 5 and Prop osition 6 , w e h a ve    − M 1 + ( − 1 + 2 n ) M 2 + ( − 1 − 2 n ) M 3 + ( − 1 + 2 n + k ) M 4 + ( − 1 − 2 n + k ) M 5 = 1 , 2 2 n M 2 + 2 2 n M 3 + 2 2( n + k ) M 4 + 2 2( n + k ) M 5 = 2 m (2 n − 1) , 2 3 n M 2 − 2 3 n M 3 + 2 3( n + k ) M 4 − 2 3( n + k ) M 5 = − 2 2 m + (2 k − 2 + 3)2 n + m . (4.7) The ab o v e Equ ations (4.5), (4.6) and (4.7) giv e M 1 = 2 n − k − 1 , M 2 = (2 n +1)2 k − 1 2 k +1 , M 3 = (2 n − 1)(2 k − 1 − 1) 2 k − 1 , M 4 = 0 , M 5 = 2 n − k − 1 2 2 k − 1 .  Remark 1. It is e asy to che ck that M 1 , M 2 and M 5 ar e always nonzer o and M 3 = 0 if and only if k = 1 . Thus, when gcd( l , n ) = 1 , the cr oss c orr elation C d ( τ ) takes exactly thr e e values, and when gcd ( l , n ) > 1 , the cr oss c orr elation takes exactly four values. Remark 2. By a c ompl ete c omputer exp eriments for up to m = 26 , Hel leseth et al liste d al l de cimation d suc h that C d ( τ ) having at most a six-value d cr oss-c orr elation function in [12] . Their the or etic al pr o ofs in [12, 13, 7] c ompletely explaine d al l de cimation d liste d in [12] and giving a thr e e-value d cr oss c orr elation, and they c onje ctur e d that the d satisfying d (2 l + 1) = 2 i (mo d 2 n − 1) and gcd( l, n ) = 1 ar e al l de cimations that give thr e e-value d cr oss c orr elation. F or the c ase of four-value d cr oss c orr elation, only p artial classes of de cimations d ar e shown to have this pr op erty [14, 6] . The ab ove the or em 1 c an explain al l de cimation d liste d in [12] and giving exactly a four-value d cr oss c orr elation, exc ept an exc eptional d = 7 for a smal l m = 8 c ase. The extensive numeric al exp eriments of He l leseth et al (up to m = 26 in [12] ) may suggest one to c onje ctur e, as Hel leseth et al do in [7] , that the c onverse of The or em 1 holds true, that is, f or m ≥ 12 , if C d ( τ ) has at most a four-value d cr oss c orr e lation, then ther e must exist some l and i such tha t d (2 l + 1) = 2 i (mo d 2 n − 1) . If this c onje ctu r e c an b e shown tr ue, then the c onje ctu r e in [7] is also true. This wil l b e an i nter esting op en pr oblem. Remark 3. When m is a p ower of 2, ther e ar e no d and l satisfying d (2 l + 1) = 2 i (mo d 2 n − 1) by L emma 1. This to gether with the c onje ctur e mentione d ab ove (if true) e xplains why ther e is no d gi v ing at most a four-value d cr oss c orr elation for m = 16 . 5. Conclusion W e hav e studied the cross correlation b et wee n an m -sequence of p er io d 2 m − 1 and the d - decimation of an m -sequence of p erio d 2 n − 1 for eve n m = 2 n . W e pro ve th at if d satisfies d (2 l + 1) = 2 i (mo d 2 n − 1) f or some l and i , then the cross correlation take s exactly either th ree 9 or four v alues, dep ending on gcd( l , n ) is equal to or larger than 1. The result enriches the wo r k of Helleseth et al. An in teresting op en problem is left. Referen ces [1] A. W. Bluher, “On x q +1 + ax + b ,” Finite Fields A pplic., vol. 10, n o. 3, pp . 285-305, July 2004. [2] H. Dobb ertin, P . F elke, T. Helleseth, and P . Rosendahl, “Niho typ e cross-correlation functions via Dickson p olynomials and Klo osterman su ms,” IEEE T rans. Inform. Theory , v ol. 52, no. 2, pp. 613-627, F eb. 2006. [3] S. W. Golom b and G. Gong, Signal Design for Go o d Correlation - F or Wireless Communicatio n, Cryptography and Radar. New Y ork: Cambridge Univ. Press, 2005. 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Helleseth, “A new family of four-va lued cross correlati on b etw een m -sequences of differen t lengths,” IEEE T rans. Inform. Theory , vol. 53, no. 11, pp. 4308-4313, N ov. 2007. [15] J. D. Olsen, R . A. Scholtz, and L. R . W elch, “Bent-function sequences,” IEEE T rans. Inform. Theory , vol. 28, no. 6, p p . 858-864, Nov. 1982. [16] P . Rosend ah l, ”Niho type cross-correlation functions and related equations,” Ph .D. dissertation, Department of Computer Science, U n iv. T u rku, T urku, Finland 2004. [17] X. Zeng, J. Q. Liu, an d L. Hu, “Generaliz ed Kasami sequences: the large set,” IEEE T rans. Inform. Theory , vol . 53, no. 7, pp. 2587-2598 , Jul. 2007. Lei Hu an d Wen fneg Jian g are with the St a te Key Labora tor y of Informa tion Securi ty, Grad - ua te Unive rsity of Chinese Academy of Sciences, Beiji ng, 100049, China. Email: hu@is.ac.cn Xiangyong Zeng and Nian Li are with the F a cul ty of Ma thema tics and Computer Science, Hubei University, W uhan,430062, China. Email: xzeng@hu bu.edu.cn