Multiplicities of Character Values of Binary Sidel'nikov-Lempel-Cohn-Eastman Sequences
aa r X i v : . [ c s . I T ] F e b Multiplicities of Character Values of BinarySidel’nikov-Lempel-Cohn-Eastman Sequences
Qi Zhang, Jing YangSeptember 22, 2018
Abstract
Binary Sidel’nikov-Lempel-Cohn-Eastman sequences (or SLCE sequences) over F have even period andalmost perfect autocorrelation. However, the evaluation of the linear complexity of these sequences is reallydifficult. In this paper, we continue the study of [1]. We first express the multiple roots of characterpolynomials of SLCE sequences into certain kinds of Jacobi sums. Then by making use of Gauss sums andJacobi sums in the “semiprimitive” case, we derive new divisibility results for SLCE sequences. Keywords:
Linear complexity, Sidel’nikov sequence, Stream cipher, Gauss sums, Jacobi sums
Let F be a finite field and s = ( s , s , s , · · · ) be a sequence over F . s is called periodic if there exists apositive integer T such that s i = s i + T for any i ≥
0. The smallest such T is called the period of s .Periodic sequences applied in stream ciphers should has some good properties, such as good balance prop-erty, low autocorrelation, large period and large linear complexity [10]. In this paper, we focus on linearcomplexity.For a sequence s , if there exists a positive integer L and c , c , · · · , c L ∈ F such that s i = − ( c s i − + c s i − + · · · + c L s i − L ) for each i ≥ L then the smallest such L is called the linear complexity of s , and corresponding polynomial c ( X ) := 1 + c X + c X + · · · + c L X L is called the minimal polynomial of s .Throughout the paper, let p denote a prime number, q = p m , F q the finite field of order q , α a primitiveelement of F q , d | ( q −
1) a prime number. Since F ∗ q is a cyclic group of order q − h α d i is the unique subgroupof index d . We define C i := α i h α d i as the i th cyclotomic coset of index d . Then the d -ary SLCE sequence s = ( s , s , s , · · · ) over F d can be defined by s n = ( i if α n + 1 ∈ C i for some i d = 2) independently. The sequence s has even period T = q −
1, perfect balance property and almostoptimal autocorrelation [21, 17]. A natural problem is to study the linear complexity of SLCE sequences.Let S ( X ) = s + s X + s X + · · · + s T − X T − . It is a basic result [10] that the minimal polynomial of s is given by c ( X ) = X T − X T − , S ( X ))Therefore, the linear complexity of s is given by L = deg c ( X ) = T − deg(gcd( X T − , S ( X ))) d = 2 implies p is a odd prime number.
1o it is enough to decide gcd( X T − , S ( X )). We emphasize that these are polynomials over F d whose charac-teristic is a divisor of T , which means we should not only determine the common roots but also compute themultiplicities of these roots.Let β be a root of X T −
1. Then β is a k th primitive root of unity over F d , where k | T and d ∤ k . Let I β ( X ) denote the minimal polynomial of β over F d . We need to determine whether I β ( X ) | S ( X ), i.e., S ( β ) = 0.Furthermore, we have to compute the multiplicities. This results are called “divisibility results” [1].[13], [16], [20], and [1] studied the linear complexity of binary SLCE sequences. [5] studied the linearcomplexity of d -ary SLCE sequences. [6] generalized the definition of SLCE sequences to the case d is a primepower and studied the linear complexity of generalized SLCE sequences over nonprime finite fields. Since itseems difficult to compute the linear complexity of d -ary SLCE sequences over F d , some researchers studied thelinear complexity and k -error linear complexity of SLCE sequences over F p . We refer the reader to [12], [11],[9], [8], [15], [22], [3], [7] and [2].In this paper, we continue the study of the linear complexity of binary SLCE sequences in [1]. [1] gave anecessary and sufficient condition of S ( β ) = 0, which is a congruence of Jacobi sums. Since Jacobi sums canbe evaluated in some specific cases, authors of [1] were able to give new divisibility results. In this paper, wefocus on the multiplicities of β . By making use of Hasse derivative and Lucas’s congruence, we can generalizethe condition in [1]. Finally, we prove that a root must be multiple root in semiprimitive case, which is a newdivisibility result.This paper is organized as follows. Section 2 briefly reviews the mathematical tools which are useful in thesequel. Section 3 gives the main results. Section 4 applies the main results to semiprimitive case. Section 5concludes this paper. A multiplicative character χ of a finite field F q is defined to be a group homomorphism from F ∗ q to C ∗ .Since F ∗ q = h α i , χ is completely determined by χ ( α ). It is easy to see χ ( α ) q − = χ ( α q − ) = χ (1) = 1, so χ ( α )must be a ( q − χ is h ζ q − i , where ζ n denotes exp(2 π √− /n ).The set of all multiplicative characters of F q is denoted by c F ∗ q . It is well-known that c F ∗ q = { η iq − : 0 ≤ i ≤ q − } where η j is determined by η j ( α ) = exp(2 π √− j ) for rational number j . Moreover, c F ∗ q forms a group isomorphicto F ∗ q under the point-wise multiplication. The identity element ε is defined by ε ( x ) = 1 for each x ∈ F ∗ q . Theinverse element of χ is χ , where χ ( x ) = χ ( x ) for each x ∈ F ∗ q .For convenience, we extend the domain of a multiplicative character χ to F q by setting χ (0) := 0.We need following results of multiplicative characters. Lemma 1.
For d | ( q − and x ∈ F ∗ q , we have d d − X i =0 η id ( x ) = ( if x ∈ h α d i if x
6∈ h α d i Lemma 2.
Let d , d | ( q − , χ a multiplicative character of order d . If gcd( d , d ) = 1 and d > , then X x ∈h α d i χ ( x ) = 0 Furthermore, for each i = 0 , , · · · , d − X x ∈ α i h α d i χ ( x ) = 02 .2 Gauss sums Let χ be a multiplication character of F q . The Gauss sum is defined as follows G ( χ ) := X x ∈ F q χ ( x ) ζ Tr( x ) p where Tr is the trace mapping from F q to F p .Gauss sums are extremely difficult to evaluate in general. But there are several special cases in which wecan get the explicit formula for Gauss sums. We need Gauss sums in quadratic case and semiprimitive case inthe sequel. Lemma 3 ([4], Theorem 11.5.4) . Let ρ = η be the quadratic multiplicative character of F q . Then G ( ρ ) = ( ( − m − √ q if p ≡ − m − i m √ q if p ≡ Lemma 4 ([4], Theorem 11.6.3) . Let χ be a multiplicative character of F q , N = ord( χ ) > . Suppose thereexists a positive integer v such that p v ≡ − N ) , with v chosen minimal. Then m = 2 vw for some positiviinteger w and G ( χ ) = ( − w − pw pv +1 N Let χ , χ be multiplicative characters of F q . The Jacobi sum is defined by J ( χ , χ ) := X x ∈ F q χ ( x ) χ (1 − x )We also denote J ( ρ, χ ) by K ( χ ) where ρ is a quadratic character.Jacobi sums are closely related to Gauss sums according to following lemma. Lemma 5 ([4], Theorem 2.1.3) . If χ χ = ε , then J ( χ , χ ) = G ( χ ) G ( χ ) G ( χ χ ) Let k be an odd positive integer, K = Q ( ζ k ) the k th cyclotomic field, O K = Z [ ζ k ] the ring of algebraicintegers in K . We need some results about the prime ideal factorization of (2) in O K . Lemma 6 ([14]) . Let f = ord k (2) , i.e., f is the smallest positive integer such that k | f − . Let P be anyprime ideal of O K over . We have • O K / P ∼ = F f . • , ζ k , · · · , ζ k − k are mutually distinct modulo P . • ∀ γ ∈ O K − P , there exists a unique k th root of unity ζ such that γ f − k ≡ ζ (mod P ) Hasse derivative is a powerful tool to study the multiplicities of a root in a polynomial over a field of finitecharacteristic. Let f ( X ) = a + a X + · · · + a n X n ∈ F [ X ], where F is a finite field. The t th Hasse derivative of f ( X ) is defined by f ( X ) ( t ) := (cid:18) tt (cid:19) a t + (cid:18) t + 1 t (cid:19) a t +1 X + · · · + (cid:18) nt (cid:19) a n X n − t Hasse derivative has following similar property to normal derivative.3 emma 7 ([18], Lemma 6.51) . Let f ( X ) ∈ F [ X ] be a nonzero polynomial, ξ ∈ F . Then the multiplicities of ξ in f ( X ) is t if and only if f ( ξ ) (0) = f ( ξ ) (1) = · · · = f ( ξ ) ( t − = 0 , f ( ξ ) ( t ) = 0 . Lucas’s congruence deals with binomial coefficients modulo a prime number.
Lemma 8 ([19]) . Let n, k be positive integers. Then (cid:18) nk (cid:19) ≡ (cid:18) n k (cid:19)(cid:18) n k (cid:19) · · · (cid:18) n l k l (cid:19) (mod 2) where n = n + n · · · + n l l and k = k + k · · · + k l l are -adic representation of n and k , respectively. From now on, we always assume d = 2, which implies p is odd. Let β be a root of X T − F , k = ord( β ), f the smallest positive integer such that k | (2 f − k is odd divisor of T = q − F ( β ) = F f . We further assume k > k = 1 implies β = 1 which was already studied in details [20].Let T = q − u T ′ where u ≥ T ′ is odd. Then β is 2 u th multiple root of X T −
1. Our aim is todetermine the multiplicities of β in S ( X ). According to lemma 7, it is enough to decide whether S ( β ) ( t ) = 0for 0 ≤ t ≤ u − Theorem 1.
Let notations be as above. For any t ∈ { , , · · · , u − } , S ( β ) ( t ) = 0 ⇔ (cid:18) T / t (cid:19) + T − X n =0 (cid:18) nt (cid:19) ρ ( α n + 1) χ ( α n ) ≡ P ) Proof.
Let K = Q ( ζ k ), O K = Z [ ζ k ] and P be a stipulated prime ideal of K over 2. By lemma 6, F ( β ) = F f ∼ = O K / P . Thus we can introduce a field isomorphism φ : F ( β ) → O K / P and S ( β ) ( t ) = 0 ⇔ β t S ( β ) ( t ) = 0 ⇔ T − X n =0 (cid:18) nt (cid:19) s n β n = 0 ⇔ φ ( T − X n =0 (cid:18) nt (cid:19) s n β n ) = 0 ⇔ T − X n =0 (cid:18) nt (cid:19) s n φ ( β ) n = 0Notice that β has order k , i.e., β = θ f − k for some θ ∈ F ∗ f . It follows from lemma 6 that there exists a unique k th primitive root of unity ζ such that φ ( β ) ≡ ζ (mod P ). Define the function χ : F ∗ q → K ∗ by χ ( α i ) := ζ i forarbitrary i . It is simple to verify that χ is a multiplicative character of F q and ord( χ ) = k . Then we have S ( β ) ( t ) = 0 ⇔ T − X n =0 (cid:18) nt (cid:19) s n ζ n ≡ P ) ⇔ T − X n =0 (cid:18) nt (cid:19) s n χ ( α n ) ≡ P )Recall that s n = ( α n + 1 ∈ α h α i α n + 1 α h α i . We introduce the quadratic character ρ = η and s n can be expressedas − ρ ( α n +1) − δ n,T/ . Thus, S ( β ) ( t ) = 0 ⇔ T − X n =0 (cid:18) nt (cid:19) s n χ ( α n ) ≡ P )4 T − X n =0 (cid:18) nt (cid:19) (1 − ρ ( α n + 1) − δ n,T/ ) χ ( α n ) ≡ P ) ⇔ T − X n =0 (cid:18) nt (cid:19) χ ( α n ) − T − X n =0 (cid:18) nt (cid:19) ρ ( α n + 1) χ ( α n ) − (cid:18) T / t (cid:19) χ ( α T/ ) ≡ P )Observe that (cid:0) nt (cid:1) ≡ n is congruent to some specific numbers modulo 2 h where h is thelength of binary representation of t by lemma 8. The first summation can be divided into several summations.Each summation has the form P n ≡ i (mod 2 h ) χ ( α n ). Since t ≤ u −
1, we have 2 h | u and P n ≡ i (mod 2 h ) χ ( α n ) = P x ∈ α i h α h i χ ( x ), which is equal to 0 due to lemma 2. On the other hand, χ ( α T/ ) = χ ( −
1) = 1 follows fromthe fact χ has odd order. Therefore, S ( β ) ( t ) = 0 ⇔ − T − X n =0 (cid:18) nt (cid:19) ρ ( α n + 1) χ ( α n ) − (cid:18) T / t (cid:19) ≡ P )which completes the proof.As we already mentioned in the proof, the summation P T − n =0 (cid:0) nt (cid:1) ρ ( α n + 1) χ ( α n ) can be divided into severalsummations, where each is summed over a cyclotomic coset. Let h = ( t = 0min { i ≥ t ≤ i } if t > t and I t = { ≤ i ≤ h − i (cid:23) t } where a (cid:23) b means each binary bit of a is great than or equal to that of b , i.e., (cid:0) ab (cid:1) ≡ (cid:0) nt (cid:1) ≡ n is congruent to some number in I t modulo 2 h . Nowwe can use I t and lemma 1 to write previous theorem into an expression involving Jacobi sums. Theorem 2.
Under the notations above, S ( β ) ( t ) = 0 is equivalent to h (cid:18) T / t (cid:19) + X i ∈ I t h − X j =0 η j h ( − ζ − ij h K ( η j h χ ) ≡ h +1 P Z [ ζ h k ]) Proof.
For a proposition Q , put δ Q = ( Q is true0 if Q is falseIt follows that S ( β ) ( t ) = 0 ⇔ (cid:18) T / t (cid:19) + T − X n =0 (cid:18) nt (cid:19) ρ ( α n + 1) χ ( α n ) ≡ P ) ⇔ (cid:18) T / t (cid:19) + T − X n =0 X i ∈ I t δ n ≡ i (mod 2 h ) ρ ( α n + 1) χ ( α n ) ≡ P ) ⇔ (cid:18) T / t (cid:19) + X i ∈ I t T − X n =0 δ α n − i ∈h α h i ρ ( α n + 1) χ ( α n ) ≡ P ) ⇔ h (cid:18) T / t (cid:19) + X i ∈ I t T − X n =0 h δ α n − i ∈h α h i ρ ( α n + 1) χ ( α n ) ≡ h +1 P Z [ ζ h k ]) ⇔ h (cid:18) T / t (cid:19) + X i ∈ I t T − X n =0 2 h − X j =0 η j h ( α n − i ) ρ ( α n + 1) χ ( α n ) ≡ h +1 P Z [ ζ h k ]) ⇔ h (cid:18) T / t (cid:19) + X i ∈ I t h − X j =0 ζ − ij h T − X n =0 ρ ( α n + 1) η j h χ ( α n ) ≡ h +1 P Z [ ζ h k ])5 h (cid:18) T / t (cid:19) + X i ∈ I t h − X j =0 ζ − ij h X x ∈ F q ρ ( x + 1) η j h χ ( x ) ≡ h +1 P Z [ ζ h k ]) ⇔ h (cid:18) T / t (cid:19) + X i ∈ I t h − X j =0 η j h χ ( − ζ − ij h X x ∈ F q ρ ( − x + 1) η j h χ ( x ) ≡ h +1 P Z [ ζ h k ]) ⇔ h (cid:18) T / t (cid:19) + X i ∈ I t h − X j =0 η j h ( − ζ − ij h J ( ρ, η j h χ ) ≡ h +1 P Z [ ζ h k ])It completes the proof.For specific t , one can obtain explicit condition by theorem 2. We give several examples to show this andomit the details of the proof. Proposition 1 ([1], Theorem 3.1) . Keep the notations as above. S ( β ) = 0 is equivalent to K ( χ ) ≡ P ) . Proposition 2.
Keep the notations as above. S ( β ) (1) = 0 is equivalent to ( K ( χ ) − K ( η χ ) ≡ P ) if q ≡ K ( χ ) + K ( η χ ) ≡ P ) if q ≡ Proposition 3.
Keep the notations as above. If q ≡ , then S ( β ) (2) = 0 is equivalent to ( K ( χ ) − (1 − ζ ) K ( η χ ) − (1 + ζ ) K ( η χ ) ≡ P Z [ ζ k ]) if q ≡ K ( χ ) + (1 − ζ ) K ( η χ ) + (1 + ζ ) K ( η χ ) ≡ P Z [ ζ k ]) if q ≡ Proposition 4.
Keep the notations as above. If q ≡ , then S ( β ) (3) = 0 is equivalent to ( K ( χ ) + ζ K ( η χ ) − K ( η χ ) − ζ K ( η χ ) ≡ P Z [ ζ k ]) if q ≡ K ( χ ) − ζ K ( η χ ) − K ( η χ ) + ζ K ( η χ ) ≡ P Z [ ζ k ]) if q ≡ β is t th multiple root of S ( X ) for general t . But we can obtain a relatively simple condition when t = 2 h . Theorem 3.
With the notations above, β has multiplicities at least h as root of S ( X ) if and only if · · · ζ − h · · · ζ − (2 h − h ... ... . . . ... ζ − (2 h − h · · · ζ − (2 h − h − h η h ( − K ( η h χ ) η h ( − K ( η h χ ) ... η h − h ( − K ( η h − h χ ) ≡ − h δ δ ... δ h − (mod 2 h +1 P Z [ ζ h k ]) where δ i = ( if T / ≡ i (mod 2 h )0 if T / i (mod 2 h ) .Proof. Consider the summation E i = T − X n =0 n ≡ i (mod 2 h ) s n β n Recall that S ( β ) ( t ) = 0 if and only if P T − n =0 (cid:0) nt (cid:1) s n β n = 0. The summation of the left hand side can break upinto several E i , and the equation becomes h − X i =0 i (cid:23) t E i = 0 ( D t )6rom lemma 7, β has multiplicities at least 2 h if and only if D t holds for every t ∈ { , , · · · , h − } .Next, we prove that equations D t where t = 0 , , · · · , h − E i = 0 where i = 0 , , · · · , h −
1. It is trivial that equations E i s implies D t s. We only need to prove that E i s can be derivedfrom D t s. Notice that D h − is actually the same as E h − = 0. Since these equations are all over a field ofcharacteristic 2, we can add D h − to other D t s to cancel the summation E h − D t s. Afterthe process, the equation D h − becomes E h − = 0 and we can use the same method to cancel E h − occurringin other D t s where 0 ≤ t ≤ h −
3. Then D h − becomes E h − = 0. Repeat these elementary operations until D becomes E = 0 and it completes the proof.Finally, we use the same techniques in proof of theorem 1 and 2 E i = 0 ⇔ T − X n =0 δ n ≡ i (mod 2 h ) s n β n = 0 ⇔ X n ≡ i (mod 2 h ) s n χ ( α n ) ≡ P ) ⇔ X n ≡ i (mod 2 h ) s n χ ( α n ) ≡ P ) ⇔ X n ≡ i (mod 2 h ) (1 − ρ ( α n + 1) − δ n,T/ ) χ ( α n ) ≡ P ) ⇔ X n ≡ i (mod 2 h ) ( − ρ ( α n + 1) − δ n,T/ ) χ ( α n ) ≡ P ) ⇔ δ i + X n ≡ i (mod 2 h ) ρ ( α n + 1) χ ( α n ) ≡ P ) ⇔ h δ i + 2 h X n ≡ i (mod 2 h ) ρ ( α n + 1) χ ( α n ) ≡ h +1 P Z [ ζ h k ]) ⇔ h δ i + T − X n =0 2 h − X j =0 η j h ( α n − i ) ρ ( α n + 1) χ ( α n ) ≡ h +1 P Z [ ζ h k ]) ⇔ h − X j =0 ζ − ij h X x ∈ F q ρ ( x + 1) η j h χ ( x ) ≡ − h δ i (mod 2 h +1 P Z [ ζ h k ]) ⇔ h − X j =0 ζ − ij h η j h ( − K ( η j h χ ) ≡ − h δ i (mod 2 h +1 P Z [ ζ h k ])It completes the proof. Theorem 4.
With the notations above, a necessary condition for β having multiplicities at least h is K ( η j h χ ) ≡ P Z [ ζ h k ]) for any j ∈ { , , · · · , h − } .Proof. Let C = · · · ζ − h · · · ζ − (2 h − h ... ... . . . ...1 ζ − (2 h − h · · · ζ − (2 h − h − h Observing C ∗ C = 2 h I , left multiply C ∗ to the congruence in theorem 3 and we obtain2 h η h ( − K ( η h χ ) η h ( − K ( η h χ )... η h − h ( − K ( η h − h χ ) ≡ − h C ∗ δ δ ... δ h − (mod 2 h +1 P Z [ ζ h k ])7 η h ( − K ( η h χ ) η h ( − K ( η h χ )... η h − h ( − K ( η h − h χ ) ≡ − C ∗ δ δ ... δ h − (mod 2 P Z [ ζ h k ])If h = u , then η h ( −
1) = − δ i = δ i, h − . If h < u , then η h ( −
1) = 1 and δ i = δ i, . In any of these twocases, the congruence can be simplified to K ( η h χ ) K ( η h χ )... K ( η h − h χ ) ≡ − − − (mod 2 P Z [ ζ h k ])It completes the proof. In this section, we apply the theorem we get to semiprimitive case. Throughout this section, we assumethat there exists a positive integer v such that 2 h k | ( p v + 1). It follows that m = 2 vw for some w . In addition,it must be the case h < u because 2 h | p v + 1 and p m − p v +1 is even. Lemma 9.
With the notations and assumptions above, K ( η i h χ ) = G ( ρ ) for any i ∈ { , , · · · , h − } .Proof. Using lemma 5, we have K ( η i h χ ) = J ( ρ, η i h χ ) = G ( ρ ) G ( η i h χ ) G ( ρη i h χ )We only need to prove G ( η i h χ ) = G ( ρη i h χ ).If i = 0 or i = 2 h − , then we need to prove G ( χ ) = G ( ρχ ). Note that both two are semiprimitive Gausssum. The smallest positive integers v , v such that k | ( p v + 1) , k | ( p v + 1) are equal. Hence, G ( χ ) = G ( ρχ )follows from lemma 4.Otherwise, the order of η i h χ is the equal to that of ρη i h χ . Thus it trivially follows from lemma 4. Lemma 10.
With the notations and assumptions above, if β is a root of S ( X ) , then the multiplicities of β is ≥ h .Proof. Notice that h < u , which implies η h ( −
1) = 1 and δ i = δ i, . By theorem 3, β has multiplicities ≥ h ifand only if · · · ζ − h · · · ζ − (2 h − h ... ... . . . ...1 ζ − (2 h − h · · · ζ − (2 h − h − h K ( η h χ ) K ( η h χ )... K ( η h − h χ ) ≡ − h (mod 2 h +1 P Z [ ζ h k ])From lemma 9, we know K ( η i h χ ) = K ( χ ) = G ( ρ ) for arbitrary i . Thus,l . h . s = · · · ζ − h · · · ζ − (2 h − h ... ... . . . ...1 ζ − (2 h − h · · · ζ − (2 h − h − h K ( χ ) K ( χ )... K ( χ ) = h K ( χ )0...0 h K ( χ ) ≡ − h (mod 2 h +1 P Z [ ζ h k ]) ⇔ K ( χ ) ≡ − P Z [ ζ h k ]) ⇔ K ( χ ) ≡ − P )The last step is because K ( χ ) = G ( ρ ) = ± p vw is an integer under our assumptions. Finally, note that K ( χ ) ≡ − P ) if and only if S ( β ) = 0 by proposition 1. Theorem 5.
Keep the notations and assumptions as above. Suppose v ′ is the smallest positive integer suchthat k | ( p v ′ + 1) and m = 2 v ′ w ′ . We have • If p ≡ , then (1 + X + · · · + X k − ) h | S ( X ) ⇔ w ′ is even. • If p ≡ , then (1 + X + · · · + X k − ) h | S ( X ) ⇔ w ′ is even or v ′ w ′ is odd.Proof. It simply follows lemma 10 and theorem 4.1 of [1].
In this paper, we generalize the main result of [1] by considering the multiplicities of character values ofSLCE sequences. And we apply our results to semiprimitive case and prove that a root must be multiple rootin semiprimitive case.There are still many open problems to be tackled in this research area. For example, can we use evaluationsof Gauss sums in other cases to get more divisibility results? Is the necessary condition strong enough to derivea contradiction which implies the multiplicities is small? Can we derive a better bound for linear complexityof SLCE sequences than [20]? How can we generalize this work to study linear complexity of d -ary SLCEsequences? Is there any method to study 1-error or even k -error linear complexity of SLCE sequences? [1] S¸aban Alaca and Goldwyn Millar. Character values of the sidelnikov-lempel-cohn-eastman sequences. Cryptography and Communications , pages 1–18, 2016.[2] H. Aly and W. Meidl. On the linear complexity and k-error linear complexity over F p of the d -ary sidel’nikovsequence. IEEE Transactions on Information Theory , 53(12):4755–4761, 2007.[3] Hassan Aly and Arne Winterhof. On the k-error linear complexity over F p of legendre and sidelnikovsequences. Designs, Codes and Cryptography , 40(3):369–374, 2006.[4] Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams.
Gauss and Jacobi sums . Canadian Mathe-matical Society Series of Monographs and Advanced Texts. John Wiley & Sons, Inc., New York, 1998. AWiley-Interscience Publication.[5] Nina Brandsttter and Wilfried Meidl.
On the Linear Complexity of Sidel’nikov Sequences over Fd , pages47–60. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006.[6] Nina Brandsttter and Wilfried Meidl. On the linear complexity of sidel’nikov sequences over nonprimefields.
Journal of Complexity , 24(56):648–659, 2008.[7] Jin-Ho Chung and Kyeongcheol Yang.
Bounds on the Linear Complexity and the 1-Error Linear Complexityover F p of M -ary Sidel’nikov Sequences , pages 74–87. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006.[8] Yu-Chang Eun, Hong-Yeop Song, and Gohar M. Kyureghyan. One-Error Linear Complexity over Fp ofSidelnikov Sequences , pages 154–165. Springer Berlin Heidelberg, Berlin, Heidelberg, 2005.99] M. Z. Garaev, F. Luca, I. E. Shparlinski, and A. Winterhof. On the lower bound of the linear complexityover fp of sidelnikov sequences.
IEEE Transactions on Information Theory , 52(7):3299–3304, 2006.[10] Solomon W. Golomb and Guang Gong.
Signal Design for Good Correlation: For Wireless Communication,Cryptography, and Radar . Cambridge University Press, New York, NY, USA, 2004.[11] T. Helleseth, M. Maas, J. E. Mathiassen, and T. Segers. Linear complexity over fp of sidel’nikov sequences.
IEEE Transactions on Information Theory , 50(10):2468–2472, 2004.[12] T. Helleseth, Kim Sang-Hyo, and No Jong-Seon. Linear complexity over fp and trace representation oflempel-cohn-eastman sequences.
IEEE Transactions on Information Theory , 49(6):1548–1552, 2003.[13] Tor Helleseth and Kyeongcheol Yang.
On Binary Sequences of Period n=pm-1 with Optimal Autocorrela-tion , pages 209–217. Springer London, London, 2002.[14] Kenneth Ireland and Michael Rosen.
A classical introduction to modern number theory.
SPRINGER, 1982.[15] Young-Sik Kim, Jung-Soo Chung, Jong-Seon No, and Habong Chung.
Linear Complexity over F p ofTernary Sidel’nikov Sequences , pages 61–73. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006.[16] Gohar M. Kyureghyan and Alexander Pott. On the linear complexity of the sidelnikov-lempel-cohn-eastmansequences. Designs, Codes and Cryptography , 29(1):149–164, 2003.[17] A. Lempel, M. Cohn, and W. Eastman. A class of balanced binary sequences with optimal autocorrelationproperties.
IEEE Transactions on Information Theory , 23(1):38–42, 1977.[18] Rudolf Lidl and Harald Niederreiter.
Finite fields . Encyclopaedia of mathematics and its applications.Cambridge University Press, New York, 1997. Cette rimpression est la version 2008.[19] E. Lucas. Sur les congruences des nombres eul´eriens et les coefficients diff´erentiels des functionstrigonom´etriques suivant un module premier.
Bull. Soc. Math. France , 6:49–54, 1878.[20] Wilfried Meidl and Arne Winterhof. Some notes on the linear complexity of sidel’nikov-lempel-cohn-eastman sequences.
Designs, Codes and Cryptography , 38(2):159–178, 2006.[21] V. M. Sidelnikov. Some k-valued pseudo-random sequences and nearly equidistant codes.