# On the 2-Adic Complexity of the Ding-Helleseth-Martinsen Binary Sequences

We determine the 2-adic complexity of the Ding-Helleseth-Martinsen (DHM) binary sequences by using cyclotomic numbers of order four, "Gauss periods" and "quadratic Gauss sum" on finite field F_q and valued in Z_2^N-1 where q ≡ 5 8 is a prime number and N=2q is the period of the DHM sequences.

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## 1 Introduction

Let be a binary sequence of period , . The autocorrelation function of the sequence is defined by

 AS(τ)=N−1∑t=0(−1)st+τ−st∈Z (0≤τ≤N−1).

For , Let

 Max AS=max{|AS(τ)|:1≤τ≤N}.

For many applications in communication, the value of Max is required as small as possible. It is easy to see that when , for all For , a binary sequence with period is called to have perfect autocorrelation if Max It is conjectured that the only perfect sequence is and up to (cyclic shift) equivalence. For , a binary sequence with period is called to have ideal autocorrelation if for all Several series of binary sequences with ideal autocorrelation have been found (-sequences, Hall sequences, Paley sequences and the twin-prime sequences, see [2]). A binary sequence with period is called to have optimal autocorrelation if Max and 3 for respectively. For a list of known binary sequences with optimal autocorrelation we refer to [2].

In the application on cryptography, binary sequences, as candidates of keys in stream cipher system, are required to have big “complexity”. There are huge works on linear complexity of binary sequences. The sequences with linear complexity can be generated by a linear shift register of length . Since the end of last century, the 2-adic complexity has been viewed as one of the important security criteria of sequences. The sequences with 2-adic complexity can be generated by a feedback (with carry) shift register of length ([6]).

Let be a binary sequence with period Let The 2-adic complexity of is defined by

 C2(S)=log2(2N−1d).

Comparing with the linear complexity, the 2-adic complexity of binary sequences with small autocorrelation has not been fully researched. The 2-adic complexity of the binary sequences with ideal autocorrelation for has been done in [12, 14, 5]. Particularly, H. Hu [5] presented a neat approach to show that for all known ideal binary sequences with period , their 2-adic complexity reaches the maximum value

For some other sequences with good autocorrelation, the 2-adic complexity is determined or estimated by a nice lower bound [4, 9-11, 13].

In this paper, we determine the 2-adic complexity of the Ding-Helleseth-Martinsen (DHM) binary sequences. The sequences has period where is a prime number, and optimal autocorrelation Max We will determine their 2-adic complexity. For doing this we use the cyclotomic numbers of order four and develop “Gauss periods” and quadratic “Gauss sum” on finite field valued in the ring .

We introduce the construction on the DHM sequences and preliminaries on cyclotomic numbers, “Gauss periods” and quadratic “Gauss sum” in Section 2. Then we present upper and lower bounds on the 2-adic complexity of the DHM sequences in Section 3. After further consideration we finally determine the exact value of the 2-adic complexity of the DHM sequences in Section 4.

## 2 Preliminaries

### 2.1 Ding-Helleseth-Martinsen (DHM) sequences

Let be a prime number, . It is well known that there exists unique up to their signs such that . Let and . Then is a subgroup of , and the cosets of in

 Dλ=θλC={θλ+4i:0≤i≤f−1} (2.1)

are called as cyclotomic classes of order four in .

Let We have the following isomorphism of rings

 φ:Z2×Zq≅Z2q=Zn,(a,b)→(q+1)b+qa.

The inverse of is

Consider the following subset of , for

 C′(i,j,l)=[{0}×(Di∪Dj)]∪[{1}×(Dl∪Dj)],C(i,j,l)=φ(C′(i,j,l)) (2.2)

Definition 2.1 Let be a prime number, be the cyclotomic classes of order four defined by (2.1). Let be the subset of defined by (2.2). The Ding-Helleseth-Martinsen (DHM) binary sequence and with period is defined by

 sλ=⎧⎪⎨⎪⎩1,if λ∈C(i,j,l)(Namely,λ=(q+1)b+qa){where} a=0,b∈Di∪Dj" {or} a=1,b∈Dl∪Dj" 0,otherwise,
 ˜sλ={1,if λ∈˜C(i,j,l),0,otherwise.

Namely, and for .

It is proved in [3] that if

 (I) t=1 and (i,j,l)=(0,1,3),(0,2,1);or(II) s=1 and (i,j,l)=(1,0,3),(0,1,2), (2.3)

then the DHM sequence has optimal autocorrelation for

If

 (˜I) t=1 % and (i,j,l)=(0,1,3),(0,2,3),(1,2,0),(1,3,0) or(˜II) s=1 and (i,j,l)=(0,1,2),(0,3,2),(1,0,3),(1,2,3).                           (˜2.3)

then the DHM sequences has optimal autocorrelations.

In Section 3 we present upper and lower bounds for the DHM sequence satisfying condition (2.3) and satisfying condition (Theorem 3.1 and 3.2). After further consideration we totally determine the value of and in Section 4 (Theorem 4.2 and 4.4).

### 2.2 Cyclotomic Numbers of Order Four

Let be a prime number, , and be the cyclotomic classes of order four in .

Definition 2.2 The cyclotomic numbers of order four in are defined by, for

 (i,j)=|(Di+1)∩Dj|=♯{(a,b):a∈Di,b∈Dj,a+1=b}.

The values of has been computed (see [1] or [8]).

Lemma 2.3 Let be a prime number, and . The values of the cyclotomic numbers of order four in are

 16(0,0)=16(2,2)=16(2,0)=A=q−7+2s,
 16(0,1)=16(1,3)=16(3,2)=B=q+1+2s−8t,
 16(1,2)=16(0,3)=16(3,1)=¯B=q+1+2s+8t,
 16(0,2)=C=q+1−6s,
 16(1,0)=16(1,1)=16(2,1)=16(3,0)=16(3,3)=16(2,3)=D=q−3−2s.

### 2.3 “Gauss periods” of order four and quadratic “Gauss sum”

Since implies we can define the following mapping

 f:Zq→Z∗2N−1,f(a)=4a

where is the group of units in the ring . is a homomorphism of groups from to , and can be viewed as an additive character of finite field valued in . Then we have the “Gauss periods” of order 4

 ηλ=∑i∈Dλ4imod(2N−1)(λ=0,1,2,3)

 G=∑i∈F∗q4iχ(i)=η0−η1+η2−η3 (mod 2N−1),

where is the unique quadratic (multiplicative) character of (the Legendre symbol). Namely, for ,

The following results show that and have some similar properties as usual Gauss periods and Gauss sums. (The proofs are also similar).

Lemma 2.4 (1). .
(2). For , where is the cyclotomic numbers of order four on and

 δλ,μ={1,if λ≡μ(mod4)0,otherwise.
###### Proof.

(1). In the following, all equality means

 G2 =q−1∑a,b=14a+bχ(ab)=q−1∑a,c=14a(1+c)χ(a2c) (taking c=ba−1) =q−1∑c=1χ(c)q−1∑a=14a(1+c).

The contribution of to the right-hand side is

 χ(q−1)q−1∑a=14aq=χ(−1)q−1∑a=11=q−1

(remark that for Therefore

 G2=q−1+q−2∑c=1χ(c)(−1+q−1∑a=04a)=q−q−1∑a=04a=q−4q−13.

(2). By the definition of ,

 ηλημ =∑a∈Dλb∈Dμ4a+b=∑c∈Fq4c∑a∈Dλc−a∈Dμ1 (c=a+b) =∑a∈Dλ−a∈Dμ1+3∑ν=0∑c∈Dν4c∑e∈Dλ−v+21+e∈Dμ−ν1 (let e=−ac for c≠0).

From we know that and
Therefore

 a∈Dλ⟺e(=−ac)∈Dλ−ν+2
 c−a∈Dμ⟺1+e(=1c(c−a))∈Dμ−ν

Then we get

 ηλημ=q−14δλ,μ+2+3∑ν=0(λ−ν+2,μ−ν)ην.

Remark From Lemma 2.4(2) we know that

 ηλ+iημ+i =q−14δλ+i,μ+i+2+3∑ν=0(λ+i−ν+2,μ+i−ν)ην =q−14δλ,μ+2+3∑ν=0(λ−ν+2,μ−ν)ην+i. (2.4)

By Lemma 2.3 we get

 16η20=Aη0+Bη1+Cη2+¯¯¯¯Bη3, 16η0η1=Dη0+Dη1+¯¯¯¯Bη2+Bη316η0η2=q−14+Aη0+Dη1+Aη2+Dη3, 16η0η3=Dη0+¯¯¯¯Bη1+Bη2+Dη3 (2.5)

From (2.4) and (2.5) we get

Lemma 2.5 Let be a prime number, Then

 16η2λ =Aηλ+Bηλ+1+Cηλ+2+¯¯¯¯Bηλ+3 =q(η0+η1+η2+η3)+(−7+2s)ηλ+(1+2s−8t)ηλ+1 +(1−6s)ηλ+2+(1+2s+8t)ηλ+3 =(4q−13−1)q+(−7+2s)ηλ+(1+2s−8t)ηλ+1+(1−6s)ηλ+2+(1+2s+8t)ηλ+3
 16ηληλ+1=(4q−13−1)q+(−3−2s)(ηλ+ηλ+1)+(1+2s+8t)ηλ+2+(1+2s−8t)ηλ+3
 16ηληλ+2=(4q−13−1)q+(−7+2s)(ηλ+ηλ+2)+(−3−2s)(ηλ+1+ηλ+3)+4(q−1)
 16ηληλ+3=16ηλ+3η(λ+3)+1.

Lemma 2.6 Assume that is a prime number and . Then

 (η0−η1)2+(η2−η3)2=−tG, 2(η0−η2)2=−(sG+q)+4q−13
 (η0−η3)2+(η2−η1)2=tG, 2(η1−η3)2=sG−q+4q−13
###### Proof.

By Lemma 2.5,

 16[(η0−η1)2+(η2−η3)2] =16(η20+η21+η22+η23−2η0η1−2η2η3) =0(η0+η1+η2+η3)−16t(η0−η1+η2−η3) =−16tG.

We get .

Similarly, we get

 (η0−η3)2+(η2−η1)2=tG

Since , by Lemma 2.5, we get

 2(η0−η2)2=−(sG+q)+4q−13, 2(η1−η3)2=sG−q+4q−13.

## 3 Upper and Lower Bounds of C2(s) and C2(˜S)

In this section we present upper and lower bounds of 2-adic complexity of DHM sequences.

Theorem 3.1 Let be the DHM binary sequence with period defined in Definition 2.1, be a prime number, If the condition (2.3) holds, then , where

###### Proof.

The 2-adic complexity of is , where , and . We need to estimate the value . From and , we know that

 d=d1d2,d1=gcd(S(2),2q−1),d2=gcd(S(2),2q+1).

By the definition of the sequence , we have

 S(2) =N−1∑λ=0sλ2λ≡∑b∈Di∪Dj2(q+1)b+∑b∈Dl∪Dj2q+(q+1)b (mod 2N−1)

Let . From we know that 2 is not a square in Therefore or 3 and

 S(2) ≡∑2b∈Di∪Dj2(q+1)2b+∑2b∈Dl∪Dj2q+(q+1)2b (mod 2N−1) ≡∑b∈Di−k∪Dj−k4b+2q∑b∈Dl−k∪Dj−k4b (mod 2N−1)  (N=2q) ≡ηi−k+ηj−k+2q(ηl−k+ηj−k) (mod 2N−1) (3.1)

(A). Firstly we prove .

The assumption implies On the other hand,

 ηλ=∑a∈Dλ4a≡∑a∈Dλ1=q−14(mod3)(λ∈{0,1,2,3}),

we get Therefore Moreover, if , then . The order of is 6, we get which implies that since is a prime number. This contradicts to assumption Thus and where

Now we prove By (3.1),

 0 ≡S(2)≡ηi−k+ηj−k−ηl−k−ηj−k≡ηi−k−ηl−k (mod 2q+1) ≡⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩η0−k−η3−kη0−k−η1−kη1−k−η3−kη0−k−η2−k(modd′2) if (i,j,l) =⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩(0,1,3)(0,2,1)(1,0,3)(0,1,2) (3.2)

where or 3. We consider the four cases of the condition (2.3) separately.

(1). For and , we have by (3.2) and or 3. Namely, or . Then or . In both cases,

 G2 ≡(η1−η0)2+(η2−η3)2(modd′2) ≡−G(modd′2) (by Lemma 2.6 and t=1)

From Lemma 2.4 we have

 G2 ≡q−4q−13(modd′2) ≡q(modd′2) (since 3∤d′2 and d′2|4q−1). (3.3)

Therefore and . We get

(2). For and , we have . Namely, or and or . In both of cases,

 q ≡G2≡(η0−η3)2+(η1−η2)2≡G(modd′2) (by Lemma 2.6 and t=1).

Therefore and we also get as in case (1).

(3). For and we have . Namely, or Therefore

 0 ≡2(η0−η2)2≡−(G+q)+4q−13(modd′2) (by Lemma 2.6 and s=1)

From we get . Therefore and Then we get .

(4). At last, for and , we have . Namely, or . Therefore

 0 ≡2(η1−η3)2≡G−q+4q−13(modd′2) (by Lemma 2.6 and s=1).

We also get .

In summary, we have proved that for all four cases.

Now we prove that . If , let be a prime divisor of . Then , or . If , we get which contradicts to . If from we know that the order of is . Therefore which contradicts to . Therefore and then for all four cases.

This implies that .

(B). Now we prove that .

From (3.1) we know that

 S(2) ≡ηi−k+ηj−k+ηl−k+ηj−k (mod 2q−1) ≡−1+ηj−k−ηλ−k (mod d1) (% where λ∈{0,1,2,3}∖{i,j,l})

Therefore

 0≡S(2)