A New Binary Sequence Family with Low ... - Semantic Scholar
1
A New Binary Sequence Family with Low Correlation and Large Size Nam Yul Yu and Guang Gong, Member, IEEE
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This work was supported by NSERC Grant RGPIN 227700-00. The authors are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (email: [email protected], [email protected]).
but larger linear span giving better potential cryptographic property. Similarly, Udaya constructed a new family of binary sequences with low correlation but large linear span for even [19]. In [7], Kim and No generalized these two constructions at the price of the decrease of maximum linear span and the increase of maximum correlation.
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The other known approaches are summarized as follows. In [2], Chang et al. showed that a binary cyclic code based on three-term sequences [13] has five-valued nonzero weight distribution, which is identical to the dual code of the triple error correcting BCH code. From these cyclic codes, equivalent families of binary sequences with six-valued correlation , family size , and maximum linear of maximum span , can be constructed. In [12], it is shown that linear binary codes become nonlinear cyclic codes after a proper permutation. From these codes, Shanbhag, Kumar, and Helleseth presented a new generalized construction of binary sequence families in [16], including Kerdock and DelsarteGoethals sequences in [5].
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>+? A9B B9Y S Z Y X O P RS'T
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OQP+RSUT >@X)EB DWV S B[EG> "!#%$'& ! OQ]@R^SUT DWV >+X
In this paper, a new family of binary sequences of period is constructed for odd and an integer with . For a given , maximum correlation of sequences in is and its family size is . Similarly, a new family of binary sequences of period is also presented for even and an integer with , where maximum correlation and family size are and , respectively. The maximum and minimum linear spans of sequences in both and are and , respectively. As increases, we obtain a new family (or ) of exponentially increased family size with maximum correlation linearly increased from its optimal value. Since the family (or ) contains (or ) as a subset, it contains (or ) as a subset. Here, is the family of Gold-like sequences constructed by Boztas and Kumar, and is the one constructed by Udaya, where -sequences are excluded in both constructions.
S
>@?_A`B BaY SGb X BcE> \ >@?'\ !d ?<e;?/hiI I \ dQfg O P R^SUT Oc]@R^SUT
S
O P RS'T
cO ]@R^S A B T B OPR T
C
O P RS'T
Oc]+RSUB T OPR T OQ]@R B T
>+?'\
S
?/e;?
OQ]@RS'T
I dQfg
A BT O P RS j B Oc]@R T
S
For a specific application, we can choose a proper value and the corresponding family (or ). For example, a small value of can be chosen if low correlation is more crucial than large family size in the application. If large family size is more important, on the other hand, we can choose a large value of . The flexibility due to allows our new sequence family to have adaptive family size and maximum correlation for practical applications. Furthermore, our new sequence family has good potential cryptographic property with large linear span. The implementation of (or ) is extremely easy by summing linear feedback shift
S
O P R^SUT
S
O ] R^SUT
OQ]@RS'T
S
OkP@RS'T
2
> "> OQIJP"? R T
B E> "H !
register (LFSR) outputs. The family with maximum and family size is a good example correlation for compromise between correlation and family size. This paper is organized as follows. In Section II, we give some preliminaries on concepts and definitions of codes and sequences. Also, we review a weight distribution of a linear cyclic subcode of the second order Reed-Muller code [11], which will be used to investigate a correlation distribution of sequences in our new family. In Section III, we present new families and of binary sequences of period for odd and even , respectively, and analyze correlation and linear span of each family. In Section IV, the distribution of correlation values of sequences in is derived in terms of the weight distribution of a linear cyclic subcode of the second order Reed-Muller code. In Section V, an is given and implementation example of sequences in based on LFSRs is discussed. Concluding remarks and some observations are given in Section VI.
OQP@RS'T
>@?QA B
D
O ] R^SUT
OFP"R > T
O P R> T
II. P RELIMINARIES The following notations will be used throughout this paper.
is the finite field with elements and , the multiplicative group of . is a vector space over
with a set of all binary -tuples. - Let be positive integers and . The trace function from to is denoted by , i.e.,
i?I
V
R T
D C
B
I V
D
C D ? I I R T E ! " )E I # $U&%$ ? I E R TNV & (' I *) ? R+ T is simply denoted as '+R T if the context is f
clear.
A. Basic concepts (a) Boolean function ! " + be a vector in with .0 -/' and 0 Let , , a function from to . Then, the function %, taking on values 0 or 1 is called a Boolean function [11]. A Boolean function consists of a sum of all possible products of -21 ’s with coefficients or [4], i.e.,
?? I T
V R f R cT
I
i?I
R cT
B
I
and @ - @ is the vector whose I th element is the product of > 6 the I th elements of @ - and @ , respectively. Indeed, +J > 6 is all zero or all one vector of length ? , and is always the dual of the extended Hamming code which is also obtained from a Sylvester-type Hadamard matrix [11]. > is given by In (2), the I th element of a codeword in + 0 " ! a Boolean function +K N K with degree of at most ! " L M the I thL M elements of @ " ! A@ , , where K NK ;are ; L M L M respectively. If we remove the I th components corresponding " ! to K >M OK L M P , then we obtain a punctured RM ; L code + for , where each codeword has length .
R DiT R B iD T
f
R DiT
? T
R ? f
?
f
? Y V Y A B f R V iD T V D >+? A4B
A BT R >"?
(c) Trace representation of a binary periodic sequence Let be a set of all binary sequences of period QD and R be a set of all functions from to . For any 0 0 function S'TR , can be represented as [4]
O
R T 0
R T
R+ T[V 3
C -%U
f
I
?*V R%W f
V - .X TNYW - '
I
I V
>? V A B
where Q"- is a coset leader of a cyclotomic coset modulo , Q and Z-N is the size of the cyclotomic coset containing . For 0 any sequence [ \ !]J- ' , there exists S'^R such that
D D
V
WO R T B ! " ] - V %R _ TN`GcV 0 where _ is a primitive element of I . Then, R T is called a trace representation of [ . The linear span of the sequence [ is equal to a - V%U c D - , or equivalently the degree of the shortest Lb 0
B linear feedback shift registers that can generate [ .
(d) Weight and exponential sum of a binary codeword or sequence " ! Let [ be a binary codeword of length ] 5] 5] X Q , or a binaryB sequence of period Q . The number of ’s in the codeword is called a (Hamming) weight. Clearly, the total V Ed of a codeword or sequence sum of additive characters [ with weight e is given by
V5R
h T f RA BT
f
X
>@?A B
3
h
-%U
f R A B T d V VQ A>
B
e )
B [ is represented by a trace representation V R_ T for0 a primitive element _ of I , then - - -21 - R%,cT V R f ? TNV 3 -21 ? 9 & !;: ofB length ? " ", @ &H 5@ B ? > are where 1 V R basis where r is a phase shift of the sequence m and the indices " ! f vectors of length ? for R 7 DiT , G f G 6 7 D , are reduced modulo Q . If m is cyclically equivalent to [ , i.e., 0
0
" !
" "
Q V 0 a R+ T , i.e., ] -
If
and
0
3
n o " ! mn o V R]-;5]- 5]7- h T , n o R+r > B. Weight distribution of a linear subcode of R 7DiT e
Q,
In this paper, we consider a componentwise sum of a pair of binary sequences, where the sum is equivalent to a codeword of a linear cyclic subcode of the punctured second order ReedMuller code. Thus, we can apply the weight distribution of the subcode for the distribution of correlation values of the sequences.
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(a) Weight distribution of a codeword set with rank For odd , we consider a codeword given by
0
R+ T[V
I1 fd NT ` (' I (3) Y = 64 Y f X is an element in I . For even E
'R T B
3
'R 6 U
where each with 6 , on the other hand, we consider a codeword given by
D V >@X
h
f 'R I 1 T E R I TN f fd 6 fd (4) U 6 f (' I Y Y i X A B I for I . With , and ? ' where each ' " ! ? = 6 respect to a basis of I , V a -% U .- *- is an f expansion of with D- ' 0 I for all0 G . Applying thisf expansion ? R a - U *- *- T is equivalent to (3) or (4), we see that +R T V to a Boolean function of degree less thanf or equal to 2, and 0
R T V
'R B
TE
3
it may be written as follows:
R f " ! ? TNV ,"! ,$# &E %^ ,$#pY, ' ?I' )( (5) % where ! is an + matrix, is D *D ? binary upper triangular ! " a binary vector in I , and ( V R . T of length D % by Obviously, ! is determined by ’s for nonzero = , and . While ,9 V +R0 f " ! N ? T runs 6 through each nonzero binary i? D B -tuple>+in?iA B I , > %R ,c> T produces each element of a codeword of 0 length in R 7DiT . Equivalently, +R T forms a codeword > > in R 7DiT for (' I . 0
R%,cT V
0
For a given nonzero ! , it is well known that the weight distribution 0 of a set% of codewords of a quadratic Boolean function %, for all is determined by a rank of a symplectic ! # [11]. 0 Equivalently, matrix , -! we 0 can consider a 0 + + / / associated symplectic form . + / 0 j with + for given ’s with [5]. We list the = 6 following fact regarding the distribution of exponential sums 0 of + .
R cT E V
E R T B E Y R E 6 Y T X R 'T
R TkV
R T
R T
0
R T B Y Y X B Y `YX >+R /? hiI T R "TV I >?/h I
Fact 1 (Theorem 6.2 in [5]): Let ’s of + in (3) or (4) 6 = be given such that at least one is nonzero for , 6 3 where 0 21 . For an integer with , if . + / 54 j has a rank , or equivalently . + / has j solutions in ' for all / ' , then the exponential sum 0 74 of takes on values of and 6 for all ' , and B its distribution is given by
X V ?I >
R T
3 f "g i
I
8 9 : E >+?/h4 A_>+?/h4
R A B T j ef g V
! 0 with +R T V .
>@?AG>I4 >I4+h E >times 4+h >I 4+h f AG>4+h f times f f times
(b) Weight distribution of a linear subcode with multiple ranks For a set of distinct nonzero ! ’s, we can consider a set of 0 % codewords given by , in (5) for all . Equivalently, we 0 can consider a set of codewords given by for distinct sets of ’s such that at least0 one is nonzero in each set 6 6 have distinct multiple of ’s for . Then, may = 6 ranks each of which corresponds to each set of given ’s. 0 6 Consequently, the exponential sum of can take on values 0 4 of and 6 for each possible from Fact 1. If further constitutes a linear subcode for the sets of ’s, we 6 may use the weight distribution of the subcode in0 order to investigate the distribution of exponential sums of . Next, we specify the known weight distributions of two lin> ear cyclic subcodes of for odd , which will be used in a later section for determining the correlation distribution of a new family of sequences. For ]Z in n@and ? I with ;= +IZ for odd , a linear cyclic subcode given by 0 A + ]* for ' has the distribution of weights and corresponding exponential sums in Table I, where . For ]D in for odd , a linear cyclic n 0 subcode given by + 9 \ %]* 2B DC # 9 for (' has the distribution of weights and corresponding exponential sums in Table II. In both Tables, the exponential 0 f sum means a fhg"i with + . There are several j different ways to establish the validity of weight distribution of n ? n , for example, see [3] and [11]. For those of , see [11].
R cT
B Y
R T
Y4X
R T
R T
>+?/h
R > 7DiT
R T
R T
D
I R 7DiTkV B E DI I R T[V 'R T 'R dQf T I V ?/h I f D R T V 'R T E 'R @T E I A BT e R R "T[V g !
'R T
#
III. N EW
FAMILY OF BINARY SEQUENCES WITH LARGE SIZE
In this section, we present new families of binary sequences with large family sizes as well as large linear spans for odd and even .
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W EIGHT DISTRIBUTION OF
Weight
W EIGHT DISTRIBUTION OF
/.+* . / .+*
! & 7$'! &
2
A
Exponential Sum
. /. /. . /.
Weight
TABLE I GIVEN BY
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- V > > >I hI is a binary sequence of trivial solution of @D-[VOJ!- for Y G bS . Thus, sequences in + > : ? A B C B >C f V R _ T for a primitive element _ of O P R^SUT represented by R T for any @ - in I with Y G b S , period with I , where > +R T , the trace> representation of >C , is given by are cyclically distinct. > The crosscorrelation of two sequences > and G in O P RS'T > +R T[V DE : % I 7 \ h jI Next, we verify that
R+ T is not aB constant & < : < : : % # associated with R T has at most L $'& L f roots in ' for polynomial of (i.e., a trivialL polynomial). I If H V , - V B Y G Y S A B from (12), and - V for S Y G Y X from all / ' . for B Y G Y S A9B because (9). Then, at least one - is nonzero for Proof: For given - ’s, the symplectic form . +R /"T B B/E H I T 0 j at least one - is nonzero. If H V , on the other hand, R associated with +R T is given by : B Y Y A B. 0 0 E 0 E E cannot be zero although - may be zero for all G S . j R+ /"TFV R+ T R /T R/T Therefore,
polynomial of with at least & L : S AGB . At this point, f I I I I [ B E E E $ $ than or equal to we need to show that it - T ) (14) & : L L L 6 can take on values of and for all ’s such that D -%U f in order to apply Fact 0 1 for determining the distribution of exponential sums of + .
6
>S A B
is every positive odd integer less than or equal to . In order to do so, we need the following fact from [11]. (Note. we slightly change the representation of the result in [11].)
B Y
?I >
Y
be a set of symplectic Fact 2 ( [11], page 454): Let 0 1, forms of (10). For some fixed integer with assume that the rank of every nonzero form in is at least . Then, the maximum size of is given by
V
>+?/e % '& h > e;?/h ! e g ! h fg ! g
for odd for even
D D
)
In the following, we consider a specific linearized polynomial and investigate a rank of the symplectic form corresponding to the polynomial.
S
B
Lemma 3: For an integer , consider a symplectic form . + / / + where a linearized polynomial + is given by
R "T[V
'R
R TT
R
TkV 3
\7h -%U
f
f R I $ V I $ V E -
R T
-
I VT
D A > > A B to S .
where
is every positive odd integer less than or equal
0
R T
V
V B R T R T A> V B B D V K R T
B
, then the linearized polynoProof: For , by H mial given in (14) (if H , it is identical to (13)) is of the form of + in Lemma 3. From Lemma 3, the exponential 0 4 sum of for H is equal to or 6 for all ’s " ! such that J0 . Thus, we can say that 74 the exponential sum of takes on values of and 6 " " for every integer such that . J Now, it suffices to show that the exponential sum takes on 74 values of 6 at least once when . Or equivalently, we need to show the symplectic form . + / j in (10) has a rank of at least once for some - ’s. Assume that this rank never occurs for all - ’s. Then, the rank of . / is at least . From j Fact 2, the maximum size of a set of such . / ’s is j . However, from (9) and (10), its actual size is , which is greater than . Therefore, the rank of occurs at least once when - ’s run through . This completes the proof of Lemma 4.
>?/h >S A K >? B > A D V K S K >+?/h D A> V > S A B R T D A >S E B > V D A5> S E K R T R T >+?/e \ h V fg >+?U\ I D A >S E B
- can be any A element of . For odd D , the rank Combining the cases 1, 2, and 3, we have the following > S E K inand I every > for possible rank . R+A > /"T E is atY5least D result on the correlation of sequences in OFP"RS'T . W > Y A B occurs at least once when - runs D S K D 5: The correlation of binary sequences in O P RS'T is through I . For even D , on the other hand, the rank is at A > S EG> and every possible rank > for D A > S EG> Y R > S Lemma n 5 B@E > "!#%$'& E > T -valued least D and maximum correlation is ! . >a Y A > D occurs at least once when - runs through I . Proof: For a trace representation R+ 0 > T of each sequence Proof: Note that the odd case in Lemma 3 is implicitly > in O P RS'T , we can consider the exponential sum of R+ T " > ? known from Theorem 16 and Corollary 17 of Chapter 15 in (8). In cases 1 and 2, the exponential sum has and in [11]. Here, we reproduce it for completeness. values. For all - ’s in case 3, from Lemma 4, the exponential >?/h74 for each integer such that D A >GY I Y S . By V sum takes on and 6 Let I be an integer with V > B ! " > 5 A B ! " M M @ Q d f S . Thus, the exponential sum takes on V \h f V h , we ' R / R+ TT where > V 7 K >? and , therefore, the I V have I V E . M R - I / V TWT . V Then, M S nonzero distinct values. Including R T V a -M U f R - # $ $ the maximum > 4 E > M f I A $'& $ is >Ie M h fg and thus the rank of overall exponential sum is R S T -valued. the degree of R R+ TT > S E5Equivalently, > T -valued. Since A > E > M Q O " P R ' S T R correlation of sequences in is . R / AG T is> atE least D I (If D is odd, the rank is at A > is determined by > S A B from M maximum value for D I K because it should beA even). least D n
A N B > I E K never occurs for Lemma 4, V A>@?/h4 @ V B[E> !^! #%$'& . For odd D , assume that the rank D B:Y Y A`B , and thus the rank of . R / T is at all - ’s with Proof of Theorem 1. The results follow directly from Lem
A > G I E M of a set I least D . Then, from Fact 2, the maximum size
mas 1 and 5. >+?/e hI / T V of such . R+ ’s is . However, its actual M g >@?<e h M when - ’s with B:Y G Y I A B run through I , Remark 1: If S V B , size is M fg Aj> I E K which is greater than . Therefore, the rank of D E 3 ' R+ fd I V T5` ' I occurs at least once when - ’s run through I . For even D , + E + R F T V ' F R @ T D (16) A> I E4> never occurs, then the minimum rank B % U if the rank D A > I E . Similarly, the maximum size of a set of such f is D which represents the Gold-like Sequences introduced by Boz > ; e / ? h e h . R / T ’s is>+?/e h V its tas and Kumar [1]. From Lemma 4, a positive odd integer f g M fg , which is A smaller > I E > than M actual size occurs less than > S is only B , so D A > V B . Hence, V ?Y I Y S when Finally, Gold-like>+? sequences AB A B A given B 6 > by '& (16) Since
B9 R+ Y T contains every R+ T ’s for correlation, or for all @ ' I . ! Y S A5B run through M I , the assertion of B - ’s with G > where
Example 1: If
Lemma 3 follows.
Using Lemma 3, we have the following result.
S Y X
D V >+X E B T X B Y R Y4 I
S
BY
and an integer with Lemma 4: For0 odd , let 8- ’s of + in (8) be given such that at least one G is nonzero , and ' . Then, the exponential 0 for 4 sum of + can take on valuesB of and 6 for an integer
R T
>?/h
S V
,
E 'RA@ B T E 3 'R D E D & +R T[V 'RF@ T L f B -U I I for ' and odd D . FromA >Lemma B 4,
IV fd T D A >
(17)
is a positive odd integer less than , so and . Thus, B . From Lemma 5, sequences given by (17) have and
?/hI
f
?/hI
D
V
K
V
7
>+?NA B A B A B > % '& A B > H ! ! & R T R T I > OPR T
6 6 six-valued correlation, or for all @ @ ' . For sequences 0 represented by D E D + L and E B ;H , the corresponding + in (8) constitutes a L linear cyclic subcode with five nonzero distinct weights if . In fact, the we assume that H can be any element in correlation distribution of can be derived from Table II. The details will be discussed in Section IV.
f & R T
I
O ] R^SUT , we R T h V 0 E f R TNV 'R T 3 'R - fd I T E f R fd I T (20) B -%U f where E 8 GcV 9 : @ E J H" I V B Y G b S 8-V @.B- J!-AB H fV d (21) B[E H I Y 4 Y X S G fd Y for @ - )J - ' I with S and 0 H' I . If - V for all B Y G Y5X , the exponentialGkb4 sum of +R T has a value of or >@? . Otherwise, we have to consider the number of solutions of . j R /Tk0 V in order to derive the distribution of exponential sums of +R T in (20). >@X and an integer S with B Y S b X , Lemma 7: For even D V 0 let - ’s of R T in (20) be given such that at least one - is BaY G Y X . Then, the symplectic form . +R /T nonzero for 0 >"I%\ roots in (' I j for all associated with +R T has at most / ' I . To investigate 0 the correlation of sequences in need to consider + given by
S V K, E D E D & D ! R+ TFV 'RA@ T 'RA@ B T L L f B E 'RF@ I C T E 3 'R+ f d I V T (18) -%U B ?/h ?/h I for ' and odd D . Similarly, V I f I B and ?/hI C . From Lemma 5, sequences given by (18) have eight>+? A B A B A B > '& A B > "H A B valued correlation, or > ! / for all @ %@ @ I ' I . 6 ! 6 ! 6 B f B. Construction of O ] RS'T for even D >@X and an integer S with 2: For even D V B Y Construction X S b , a family O ] R^SUT of binary sequences is defined by Proof: We have " " \h h Oc]@RS'T[V ) > ? V AR @ %@ f 5T B@ - ' I I VT E I f I$ V I$ V E B . R / k T V / 3 R 8 + " ! I hI j > where > V > > is a binary sequence of -%U + > : ? A B C f f B period with >C V R_ T for a primitive element _ of V R / R TJT ) I , where > +R T , the trace> representation of >C , is given by I Thus, . R+ /"T[V for all / ' \h if and only if R T V . V j f I E Using the same approach as we did in the proof of Lemma 2, > R+ TFV 'RA@ T 3 '[email protected] fd T we obtain
R T identical to (13) and the equation (15). B -%U fV (19) Following the same steps as in the proof of Lemma 2, we h E 3 f ' R fd I T E +R f d I T can derive that the number of solutions of . R /TkV is at >I%\ . j f \ most -%U >@X and an integer S with B Y S b X , for (' I . Lemma 8: 0 For even D V >+X and an integer S with B:Y S b let 8- ’s of R B YT in Y5 (20) be given such that at least one - is X , and ' I . Then, the exponential Theorem 2: For even DaV X , the @ > ' ? \ G nonzero0 for >? a ? A B > E sum of R+ T can take on valuesB of and 6 for an integer T - where D Aj> is zero or every positive even integer of period . The correlation of sequences is R S \ @ B E > less than !"d . Therefore, O ] R^SUT > S . Hence, the correlation of binary sequences valued and maximum correlation is + > ? 4 A B @ > U ? \ [ B E > \ or equal to in !d T signal set. constitutes a R cO [B ]@ER^SU> T is \ R > S E T -valued and maximum correlation n is !d . In order to prove Theorem 2, we need the following lemmas. B Y Since their proofs are similar to those for O P RSUT , we omit the Proof: For even D , we see that while each - with details. Gkb4S runs through I , each - of R T in (13) runs through I . From Lemma 6: All sequences in OQ]@R^SUT are cyclically distinct. 0 Lemma 3, therefore, we obtain >?/7h that 4 the exponential >@?U\ . sum of +R T takes on values of and 6 for every integer Hence, the family size of OQ]+RSUT is A > V > " ! > S AG> . such that D A > Proof: Similar to the proof of Lemma 1, +R TV that the cases D V and D A > V Next, we will show > all in I if and only if (7) is achieved. If > S also occur. If all - ’s of R T in (13) are zero, then R T[V GR;H B T E for > - >+?9A B T V B , then is not a factor of ; - h >+? A5B T since ; - B E5> - h T V B for has at most four solutions, i.e., V ! f ! , and f ; I B Y Y X V occurs at least S G have at least two equations of H fd in f V . Hence, the case D X for even D . Thus, (7)V has, for (7) because S b a unique solution once.fd Applying Fact 2, we also obtain that exponential B . Hence, all sequences in O ] RSUT are cyclically sum of 0 +R T can take on values of 6 >?/h 4 at the given by H V least once such A > V > S in the similar way to the proof of Lemma 4. distinct. that D Example 2: If
8
0
R T takes on values of D A> >E V J > " ! > S . R S T -valued. FurV ?I A S .
Therefore, the exponential sum of + 74 and 6 for an integer where Including n and , the correlation is thermore, for
>+?/h
>+?
V B[E4> !' d \
Proof of Theorem 2. The results follow directly from Lemma 6 and Lemma 8.
S V B, h IV E E f DE@R T[V 'RF@ T 3 'R fd T B -%U f
TABLE III L INEAR SPAN AND
+*,2 -/ .1 0 +*, 2 /.10 +*, 2 $ 0 .. .
Remark 2: If
13 )
THE NUMBER OF CORRESPONDING SEQUENCES IN
+*; 2 2 (-/.10
( OR
- ( .% 7 ( .% 7 (
Number of sequences
(. 2(
2.
.. .
I T f d (22) Since =M@ - ’s are and RS A =T @ - ’s are nonzero, the number of f \ R >+? AB T \h . corresponding sequences given by R+ T is (' I 6 Y > 6 Y S , we obtain = = Applying this result to each with which represents the sequences constructed by ? Udaya? [19]. A > V and > . Hence, V I and I AGB . the linear span R^SUT of binary sequences in OFP@RS'T with the From Lemma 8, D Finally, the sequences given by (22) has six-valued correlation, distribution shown in Table III. Using the similar approach to >+? AB A B A B 6 > A B 6 > for all @ ' I . odd case, we see that the linear span of sequences in O ] R^SUT is or ! !'dQf the same as OQPR^SUT . B >, Example 3: If S V Corollary 1: The maximum and minimum linear E ?/e ? and ?/e;?<spans hI%\ of, D E D & +R TkV 'AR @ T 'FR @ B T P sequences in O R ' S T and Q O @ ] R ' S T are I dQfg I dQfg L f B E 3 'R+ f d I V T E R+ fd I T (23) respectively. f -%U I I A > > D. Comparison of families of binary sequences D D V for (' and even . From Lemma 8, In Table IV, we give some comparisons of the families V ? I ? I A B ? I A > . Hence, and thus, the correlation of O P R^SUT and Oc]@R^SUT with well known families of binary sequences >?QA B A B A B > A B sequences given by (23) belongs to > A B 6 > 8I for all @ %@ ' I . 6 ! 6 with low correlation. In Table IV, we include a family of !dQf !d binary sequences constructed from Chang et al.’s investigation B f 0 of a binary cyclic code based on three-term sequences [2]. For Remark 3: Contrary to odd D , R T in (20) does not con>@X@EGB , each sequence in the family is represented by Y X Gkb may odd D V stitute a linear cyclic subcode because - with S E 'RA@ C T E 'R+ C ! TN V > dQf EB not run through all elements in I . Therefore, we should use DE D R+ T[V 'RA@ T & L f B I different methods from odd D to investigate the correlation I > for (' and all @ @ f in . On the other E hand, a familyE distribution in O ] R T , which is a problem we are working on. 'RF@ B T of sequences defined B by D E D R+ T[V 'RA@ T & B correcting BCHf code, '+R $C7T from the dual of the L triple n error C. Linear spans of O P R^SUT and Oc]@R^SUT has the same period, family size, , and maximum linear The linear spans of binary sequences in OFP"RS'T and O ] R^SUT span as those of Chang et al.’s for all @ %@ in I for odd Y B f GQb4S . D [11]. are determined by the number of nonzero @Z- ’s with In terms of periods and maximum linear spans, new famTheorem 3: In the family O P R^SUT (or Oc]@RS'T ), consider a ilies O P R^SUT and Oc]+RSUT are identical to the families of Goldsequence represented by R+ T where = @ - ’s in ? V like sequences ! " \h > and Udaya’s, respectively. At the expense of FR @ @ T are equal to . Let 6 RSUT be the linear span maximum correlation n 5 f , however, we have larger family of B the sequence. Then, P sizes in O ^ R U S T and Q O ) ] ^ R U S T than in those families. Basically, we A > = EB T k D R D Y Y P obtain and with exponentially increased family O R ' S T Q O @ ] R ' S T RS'T[V > ` = S n 5 6 sizes by the linear increase of from optimal correlation. \ >@? AGB \7h and there are R T 6 sequences in O P RS'T (or Oc]@R^SUT ) Furthermore, we can choose S and the corresponding family 6 RSUT . From this result, the linear span of O P R^SUT (or Oc]@R^SUT ) for its specific application. For example, if having linear span 6 sequences in OcP"RS'T (or O ] R^SUT ) and its distribution are shown low correlation is more crucial than a large family size in the application, then a small value of S is chosen. Otherwise, in Table III. we choose a large value of S in order to get a large family Proof: Firstly, consider the linear span of sequences in size. Considering the flexibility due to S , OFP@R^SUT and O ] RS'T have O P R^SUT . In Construction a sequence represented by +R T larger family sizes than any other families of binary sequences X E B V ? dQI f 1,trace > has a total terms and each trace term has in Table IV. the linear span of D . If =&@ - ’s of the sequence are equal to , ? I A =+T nonzero trace terms and the corresponding In OcP"RSUT and O ] R^SUT , C -sequences are not included for it has R dQ f increasing their minimum linear spans. Thus, they have larger linear span of the sequence is given by minimum linear spans than any other binary sequence families EB A A> = E B T D k D R D Y Y in Table IV except for bent sequences, which will be good RS'T[ V > = D V > ` = S ) property for potential cryptographic applications. 6 +R
9
TABLE IV C OMPARISON OF THE Family of Sequences
FAMILIES OF BINARY SEQUENCES WITH LOW CORRELATION
Period
Family Size
Linear Span (Maximum, Minimum)
'! & $ 2 ! ! / . ( 2 ! ! or ! - ! Bent [14] .10 ! @*;2 -/.10 ' & Boztas and Kumar [1] ! @*;2 -/.10 - ! -/. Udaya [19] 2 - ! H Chang et al. (triplet) [2] [11] 5 2 - ! H Rothaus [15] 5 2 ' & 0 / . Kerdock [16] ! 2 H 20 / . ! Delsarte-Goethals [16] 2
@*;2 -/.10 +*;2 $ 0 ! H New Family 1 2 2
@*;2 -/.10 +*;2 $ 0 - ! New Family 31
( @*;2 -/.10 +*; 2 2 (-+X E B
OQP"R > T
OQPR > T
O P R > NT V ) > ? V RF@ @ f T5 @ - ' I " ! I hIB > where > V > > is a binary sequence of >+? A B with B f >C V DE D R_ C T for a primitive element period L & by _ of I , where DE D & R T is given L IV E E D E D & R TkV 'RF@ T 'RF@ B T 3 'R+ fd T L f B -%U I I for (' . > From Theorem 1, we know that the family O P R T has >IJ? cyclically distinct binary sequences of period >"?A B . > The correlation of sequences in O P R T is six-valued and its B E > H . Consequently, OQP+R > T constitutes a maximum is >R @?:AGB >IJ? BcE> % H ! T signal > set. This data of OQPR T is listed !
in Table IV. Next, we will investigate the distribution of correlation values of sequences in . The correlation of a pair of is derived from the exponential sum of sequences in
0
OQP"R > T > OQP@R T E J HT; T E 'RJRF@ E J H B TE B T R T[V 'RJRF@ f f B B E 3 'RJR B[E H f d I V TE fd I V T -%U I
for
@
f
E@
of
%@ )J 9J ' B J )H B f , and B ff 9 0
R+ T V
odd even even even odd even odd odd odd odd odd even odd even
I and H V ' I . With ] VO@ V QB E H f d I for >Y G Y4X , 0 R T B TE
'R]*
I
'R B
I
TE
3
B
-%U
I
'R - 9
E
V
J H"
has B a form
IV fd T
(24)
' . Depending on - ’s, the for ]D ' and 0 - ' 9 exponential sum of 9 can be classified into two exclusive cases. To facilitate the analysis, assume that - can be any 9 element in .
I -cV
R T
> Y
Y X
0
R T V > 'R T E R iD T
G Case 1. for . In this case, + > ]* 9 B constitutes a linear cyclic subcode of for ]Z0 ' . Thus, the distribution of the exponential sums of + is identical to Table I with . G0 for Case 2. At least one - . In this case, the 9 distribution of the exponential sums of + follows from the following lemma.
'R )T I R T
V
V X ? V Y f Y4 R T
Lemma 9: For given - ’s with at least one nonzero 9 in (24) has five-valued exponential sum for all ]Z in the distribution is
R A B kT j e f g f g"i !8 >@?@?"I 4 h >4 h ? A>"I54 f RR >+?A & >"f I6 4 & T E & f I T R >@times f I R >"I 4 ! h f 6 >4 ! h f T times ! T times (25) where V >+?
A New Binary Sequence Family with Low Correlation and Large Size Nam Yul Yu and Guang Gong, Member, IEEE
+ *,2 -/.10 "!! #%$'& )( 3 4 567 98 +*;)2 -@? AB
C
D
D
D
This work was supported by NSERC Grant RGPIN 227700-00. The authors are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (email: [email protected], [email protected]).
but larger linear span giving better potential cryptographic property. Similarly, Udaya constructed a new family of binary sequences with low correlation but large linear span for even [19]. In [7], Kim and No generalized these two constructions at the price of the decrease of maximum linear span and the increase of maximum correlation.
D
The other known approaches are summarized as follows. In [2], Chang et al. showed that a binary cyclic code based on three-term sequences [13] has five-valued nonzero weight distribution, which is identical to the dual code of the triple error correcting BCH code. From these cyclic codes, equivalent families of binary sequences with six-valued correlation , family size , and maximum linear of maximum span , can be constructed. In [12], it is shown that linear binary codes become nonlinear cyclic codes after a proper permutation. From these codes, Shanbhag, Kumar, and Helleseth presented a new generalized construction of binary sequence families in [16], including Kerdock and DelsarteGoethals sequences in [5].
BFEG> % H !
K"D
>+? A9B B9Y S Z Y X O P RS'T
>"IJ?
LNM
OQP+RSUT >@X)EB DWV S B[EG> "!#%$'& ! OQ]@R^SUT DWV >+X
In this paper, a new family of binary sequences of period is constructed for odd and an integer with . For a given , maximum correlation of sequences in is and its family size is . Similarly, a new family of binary sequences of period is also presented for even and an integer with , where maximum correlation and family size are and , respectively. The maximum and minimum linear spans of sequences in both and are and , respectively. As increases, we obtain a new family (or ) of exponentially increased family size with maximum correlation linearly increased from its optimal value. Since the family (or ) contains (or ) as a subset, it contains (or ) as a subset. Here, is the family of Gold-like sequences constructed by Boztas and Kumar, and is the one constructed by Udaya, where -sequences are excluded in both constructions.
S
>@?_A`B BaY SGb X BcE> \ >@?'\ !d ?<e;?/hiI I \ dQfg O P R^SUT Oc]@R^SUT
S
O P RS'T
cO ]@R^S A B T B OPR T
C
O P RS'T
Oc]+RSUB T OPR T OQ]@R B T
>+?'\
S
?/e;?
OQ]@RS'T
I dQfg
A BT O P RS j B Oc]@R T
S
For a specific application, we can choose a proper value and the corresponding family (or ). For example, a small value of can be chosen if low correlation is more crucial than large family size in the application. If large family size is more important, on the other hand, we can choose a large value of . The flexibility due to allows our new sequence family to have adaptive family size and maximum correlation for practical applications. Furthermore, our new sequence family has good potential cryptographic property with large linear span. The implementation of (or ) is extremely easy by summing linear feedback shift
S
O P R^SUT
S
O ] R^SUT
OQ]@RS'T
S
OkP@RS'T
2
> "> OQIJP"? R T
B E> "H !
register (LFSR) outputs. The family with maximum and family size is a good example correlation for compromise between correlation and family size. This paper is organized as follows. In Section II, we give some preliminaries on concepts and definitions of codes and sequences. Also, we review a weight distribution of a linear cyclic subcode of the second order Reed-Muller code [11], which will be used to investigate a correlation distribution of sequences in our new family. In Section III, we present new families and of binary sequences of period for odd and even , respectively, and analyze correlation and linear span of each family. In Section IV, the distribution of correlation values of sequences in is derived in terms of the weight distribution of a linear cyclic subcode of the second order Reed-Muller code. In Section V, an is given and implementation example of sequences in based on LFSRs is discussed. Concluding remarks and some observations are given in Section VI.
OQP@RS'T
>@?QA B
D
O ] R^SUT
OFP"R > T
O P R> T
II. P RELIMINARIES The following notations will be used throughout this paper.
is the finite field with elements and , the multiplicative group of . is a vector space over
with a set of all binary -tuples. - Let be positive integers and . The trace function from to is denoted by , i.e.,
i?I
V
R T
D C
B
I V
D
C D ? I I R T E ! " )E I # $U&%$ ? I E R TNV & (' I *) ? R+ T is simply denoted as '+R T if the context is f
clear.
A. Basic concepts (a) Boolean function ! " + be a vector in with .0 -/' and 0 Let , , a function from to . Then, the function %, taking on values 0 or 1 is called a Boolean function [11]. A Boolean function consists of a sum of all possible products of -21 ’s with coefficients or [4], i.e.,
?? I T
V R f R cT
I
i?I
R cT
B
I
and @ - @ is the vector whose I th element is the product of > 6 the I th elements of @ - and @ , respectively. Indeed, +J > 6 is all zero or all one vector of length ? , and is always the dual of the extended Hamming code which is also obtained from a Sylvester-type Hadamard matrix [11]. > is given by In (2), the I th element of a codeword in + 0 " ! a Boolean function +K N K with degree of at most ! " L M the I thL M elements of @ " ! A@ , , where K NK ;are ; L M L M respectively. If we remove the I th components corresponding " ! to K >M OK L M P , then we obtain a punctured RM ; L code + for , where each codeword has length .
R DiT R B iD T
f
R DiT
? T
R ? f
?
f
? Y V Y A B f R V iD T V D >+? A4B
A BT R >"?
(c) Trace representation of a binary periodic sequence Let be a set of all binary sequences of period QD and R be a set of all functions from to . For any 0 0 function S'TR , can be represented as [4]
O
R T 0
R T
R+ T[V 3
C -%U
f
I
?*V R%W f
V - .X TNYW - '
I
I V
>? V A B
where Q"- is a coset leader of a cyclotomic coset modulo , Q and Z-N is the size of the cyclotomic coset containing . For 0 any sequence [ \ !]J- ' , there exists S'^R such that
D D
V
WO R T B ! " ] - V %R _ TN`GcV 0 where _ is a primitive element of I . Then, R T is called a trace representation of [ . The linear span of the sequence [ is equal to a - V%U c D - , or equivalently the degree of the shortest Lb 0
B linear feedback shift registers that can generate [ .
(d) Weight and exponential sum of a binary codeword or sequence " ! Let [ be a binary codeword of length ] 5] 5] X Q , or a binaryB sequence of period Q . The number of ’s in the codeword is called a (Hamming) weight. Clearly, the total V Ed of a codeword or sequence sum of additive characters [ with weight e is given by
V5R
h T f RA BT
f
X
>@?A B
3
h
-%U
f R A B T d V VQ A>
B
e )
B [ is represented by a trace representation V R_ T for0 a primitive element _ of I , then - - -21 - R%,cT V R f ? TNV 3 -21 ? 9 & !;: ofB length ? " ", @ &H 5@ B ? > are where 1 V R basis where r is a phase shift of the sequence m and the indices " ! f vectors of length ? for R 7 DiT , G f G 6 7 D , are reduced modulo Q . If m is cyclically equivalent to [ , i.e., 0
0
" !
" "
Q V 0 a R+ T , i.e., ] -
If
and
0
3
n o " ! mn o V R]-;5]- 5]7- h T , n o R+r > B. Weight distribution of a linear subcode of R 7DiT e
Q,
In this paper, we consider a componentwise sum of a pair of binary sequences, where the sum is equivalent to a codeword of a linear cyclic subcode of the punctured second order ReedMuller code. Thus, we can apply the weight distribution of the subcode for the distribution of correlation values of the sequences.
>
DaV >+X E B
(a) Weight distribution of a codeword set with rank For odd , we consider a codeword given by
0
R+ T[V
I1 fd NT ` (' I (3) Y = 64 Y f X is an element in I . For even E
'R T B
3
'R 6 U
where each with 6 , on the other hand, we consider a codeword given by
D V >@X
h
f 'R I 1 T E R I TN f fd 6 fd (4) U 6 f (' I Y Y i X A B I for I . With , and ? ' where each ' " ! ? = 6 respect to a basis of I , V a -% U .- *- is an f expansion of with D- ' 0 I for all0 G . Applying thisf expansion ? R a - U *- *- T is equivalent to (3) or (4), we see that +R T V to a Boolean function of degree less thanf or equal to 2, and 0
R T V
'R B
TE
3
it may be written as follows:
R f " ! ? TNV ,"! ,$# &E %^ ,$#pY, ' ?I' )( (5) % where ! is an + matrix, is D *D ? binary upper triangular ! " a binary vector in I , and ( V R . T of length D % by Obviously, ! is determined by ’s for nonzero = , and . While ,9 V +R0 f " ! N ? T runs 6 through each nonzero binary i? D B -tuple>+in?iA B I , > %R ,c> T produces each element of a codeword of 0 length in R 7DiT . Equivalently, +R T forms a codeword > > in R 7DiT for (' I . 0
R%,cT V
0
For a given nonzero ! , it is well known that the weight distribution 0 of a set% of codewords of a quadratic Boolean function %, for all is determined by a rank of a symplectic ! # [11]. 0 Equivalently, matrix , -! we 0 can consider a 0 + + / / associated symplectic form . + / 0 j with + for given ’s with [5]. We list the = 6 following fact regarding the distribution of exponential sums 0 of + .
R cT E V
E R T B E Y R E 6 Y T X R 'T
R TkV
R T
R T
0
R T B Y Y X B Y `YX >+R /? hiI T R "TV I >?/h I
Fact 1 (Theorem 6.2 in [5]): Let ’s of + in (3) or (4) 6 = be given such that at least one is nonzero for , 6 3 where 0 21 . For an integer with , if . + / 54 j has a rank , or equivalently . + / has j solutions in ' for all / ' , then the exponential sum 0 74 of takes on values of and 6 for all ' , and B its distribution is given by
X V ?I >
R T
3 f "g i
I
8 9 : E >+?/h4 A_>+?/h4
R A B T j ef g V
! 0 with +R T V .
>@?AG>I4 >I4+h E >times 4+h >I 4+h f AG>4+h f times f f times
(b) Weight distribution of a linear subcode with multiple ranks For a set of distinct nonzero ! ’s, we can consider a set of 0 % codewords given by , in (5) for all . Equivalently, we 0 can consider a set of codewords given by for distinct sets of ’s such that at least0 one is nonzero in each set 6 6 have distinct multiple of ’s for . Then, may = 6 ranks each of which corresponds to each set of given ’s. 0 6 Consequently, the exponential sum of can take on values 0 4 of and 6 for each possible from Fact 1. If further constitutes a linear subcode for the sets of ’s, we 6 may use the weight distribution of the subcode in0 order to investigate the distribution of exponential sums of . Next, we specify the known weight distributions of two lin> ear cyclic subcodes of for odd , which will be used in a later section for determining the correlation distribution of a new family of sequences. For ]Z in n@and ? I with ;= +IZ for odd , a linear cyclic subcode given by 0 A + ]* for ' has the distribution of weights and corresponding exponential sums in Table I, where . For ]D in for odd , a linear cyclic n 0 subcode given by + 9 \ %]* 2B DC # 9 for (' has the distribution of weights and corresponding exponential sums in Table II. In both Tables, the exponential 0 f sum means a fhg"i with + . There are several j different ways to establish the validity of weight distribution of n ? n , for example, see [3] and [11]. For those of , see [11].
R cT
B Y
R T
Y4X
R T
R T
>+?/h
R > 7DiT
R T
R T
D
I R 7DiTkV B E DI I R T[V 'R T 'R dQf T I V ?/h I f D R T V 'R T E 'R @T E I A BT e R R "T[V g !
'R T
#
III. N EW
FAMILY OF BINARY SEQUENCES WITH LARGE SIZE
In this section, we present new families of binary sequences with large family sizes as well as large linear spans for odd and even .
D
4
W EIGHT DISTRIBUTION OF
Weight
W EIGHT DISTRIBUTION OF
/.+* . / .+*
! & 7$'! &
2
A
Exponential Sum
. /. /. . /.
Weight
TABLE I GIVEN BY
#
- V > > >I hI is a binary sequence of trivial solution of @D-[VOJ!- for Y G bS . Thus, sequences in + > : ? A B C B >C f V R _ T for a primitive element _ of O P R^SUT represented by R T for any @ - in I with Y G b S , period with I , where > +R T , the trace> representation of >C , is given by are cyclically distinct. > The crosscorrelation of two sequences > and G in O P RS'T > +R T[V DE : % I 7 \ h jI Next, we verify that
R+ T is not aB constant & < : < : : % # associated with R T has at most L $'& L f roots in ' for polynomial of (i.e., a trivialL polynomial). I If H V , - V B Y G Y S A B from (12), and - V for S Y G Y X from all / ' . for B Y G Y S A9B because (9). Then, at least one - is nonzero for Proof: For given - ’s, the symplectic form . +R /"T B B/E H I T 0 j at least one - is nonzero. If H V , on the other hand, R associated with +R T is given by : B Y Y A B. 0 0 E 0 E E cannot be zero although - may be zero for all G S . j R+ /"TFV R+ T R /T R/T Therefore,
polynomial of with at least & L : S AGB . At this point, f I I I I [ B E E E $ $ than or equal to we need to show that it - T ) (14) & : L L L 6 can take on values of and for all ’s such that D -%U f in order to apply Fact 0 1 for determining the distribution of exponential sums of + .
6
>S A B
is every positive odd integer less than or equal to . In order to do so, we need the following fact from [11]. (Note. we slightly change the representation of the result in [11].)
B Y
?I >
Y
be a set of symplectic Fact 2 ( [11], page 454): Let 0 1, forms of (10). For some fixed integer with assume that the rank of every nonzero form in is at least . Then, the maximum size of is given by
V
>+?/e % '& h > e;?/h ! e g ! h fg ! g
for odd for even
D D
)
In the following, we consider a specific linearized polynomial and investigate a rank of the symplectic form corresponding to the polynomial.
S
B
Lemma 3: For an integer , consider a symplectic form . + / / + where a linearized polynomial + is given by
R "T[V
'R
R TT
R
TkV 3
\7h -%U
f
f R I $ V I $ V E -
R T
-
I VT
D A > > A B to S .
where
is every positive odd integer less than or equal
0
R T
V
V B R T R T A> V B B D V K R T
B
, then the linearized polynoProof: For , by H mial given in (14) (if H , it is identical to (13)) is of the form of + in Lemma 3. From Lemma 3, the exponential 0 4 sum of for H is equal to or 6 for all ’s " ! such that J0 . Thus, we can say that 74 the exponential sum of takes on values of and 6 " " for every integer such that . J Now, it suffices to show that the exponential sum takes on 74 values of 6 at least once when . Or equivalently, we need to show the symplectic form . + / j in (10) has a rank of at least once for some - ’s. Assume that this rank never occurs for all - ’s. Then, the rank of . / is at least . From j Fact 2, the maximum size of a set of such . / ’s is j . However, from (9) and (10), its actual size is , which is greater than . Therefore, the rank of occurs at least once when - ’s run through . This completes the proof of Lemma 4.
>?/h >S A K >? B > A D V K S K >+?/h D A> V > S A B R T D A >S E B > V D A5> S E K R T R T >+?/e \ h V fg >+?U\ I D A >S E B
- can be any A element of . For odd D , the rank Combining the cases 1, 2, and 3, we have the following > S E K inand I every > for possible rank . R+A > /"T E is atY5least D result on the correlation of sequences in OFP"RS'T . W > Y A B occurs at least once when - runs D S K D 5: The correlation of binary sequences in O P RS'T is through I . For even D , on the other hand, the rank is at A > S EG> and every possible rank > for D A > S EG> Y R > S Lemma n 5 B@E > "!#%$'& E > T -valued least D and maximum correlation is ! . >a Y A > D occurs at least once when - runs through I . Proof: For a trace representation R+ 0 > T of each sequence Proof: Note that the odd case in Lemma 3 is implicitly > in O P RS'T , we can consider the exponential sum of R+ T " > ? known from Theorem 16 and Corollary 17 of Chapter 15 in (8). In cases 1 and 2, the exponential sum has and in [11]. Here, we reproduce it for completeness. values. For all - ’s in case 3, from Lemma 4, the exponential >?/h74 for each integer such that D A >GY I Y S . By V sum takes on and 6 Let I be an integer with V > B ! " > 5 A B ! " M M @ Q d f S . Thus, the exponential sum takes on V \h f V h , we ' R / R+ TT where > V 7 K >? and , therefore, the I V have I V E . M R - I / V TWT . V Then, M S nonzero distinct values. Including R T V a -M U f R - # $ $ the maximum > 4 E > M f I A $'& $ is >Ie M h fg and thus the rank of overall exponential sum is R S T -valued. the degree of R R+ TT > S E5Equivalently, > T -valued. Since A > E > M Q O " P R ' S T R correlation of sequences in is . R / AG T is> atE least D I (If D is odd, the rank is at A > is determined by > S A B from M maximum value for D I K because it should beA even). least D n
A N B > I E K never occurs for Lemma 4, V A>@?/h4 @ V B[E> !^! #%$'& . For odd D , assume that the rank D B:Y Y A`B , and thus the rank of . R / T is at all - ’s with Proof of Theorem 1. The results follow directly from Lem
A > G I E M of a set I least D . Then, from Fact 2, the maximum size
mas 1 and 5. >+?/e hI / T V of such . R+ ’s is . However, its actual M g >@?<e h M when - ’s with B:Y G Y I A B run through I , Remark 1: If S V B , size is M fg Aj> I E K which is greater than . Therefore, the rank of D E 3 ' R+ fd I V T5` ' I occurs at least once when - ’s run through I . For even D , + E + R F T V ' F R @ T D (16) A> I E4> never occurs, then the minimum rank B % U if the rank D A > I E . Similarly, the maximum size of a set of such f is D which represents the Gold-like Sequences introduced by Boz > ; e / ? h e h . R / T ’s is>+?/e h V its tas and Kumar [1]. From Lemma 4, a positive odd integer f g M fg , which is A smaller > I E > than M actual size occurs less than > S is only B , so D A > V B . Hence, V ?Y I Y S when Finally, Gold-like>+? sequences AB A B A given B 6 > by '& (16) Since
B9 R+ Y T contains every R+ T ’s for correlation, or for all @ ' I . ! Y S A5B run through M I , the assertion of B - ’s with G > where
Example 1: If
Lemma 3 follows.
Using Lemma 3, we have the following result.
S Y X
D V >+X E B T X B Y R Y4 I
S
BY
and an integer with Lemma 4: For0 odd , let 8- ’s of + in (8) be given such that at least one G is nonzero , and ' . Then, the exponential 0 for 4 sum of + can take on valuesB of and 6 for an integer
R T
>?/h
S V
,
E 'RA@ B T E 3 'R D E D & +R T[V 'RF@ T L f B -U I I for ' and odd D . FromA >Lemma B 4,
IV fd T D A >
(17)
is a positive odd integer less than , so and . Thus, B . From Lemma 5, sequences given by (17) have and
?/hI
f
?/hI
D
V
K
V
7
>+?NA B A B A B > % '& A B > H ! ! & R T R T I > OPR T
6 6 six-valued correlation, or for all @ @ ' . For sequences 0 represented by D E D + L and E B ;H , the corresponding + in (8) constitutes a L linear cyclic subcode with five nonzero distinct weights if . In fact, the we assume that H can be any element in correlation distribution of can be derived from Table II. The details will be discussed in Section IV.
f & R T
I
O ] R^SUT , we R T h V 0 E f R TNV 'R T 3 'R - fd I T E f R fd I T (20) B -%U f where E 8 GcV 9 : @ E J H" I V B Y G b S 8-V @.B- J!-AB H fV d (21) B[E H I Y 4 Y X S G fd Y for @ - )J - ' I with S and 0 H' I . If - V for all B Y G Y5X , the exponentialGkb4 sum of +R T has a value of or >@? . Otherwise, we have to consider the number of solutions of . j R /Tk0 V in order to derive the distribution of exponential sums of +R T in (20). >@X and an integer S with B Y S b X , Lemma 7: For even D V 0 let - ’s of R T in (20) be given such that at least one - is BaY G Y X . Then, the symplectic form . +R /T nonzero for 0 >"I%\ roots in (' I j for all associated with +R T has at most / ' I . To investigate 0 the correlation of sequences in need to consider + given by
S V K, E D E D & D ! R+ TFV 'RA@ T 'RA@ B T L L f B E 'RF@ I C T E 3 'R+ f d I V T (18) -%U B ?/h ?/h I for ' and odd D . Similarly, V I f I B and ?/hI C . From Lemma 5, sequences given by (18) have eight>+? A B A B A B > '& A B > "H A B valued correlation, or > ! / for all @ %@ @ I ' I . 6 ! 6 ! 6 B f B. Construction of O ] RS'T for even D >@X and an integer S with 2: For even D V B Y Construction X S b , a family O ] R^SUT of binary sequences is defined by Proof: We have " " \h h Oc]@RS'T[V ) > ? V AR @ %@ f 5T B@ - ' I I VT E I f I$ V I$ V E B . R / k T V / 3 R 8 + " ! I hI j > where > V > > is a binary sequence of -%U + > : ? A B C f f B period with >C V R_ T for a primitive element _ of V R / R TJT ) I , where > +R T , the trace> representation of >C , is given by I Thus, . R+ /"T[V for all / ' \h if and only if R T V . V j f I E Using the same approach as we did in the proof of Lemma 2, > R+ TFV 'RA@ T 3 '[email protected] fd T we obtain
R T identical to (13) and the equation (15). B -%U fV (19) Following the same steps as in the proof of Lemma 2, we h E 3 f ' R fd I T E +R f d I T can derive that the number of solutions of . R /TkV is at >I%\ . j f \ most -%U >@X and an integer S with B Y S b X , for (' I . Lemma 8: 0 For even D V >+X and an integer S with B:Y S b let 8- ’s of R B YT in Y5 (20) be given such that at least one - is X , and ' I . Then, the exponential Theorem 2: For even DaV X , the @ > ' ? \ G nonzero0 for >? a ? A B > E sum of R+ T can take on valuesB of and 6 for an integer T - where D Aj> is zero or every positive even integer of period . The correlation of sequences is R S \ @ B E > less than !"d . Therefore, O ] R^SUT > S . Hence, the correlation of binary sequences valued and maximum correlation is + > ? 4 A B @ > U ? \ [ B E > \ or equal to in !d T signal set. constitutes a R cO [B ]@ER^SU> T is \ R > S E T -valued and maximum correlation n is !d . In order to prove Theorem 2, we need the following lemmas. B Y Since their proofs are similar to those for O P RSUT , we omit the Proof: For even D , we see that while each - with details. Gkb4S runs through I , each - of R T in (13) runs through I . From Lemma 6: All sequences in OQ]@R^SUT are cyclically distinct. 0 Lemma 3, therefore, we obtain >?/7h that 4 the exponential >@?U\ . sum of +R T takes on values of and 6 for every integer Hence, the family size of OQ]+RSUT is A > V > " ! > S AG> . such that D A > Proof: Similar to the proof of Lemma 1, +R TV that the cases D V and D A > V Next, we will show > all in I if and only if (7) is achieved. If > S also occur. If all - ’s of R T in (13) are zero, then R T[V GR;H B T E for > - >+?9A B T V B , then is not a factor of ; - h >+? A5B T since ; - B E5> - h T V B for has at most four solutions, i.e., V ! f ! , and f ; I B Y Y X V occurs at least S G have at least two equations of H fd in f V . Hence, the case D X for even D . Thus, (7)V has, for (7) because S b a unique solution once.fd Applying Fact 2, we also obtain that exponential B . Hence, all sequences in O ] RSUT are cyclically sum of 0 +R T can take on values of 6 >?/h 4 at the given by H V least once such A > V > S in the similar way to the proof of Lemma 4. distinct. that D Example 2: If
8
0
R T takes on values of D A> >E V J > " ! > S . R S T -valued. FurV ?I A S .
Therefore, the exponential sum of + 74 and 6 for an integer where Including n and , the correlation is thermore, for
>+?/h
>+?
V B[E4> !' d \
Proof of Theorem 2. The results follow directly from Lemma 6 and Lemma 8.
S V B, h IV E E f DE@R T[V 'RF@ T 3 'R fd T B -%U f
TABLE III L INEAR SPAN AND
+*,2 -/ .1 0 +*, 2 /.10 +*, 2 $ 0 .. .
Remark 2: If
13 )
THE NUMBER OF CORRESPONDING SEQUENCES IN
+*; 2 2 (-/.10
( OR
- ( .% 7 ( .% 7 (
Number of sequences
(. 2(
2.
.. .
I T f d (22) Since =M@ - ’s are and RS A =T @ - ’s are nonzero, the number of f \ R >+? AB T \h . corresponding sequences given by R+ T is (' I 6 Y > 6 Y S , we obtain = = Applying this result to each with which represents the sequences constructed by ? Udaya? [19]. A > V and > . Hence, V I and I AGB . the linear span R^SUT of binary sequences in OFP@RS'T with the From Lemma 8, D Finally, the sequences given by (22) has six-valued correlation, distribution shown in Table III. Using the similar approach to >+? AB A B A B 6 > A B 6 > for all @ ' I . odd case, we see that the linear span of sequences in O ] R^SUT is or ! !'dQf the same as OQPR^SUT . B >, Example 3: If S V Corollary 1: The maximum and minimum linear E ?/e ? and ?/e;?<spans hI%\ of, D E D & +R TkV 'AR @ T 'FR @ B T P sequences in O R ' S T and Q O @ ] R ' S T are I dQfg I dQfg L f B E 3 'R+ f d I V T E R+ fd I T (23) respectively. f -%U I I A > > D. Comparison of families of binary sequences D D V for (' and even . From Lemma 8, In Table IV, we give some comparisons of the families V ? I ? I A B ? I A > . Hence, and thus, the correlation of O P R^SUT and Oc]@R^SUT with well known families of binary sequences >?QA B A B A B > A B sequences given by (23) belongs to > A B 6 > 8I for all @ %@ ' I . 6 ! 6 with low correlation. In Table IV, we include a family of !dQf !d binary sequences constructed from Chang et al.’s investigation B f 0 of a binary cyclic code based on three-term sequences [2]. For Remark 3: Contrary to odd D , R T in (20) does not con>@X@EGB , each sequence in the family is represented by Y X Gkb may odd D V stitute a linear cyclic subcode because - with S E 'RA@ C T E 'R+ C ! TN V > dQf EB not run through all elements in I . Therefore, we should use DE D R+ T[V 'RA@ T & L f B I different methods from odd D to investigate the correlation I > for (' and all @ @ f in . On the other E hand, a familyE distribution in O ] R T , which is a problem we are working on. 'RF@ B T of sequences defined B by D E D R+ T[V 'RA@ T & B correcting BCHf code, '+R $C7T from the dual of the L triple n error C. Linear spans of O P R^SUT and Oc]@R^SUT has the same period, family size, , and maximum linear The linear spans of binary sequences in OFP"RS'T and O ] R^SUT span as those of Chang et al.’s for all @ %@ in I for odd Y B f GQb4S . D [11]. are determined by the number of nonzero @Z- ’s with In terms of periods and maximum linear spans, new famTheorem 3: In the family O P R^SUT (or Oc]@RS'T ), consider a ilies O P R^SUT and Oc]+RSUT are identical to the families of Goldsequence represented by R+ T where = @ - ’s in ? V like sequences ! " \h > and Udaya’s, respectively. At the expense of FR @ @ T are equal to . Let 6 RSUT be the linear span maximum correlation n 5 f , however, we have larger family of B the sequence. Then, P sizes in O ^ R U S T and Q O ) ] ^ R U S T than in those families. Basically, we A > = EB T k D R D Y Y P obtain and with exponentially increased family O R ' S T Q O @ ] R ' S T RS'T[V > ` = S n 5 6 sizes by the linear increase of from optimal correlation. \ >@? AGB \7h and there are R T 6 sequences in O P RS'T (or Oc]@R^SUT ) Furthermore, we can choose S and the corresponding family 6 RSUT . From this result, the linear span of O P R^SUT (or Oc]@R^SUT ) for its specific application. For example, if having linear span 6 sequences in OcP"RS'T (or O ] R^SUT ) and its distribution are shown low correlation is more crucial than a large family size in the application, then a small value of S is chosen. Otherwise, in Table III. we choose a large value of S in order to get a large family Proof: Firstly, consider the linear span of sequences in size. Considering the flexibility due to S , OFP@R^SUT and O ] RS'T have O P R^SUT . In Construction a sequence represented by +R T larger family sizes than any other families of binary sequences X E B V ? dQI f 1,trace > has a total terms and each trace term has in Table IV. the linear span of D . If =&@ - ’s of the sequence are equal to , ? I A =+T nonzero trace terms and the corresponding In OcP"RSUT and O ] R^SUT , C -sequences are not included for it has R dQ f increasing their minimum linear spans. Thus, they have larger linear span of the sequence is given by minimum linear spans than any other binary sequence families EB A A> = E B T D k D R D Y Y in Table IV except for bent sequences, which will be good RS'T[ V > = D V > ` = S ) property for potential cryptographic applications. 6 +R
9
TABLE IV C OMPARISON OF THE Family of Sequences
FAMILIES OF BINARY SEQUENCES WITH LOW CORRELATION
Period
Family Size
Linear Span (Maximum, Minimum)
'! & $ 2 ! ! / . ( 2 ! ! or ! - ! Bent [14] .10 ! @*;2 -/.10 ' & Boztas and Kumar [1] ! @*;2 -/.10 - ! -/. Udaya [19] 2 - ! H Chang et al. (triplet) [2] [11] 5 2 - ! H Rothaus [15] 5 2 ' & 0 / . Kerdock [16] ! 2 H 20 / . ! Delsarte-Goethals [16] 2
@*;2 -/.10 +*;2 $ 0 ! H New Family 1 2 2
@*;2 -/.10 +*;2 $ 0 - ! New Family 31
( @*;2 -/.10 +*; 2 2 (-+X E B
OQP"R > T
OQPR > T
O P R > NT V ) > ? V RF@ @ f T5 @ - ' I " ! I hIB > where > V > > is a binary sequence of >+? A B with B f >C V DE D R_ C T for a primitive element period L & by _ of I , where DE D & R T is given L IV E E D E D & R TkV 'RF@ T 'RF@ B T 3 'R+ fd T L f B -%U I I for (' . > From Theorem 1, we know that the family O P R T has >IJ? cyclically distinct binary sequences of period >"?A B . > The correlation of sequences in O P R T is six-valued and its B E > H . Consequently, OQP+R > T constitutes a maximum is >R @?:AGB >IJ? BcE> % H ! T signal > set. This data of OQPR T is listed !
in Table IV. Next, we will investigate the distribution of correlation values of sequences in . The correlation of a pair of is derived from the exponential sum of sequences in
0
OQP"R > T > OQP@R T E J HT; T E 'RJRF@ E J H B TE B T R T[V 'RJRF@ f f B B E 3 'RJR B[E H f d I V TE fd I V T -%U I
for
@
f
E@
of
%@ )J 9J ' B J )H B f , and B ff 9 0
R+ T V
odd even even even odd even odd odd odd odd odd even odd even
I and H V ' I . With ] VO@ V QB E H f d I for >Y G Y4X , 0 R T B TE
'R]*
I
'R B
I
TE
3
B
-%U
I
'R - 9
E
V
J H"
has B a form
IV fd T
(24)
' . Depending on - ’s, the for ]D ' and 0 - ' 9 exponential sum of 9 can be classified into two exclusive cases. To facilitate the analysis, assume that - can be any 9 element in .
I -cV
R T
> Y
Y X
0
R T V > 'R T E R iD T
G Case 1. for . In this case, + > ]* 9 B constitutes a linear cyclic subcode of for ]Z0 ' . Thus, the distribution of the exponential sums of + is identical to Table I with . G0 for Case 2. At least one - . In this case, the 9 distribution of the exponential sums of + follows from the following lemma.
'R )T I R T
V
V X ? V Y f Y4 R T
Lemma 9: For given - ’s with at least one nonzero 9 in (24) has five-valued exponential sum for all ]Z in the distribution is
R A B kT j e f g f g"i !8 >@?@?"I 4 h >4 h ? A>"I54 f RR >+?A & >"f I6 4 & T E & f I T R >@times f I R >"I 4 ! h f 6 >4 ! h f T times ! T times (25) where V >+?
