Some Upper Bounds and Exact Values on Linear Complexities Over F M of Sidelnikov Sequences for M = 2 and 3

—Sidelnikov sequences, a kind of cyclotomic 1 sequences with many desired properties such as low correlation 2 and variable alphabet sizes, can be employed to construct a 3 polyphase sequence family that has many applications in high-4 speed data communications. Recently, cyclotomic numbers have 5 been used to investigate the linear complexity of Sidelnikov 6 sequences, mainly about binary ones, although the limitation on 7 the orders of the available cyclotomic numbers makes it difﬁcult. 8 This paper continues to study the linear complexity over F M 9 of M -ary Sidelnikov sequence of period q − 1 using Hasse 10 derivative, which implies q = p m , m ≥ 1 and M | ( q − 1) . 11 The t th Hasse derivative formulas are presented in terms of 12 cyclotomic numbers, and some upper bounds on the linear 13 complexity for M = 2 and 3 are obtained only with some 14 additional restrictions on q . Furthermore, concrete illustrations 15 for several families of these sequences, such as q ≡ 1 (mod 2) 16 and q ≡ 1 (mod 3) , show these upper bounds are tight and 17 reachable; especially for q = 2 × 3 λ +1(1 ≤ λ ≤ 20) , the exact 18 linear complexities over F 3 of the ternary Sidelnikov sequences 19 are determined; and it turns out that all the linear complexities 20 of the sequences considered are very close to their periods. 21

For q = p m where p is an odd prime and m is a positive integer, Sidelnikov [6] introduced a kind of cyclotomic sequence called the M -ary Sidelnikov sequence of period q − 1 where M |(q − 1).Soon afterwards, Lempel, Cohn and Eastman [7] re-introduced its binary form called Sidelnikov-Lempel-Cohn-Eastman sequence.In the last two decades, a lot of attention has been devoted to this binary sequence.For example, using the cyclotomic numbers, Helleseth and Yang [8] originally investigated the autocorrelation function and linear complexity over F 2 of the binary Sidelnikov sequences.Later on, Kyureghan and Pott [9], and Meidl and Winterhof [10] determined the exact linear complexity over F 2 of some of these sequences with well-known results on cyclotomic numbers; a lower bound on the linear complexity profile of these sequences was also introduced in [10], which is the desirable important property of applications.Then, Wang [11] and Su [12] studied the linear complexity of binary cyclotomic sequences of order 6 and Legendre-Sidelnikov sequences of period p(q − 1), respectively.Ye et al. [13] further studied the linear complexity of a new kind of binary cyclotomic sequence, with length p r , and Liang et al. [14] computed the linear complexity of Ding-Helleseth generalized cyclotomic sequences by using cyclotomic numbers of order 8.Following the footsteps of these pioneers, Zeng et al. [15] discussed the F M -linear complexity of M -ary Sidelnikov sequences of period p−1 = f ×M λ where M is not just equal to 2. On the other hand, using the discrete Fourier transform (i.e., DFT), Helleseth et al. [16], [17] also determined the linear complexity over F p of binary Sidelnikov sequences, and Garaev et al. [18] derived the lower bound of the linear complexity over F p .For the k-error linear complexity over F p of the d-ary Sidelnikov sequence, Chung and Yang [19] presented many results of interest, then Aly and Meidl [20] further complemented these results.
In this paper, we continue to study the linear complexity over F M of M -ary Sidelnikov sequence of period q − 1 using Hasse derivative, where q = p m , m ≥ 1 and M |(q − 1).Some upper bounds on the linear complexity for M = 2 and 3 are obtained, and some exact values of the linear complexity for several families of these sequences, such as q ≡ 1 (mod 2) and q ≡ 1 (mod 3), illustrate these upper bounds are tight and reachable.In particular, the exact linear complexities over This work is licensed under a Creative Commons Attribution 4.0 License.For more information, see https://creativecommons.org/licenses/by/4.0/F 3 of the ternary Sidelnikov sequences are determined for q = 2 × 3 λ + 1(1 ≤ λ ≤ 20), and it turns out that all the linear complexities of the sequences considered are very close to their periods.Some examples over F 2 have been confirmed for binary Sidelnikov sequences by Helleseth and Yang.
The rest of this paper is organized as follows.In Section II, after reviewing some notations and definitions, we present formulas for the tth Hasse derivative of the generating function S(x) of the M -ary Sidelnikov sequence {s n } n≥0 in terms of the cyclotomic numbers.In Section III, the multiplicities of some rth primitive roots of unity over F M as roots of S(x) are determined using the Hasse derivative to estimate the F M -linear complexity of M -ary Sidelnikov sequences for the two cases of q ≡ 1 (mod 2) and q ≡ 1 (mod 3).Some special examples are listed in Table I for q = 2 × 3 λ + 1(1 ≤ λ ≤ 20).Note that this section extends our conference version [15] by adding Subsection III-A on the case q ≡ 1 (mod 2) which includes Theorem 1 and Examples 4 and 5, by supplementing a main result on the case q ≡ 1 (mod 3) in Theorem 2 of Subsection III-B, and by giving all proofs of the relevant results here.In Section IV, there are some concluding remarks.In addition, some known cyclotomic numbers of orders 2, 2r, 3, 6 and 9 are displayed in Chapter IV due to the need to prove the results of this paper.

II. PRELIMINARIES
In this section, after some notations are listed, the M -ary Sidelnikov sequence, the F M -linear complexity and the cyclotomic number are defined in Definitions 1, 3 and 4, respectively.Lemma 3 presents Hasse derivates in terms of cyclotomic numbers, and will be used to determine the F M -linear complexity of the M -ary Sidelnikov sequence.
• p: an odd prime.
• q: an odd prime power p m with m ≥ 1.
• F p and F q : the finite fields with p and q elements, respectively.
• M : M is a prime with M |(q − 1).
• α: a fixed primitive element of F q .
• F M [x]: the polynomial ring over finite field F M .
• R(γ): the multiplicity of a primitive rth root γ of unity over F M as a root of S(x), where γ = e j 2π r and j = √ −1.
• LC(•): the F M -linear complexity of a sequence.It is written as LC for short if the context is clear.
• Ind x: the index of x ∈ F q to the base g modulo q [21].
The M -ary Sidelnikov sequence is defined as follows.
Definition 1 ( [22]): For a fixed primitive element α of F q and M |(q − 1), let D The M -ary Sidelnikov sequence {s n } n≥0 of period q − 1 is defined as (1) Equivalently, 0 , and let 129 log α (0) = 0. 130 Example 1: Let q = 7 and M = 3.For α = 3, we have 131 a ternary Sidelnikov sequence {s n } n≥0 of period 6, that is, 132 (3) 138 Definition 3: The linear complexity over 145 Therefore, the F M -linear complexity of the M -ary Sidel-146 nikov sequence {s n } n≥0 can be determined by (4) 148 where S(x) is by (1) 149 (5) 150 Example 2: The F 3 -linear complexity of the ternary Sidel-151 nikov sequence {s n } n≥0 of period 6 in Example 1 is 5 since 152 gcd( 153 Similar to Example 2, in order to evaluate LC({s n } n≥0 ) 154 from (4), we will determine the multiplicity of γ as a root of 155 S(x), where γ is also a (q − 1)-th root of unity over F M or 156 in an extension field of F M , by using the cyclotomic numbers 157 defined as follows.
Let γ be a primitive 175 rth root of unity over F M or in an extension field of F M where 176 r|f .Then the multiplicity of γ as a root of S(x) is i, i.e., 177 where S(x) (t) (t = 0, 1, . . ., i) is the tth Hasse derivative of 180 S(x) [24], and defined as 182 where the binomial coefficients n t modulo M can be evalu-183 ated with the following Corollary 1.
189 However, since there exists a convention that if x < y then

200
where i t = 0 if i < t.

201
Proof: First, we prove Lucas' theorem is still true if there 202 exists j such that b j > a j .To the end, we need to compare 203 the coefficients of binomial expansion of (1 + x) a , where 204 a = l−1 j=0 a j M j and a 0 , . . ., a l−1 are the digits in the M -ary 205 representation of a.

206
Since M is a prime, it follows that 207 208 Then we have 209 the left and right sides of (10) should be equal by using the unique M -ary representation property, Thus, if b j ≤ a j for all 0 ≤ j ≤ l − 1, the coefficient on the left of ( 11) must be congruent modulo M to the coefficient on the right, which is exactly the result of Lucas' theorem.
Otherwise, if there exists 0 ≤ j ≤ l − 1 such that b j > a j , then aj bj = 0, which means there is no item of x bj M j on the right of ( 11), leading to there is no item of x b .Then, the coefficients on both sides of ( 11) are equal to 0, that is to say, the Lucas' theorem is also true for b j > a j (0 ≤ j ≤ l − 1).So, ( 7) is true.
. ., n l and t 0 , . . ., t l−1 be the digits in the M -ary representations of n and t, respectively.Then n 7), it is easy to see that Thus, the proof completes.
Next, the tth Hasse derivatives (t = 0, 1, . . . ) in terms of cyclotomic numbers are listed in the following lemma that will be used to determine the multiplicities of all the f th roots of unity, as the roots of S(x).
Lemma 3: [15] Let q = p m ≡ 1 (mod M ) where p is an odd prime and M is prime.Let γ be a primitive rth root of unity over F M or in an extension field of F M .S(x) is the generating function of an M -ary Sidelnikov sequence {s n } 0≤n≤q−2 .Then the tth Hasse derivatives S(x) (t) ∈ F M [x] (t = 0, 1, . . . ) satisfy the following identities.

1)
where n ≡ u (mod M ); 2) where n ≡ i (mod M l ), and l = log 3) (see (14a) and (14b), as shown at the bottom of the next page;) 4) (see (15a) and (15b), as shown at the bottom of the next page.) Remark 1: 1) The Hasse derivative in Lemma 3 is a bridge across the cyclotomic number and the linear complexity.
Using this technique, one can determine the exact F M -linear complexity of an M -ary Sidelnikov sequence according to certain cyclotomic numbers.However, the well-known results on cyclotomic numbers are now just limited to the orders e ≤ 24.This limitation hinders our ability to calculate the multiplicity of γ if r is large.So, it seems difficult to determine the exact F M -linear complexity.2) For the proof details of Lemma 3, please refer to [10], [15].

III. UPPER BOUNDS AND SOME EXACT VALUES
This section investigates the F M -linear complexities of the M -ary Sidelnikov sequences.In the case of q ≡ 1 (mod 2), Theorem 1 shows that the F 2 -linear complexities of binary Sidelnikov sequences of period q − 1 are upper bounded by q − 2r if r satisfies certain conditions.In the case of q ≡ 1 (mod 3), for the trivial root 1, the primitive 2nd root and the primitive 3rd root of unity over F 3 or in an extension field of F 3 , the multiplicities of them as the roots of S(x) are determined in Propositions 1, 2 and 3, respectively.Furthermore, the F 3 -linear complexities of the ternary Sidelnikov sequences are presented in Theorem 2 and Corollary 2.
Note that, for the detailed meanings of the capital letters such as "A", "B", "C", etc. in this section, please refer to Appendices IV.

A. Binary Case
The linear complexity of binary Sidelnikov sequence was originally investigated by Helleseth and Yang, and later extended by Kyureghyan and Pott, and Meidl and Winterhof.
In the following theorem, we continue to estimate the linear complexity of the Sidelnikov sequence for q ≡ 1 (mod 2) using the technique introduced by Meidl and Winterhof, and the result is an extension of that in [10].
Theorem 1: Let q = p m ≡ 1 (mod 2r) for m = uv, where p and r are both odd primes, u ≥ 1, v is the order of p modulo r, and v is even.Let 2 be a primitive root modulo r 296 and {s n } n≥0 be a binary Sidelnikov sequence of period q − 1. 297 Then the linear complexity of {s n } n≥0 over F 2 is less than 298 or equal to q − 2r if 299 1) u is even; or 300 2) u is odd, and 4 v with p ≡ 3 (mod 4).

Proof:
Let S(x) be the generating function of 302 {s n } 0≤n≤q−2 .The multiplicities of 1 as a root of S(x) have 303 been intensively discussed in [9] and [10].Here we consider 304 the multiplicity of γ as a root of S(x) where γ( = 1) is a 305 primitive rth root of unity in an extension field of F 2 .Note 306 that (x 2r − 1)|(x q−1 − 1) since 2r|(q − 1).
where * means that for a fixed h, one and only one of h and 324 h + r is odd, and is taken once by 2j + 1 when j runs from 325 0 to r − 1. 326 where n = h (mod r); where l = log M (t) + 1 if t ≥ 1, and u(i, h) is (by the Chinese-Remainder-Theorem) the unique integer u satisfying Since γ, . . ., γ r−1 are linear independent over F 2 , it follows 327 that S(γ) = 0 if and only if 332 Then, we can get that 333 1) if u is even and v is even, then p (uv)/2 ≡ 1 (mod 4), 334 2) if u is odd, and v is even and 4 v, then p (uv)/2 ≡ 3 335 (mod 4) if p ≡ 3 (mod 4).

336
So, in these two cases, γ is a root of S(x), and 338 Second, we consider whether γ is a double root of S(x) in the above cases.According to Lemma 3, let l = 1 since , and u ≡ 1 (mod 2).Then we have

352
For any fixed h, u(1, h) is always odd, and can be equal

364
From the proof of Theorem Example 5: Let p = 19, u = 1, v = 2, and r = 5.Then we have a binary Sidelnikov sequence {s n } n≥0 of period 360.Similar to example 4, it is clear that 1 is a root of S(x), and for any γ( = 1) being a primitive 5th root of unity, γ is a double root of S(x), which means (x 4 + x 3 + x 2 + x + 1) 2 |S(x).Thus, the linear complexity of {s n } n≥0 over F 2 is LC({s n } n≥0 ) ≤ 351 from Theorem 1 2).In addition, from Proposition 1 in [10], it is clear that the multiplicity of 1 as a root of S(x) is 2, so, LC({s n } n≥0 ) ≤ 350.In fact, it follows that gcd( Remark 2: 1) In the proof of Theorem 1, we make full use of the formulas in Appendix B for the cyclotomic numbers of order 2r over F q with q = p uv ≡ 1 (mod 2r), where the order v of p modulo r is only even.Unfortunately, when v is odd, the cyclotomic problem is more intricate [27].
2) In general, the determination of cyclotomic numbers of order e is difficult if e is not small [10], meaning that we can only utilize these formulas for small r.Here we consider the cases r = 3 and 5 as examples.

B. Ternary Case
In this subsection, let q ≡ 1 (mod 3) where q is a prime.For the trivial root 1, the primitive 2nd and 3rd roots of unity, the multiplicities of them as the roots of S(x) are determined in Propositions 1, 2 and 3, respectively, by using the cyclotomic numbers of orders e's (e.g, 3, 6 and 9).However, if e is not small, it is very difficult to calculate the cyclotomic numbers, so the determined values of the multiplicity R are not very large in these propositions.For all that, Theorem 2 and Corollary 2 present the F 3 -linear complexities of the ternary Sidelnikov sequences, especially in the case of q = 2 × 3 λ + 1 where λ is a positive integer.
Firstly, we determine the multiplicities of the trivial root 1 of unity, as a root of S(x), in the case of Ind 3 ≡ 0 (mod 3).
We will determine the multiplicities of γ as a root of S(x) using Appendix D. 1) First, consider the case R(γ) = 1.From (14a), it follows that ≡ 0 (mod 3) for all cases if b ≡ 0 (mod 3), which implies that γ is a single root of S(x) if and only if b ≡ 0 (mod 3).

572
Thirdly, it is worth noting from Proposition 1 that q ≡ 1 573 (mod 9) is one of necessary and sufficient conditions of 574 1 as a multiple root of S(x).Then, we are interested in the 575 multiplicity of γ (a primitive 3rd root of unity in an extension 576 field of F 3 ) as a root of S(x).577 Proposition 3: Let q ≡ 1 (mod 9) be a prime 578 where 4q = c 2 + 27d 2 and c ≡ 7 (mod 9).579 if Ind 2 ≡ 0 (mod 6), a ≡ b (mod 9) and b ≡ 0 (mod 3) if Ind 2 ≡ 2 or 5 (mod 6), a ≡ −b (mod 9) and b ≡ 0 (mod 3) if Ind 2 ≡ 1 or 4 (mod 6), where Ind 2 means the index of 2 to a base g modulo q.
Finally, for the special case of q = 2× 3 λ + 1 where λ is an integer, the following corollary can be obtained, and Table I lists all examples for 1 ≤ λ ≤ 20.From this table, it is easy to see that the linear complexities of all sequences considered are extremely close to their periods.

1 j=0
b j M j where b 0 , . . ., b l−1 are the digits in the 214 M -ary representation of b.It is clear that the items of x b on 215