Binary Sequences with Optimal Autocorrelation

Transcription

1 Cunsheng DING, HKUST, Kowloon, HONG KONG, CHINA September 2008

2 Outline of this talk Difference sets and almost difference sets Cyclotomic classes Introduction of binary sequences with optimal autocorrelation Specific constructions: the case N 1 (mod 4) Specific constructions: the case N 2 (mod 4) Concluding remarks Remark: The cases N 3 (mod 4) and N 0 (mod 4) will not be covered in this talk. Page 1 September 2008

3 Part I: (Almost) Difference Sets Page 2 September 2008

4 Difference Sets Definition: Let (A,+) be an abelian group of order n. Let C be a k-subset of A. Define the difference function as d C (w) = (C+ w) C. The set C is an (n,k,λ) difference set (DS) of A if d C (w) = λ for every nonzero element of A. Necessary condition: k(k 1) = (n 1)λ. Example: Let p = 7. The set of quadratic residues modulo 7, C = {1,2,4}, is a (7,3,1) difference set of (Z 7,+). D. Jungnickel, A. Pott, Difference sets: an introduction, in Difference Sets, Sequences and their Correlation Properties, eds., A. Pott, P.V. Kumar, T. Helleseth and D. Jungnickel, pp Amsterdam: Kluwer, Page 3 September 2008

5 Almost Difference Sets Definition: Let (A,+) be an abelian group of order n. A k-subset C of A is called an (n,k,λ,t) almost difference set (ADS) of A if d C (w) takes on λ altogether t times and λ+1 altogether n 1 t times when w ranges over all the nonzero elements of A. Necessary condition: k(k 1) = tλ+(n 1 t)(λ+1). Example: The set of quadratic residues modulo 13, C = {1,3,4,9,10,12}, is a (13,6,2,6) ADS of (Z 13,+), becuase C C = { 0 6,11 3,12 2,1 2,2 3,3 2,4 2,5 3,6 3,7 3,8 3,9 2,10 2 }. K.T. Arasu, C. Ding, T. Helleseth, P.V. Kumar, H. Martinsen, Almost difference sets and their sequences with optimal autocorrelation, IEEE Trans. Information Theory 47 (2001) Page 4 September 2008

6 Part II: Cyclotomic Classes Page 5 September 2008

7 Cyclotomic Classes Let q = d f + 1 be a power of a prime, θ a fixed primitive element of GF(q). Define D (d,q) i = θ i θ d. The cosets D (d,q) l are called the index classes or cyclotomic classes of order d with respect to GF(q). Clearly GF(q) \ {0} = d 1 i=0 D(d,q) i. Define (l,m) d = ( ) l + 1 D (d,q) D (d,q) m These (l,m) d are called cyclotomic numbers of order d with respect to GF(q). Remark: They are basic building blocks in many systems! Applications: Sequences, coding theory, cryptography, combinatorics.. Page 6 September 2008

8 Part III: Introduction of Binary Sequences with Optimal Autocorrelation Page 7 September 2008

9 The Autocorrelation Function The autocorrelation function of a binary sequence {s(t)} of period N at shift w is AC s (w) = N 1 t=0 ( 1) s(t+w) s(t). The set C s = {0 i N 1 : s(i) = 1} is the support of {s(t)}; and {s(t)} is the characteristic sequence of C s Z N. The weight of {s(t)} is defined to be C s. This is a one-to-one correspondence. Studying binary sequences of period N is equivalent to that of subsets of Z N. {s(t)} is balanced if the weight is N/2 for even N, and (N ± 1)/2 for odd N. Page 8 September 2008

10 The Autocorrelation and Difference Functions Let {s(t)} be a binary sequence of period N. Define C = {0 i N 1 : s(i) = 1}, d C (w) = (w+c) C, which is the difference function of C Z N defined before. Then AC s (w) = N 4(k d C (w)), where k = C. It is a bridge between binary sequences and combinatorial designs! Page 9 September 2008

11 The Best Possible Autocorrelation Values (I) (1) AC s (w) = 1 for all 1 w N 1 (ideal autocorrelation), if N 3 (mod 4); (2) AC s (w) = 1 for all 1 w N 1, if N 1 (mod 4); (3) AC s (w) = 2 or AC s (w) = 2 for all 1 w N 1, if N 2 (mod 4); (4) AC s (w) = 0 for all 1 w N 1, if N 0 (mod 4). Fundamental problem: Is there any binary sequence {s(t)} of period N with such uniform out-of-phase autocorrelation value? Page 10 September 2008

12 The Best Possible Autocorrelation Value: the Case N 3 (mod 4) Fundamental problem: Is there any binary sequence {s(t)} of period N with AC s (τ) = 1 for all 1 τ N 1? Answer: There are many constructions. Details will not be covered in this talk. Page 11 September 2008

13 The Best Possible Autocorrelation Value: the Case N 1 (mod 4) Fundamental problem: Is there any binary sequence {s(t)} of period N with AC s (τ) = 1 for all 1 τ N 1? It is easily proved that no balanced binary sequence of period N 1 (mod 4) with only out-of-phase autocorrelation value 1 exists. There is evidence that no example with N > 13 can exist. Open problem 1: Prove or disprove that no binary sequence of period N 1 (mod 4) > 13 with only out-of-phase autocorrelation value 1 exists. D. Jungnickel, A. Pott, Perfect and almost perfect sequences, Discrete Applied Mathematics 95 (1999) C. Carlet, C. Ding, Highly nonlinear functions, J. Complexity 20 (2004) Page 12 September 2008

14 The Best Possible Autocorrelation Value: the Case N 0 (mod 4) Fundamental problem: Is there any binary sequence {s(t)} of period N with AC s (τ) = 0 for all 1 τ N 1? It is easily proved that no balanced binary sequence of period N 0 (mod 4) with only autocorrelation value 0 exists. There is strong evidence that no example with N > 4 can exist. Open problem 2: Prove or disprove that no binary sequence of period N 0 (mod 4) > 4 with only out-of-phase autocorrelation value 0 exists. D. Jungnickel, A. Pott, Perfect and almost perfect sequences, Discrete Applied Mathematics 95 (1999) Page 13 September 2008

15 The Best Possible Autocorrelation Value: the Case N 2 (mod 4) Fundamental problem: Is there any binary sequence {s(t)} of period N with AC s (τ) = 2 (respectively AC s (τ) = 2) for all 1 τ N 1? It is easily proved that no balanced binary sequence of period N 2 (mod 4) with only autocorrelation value 2 (respectively 2) exists. It looks that no example exists. Open problem 3: Prove or disprove that no binary sequence of period N 2 (mod 4) with only out-of-phase autocorrelation 2 (respectively 2) exists. Page 14 September 2008

16 The Optimal Autocorrelation Values (1) AC s (w) = 1 for all 1 w N 1, if N 3 (mod 4); (2) AC s (w) {1, 3} for all 1 w N 1, if N 1 (mod 4); (3) AC s (w) {2, 2} for all 1 w N 1, if N 2 (mod 4); (4) AC s (w) {0, 4} or AC s (w) {0,4} or AC s (w) { 4,0,4} for all 1 w N 1, if N 0 (mod 4). Remark: If we are interested in binary sequence of period N with Hamming weight (N ± δ)/2, where 0 δ 3, these are indeed optimal autocorrelation values (the proof is trivial). Page 15 September 2008

17 Optimal Autocorrelation and Combinatorial Designs (1) Let N 3 (mod 4). Then AC s (w) = 1 for all 1 w N 1, iff C is an (N,(N + 1)/2,(N + 1)/4) or (N,(N 1)/2,(N 3)/4) DS of Z N. (2) Let N 1 (mod 4). Then AC s (w) {1, 3} for all 1 w N 1, iff C is an (N,k,k (N + 3)/4,Nk k 2 (N 1) 2 /4) ADS. (3) Let N 2 (mod 4). Then AC s (w) {2, 2} for all 1 w N 1, iff C is an (N,k,k (N + 2)/4,Nk k 2 (N 1)(N 2)/4) ADS. (4) Let N 0 (mod 4). Then AC s (w) {0, 4} for all 1 w N 1, iff C is an (N,k,k (N + 4)/4,Nk k 2 (N 1)N/4) ADS. Remark: Constructing binary sequences with optimal autocorrelation becomes that of cyclic (almost) difference sets. Arasu, Ding, Helleseth, Kumer, Martinsen, Almost difference sets and their sequences with optimal autocorrelation, IEEE IT 47(7) (2001) Page 16 September 2008

18 The Equivalence of Binary Sequences Definition: Let {s 1 (t)} and {s 2 (t)} be two binary sequences of period N. If there are a nonnegative integer u with gcd(u,n) = 1, an integer v, and a constant l {0,1} such that the two sequences are said equivalent. s 1 (t) = s 2 (ut + v)+l for all t, Equivalent sequences have the same set of autocorrelation values. A sequence is equivalent to its complement. Page 17 September 2008

19 Part IV: Construction of Optimal Sequences the Case N 1 (mod 4) Legendre (1798) Jensen-Jensen-Høholdt (1991) Ding (1998), Mertens and Bessenrodt (1998) Ding-Helleseth-Lam (1999) Page 18 September 2008

20 Legendre Sequences Let p 1 (mod 4) be a prime. The Legendre sequence is defined by 1, if t mod p is a quadratic residue; s(t) = 0, otherwise. Remark: Linear complexity by Turyn, rediscovered by Ding, Helleseth and Shan. C. Ding, Pattern distribution of Legendre sequences, IEEE Trans. Information Theory 44 (1998) R. Turyn, The linear generation of the Legendre sequences, J. Soc. Ind. Appl. Math. 12(1) (1964) C. Ding, T. Helleseth, W. Shan, On the linear complexity of Legendre sequences, IEEE Trans. Information Theory 44 (1998) Page 19 September 2008

21 The Two-Prime Sequences (I) Let p and q be two distinct primes. The two-prime sequence (generalized cyclotomic sequence of order 2) is defined by s i = F(i mod pq) with 0, j {0,q,2q,,(p 1)q}; F( j) = 1, j {p,2p,,(q 1)p}; ( )) 1 ( j j p)( q /2, otherwise where ( ap ) denotes the Legendre symbol. The sequences are different from the Jacobi sequences in literature. They are a generalization of the twin-prime sequences (i.e., when q p = 2) [Whiteman 1962, Illinois J. Math 6 (1962) ]. It has optimal autocorrelation { 3,1} when q p = 4. Page 20 September 2008

22 They were described in e.g.: The Two-Prime Sequences (II) J.M. Jensen, H.E. Jensen, T. Høholdt, The merit factor of binary sequences realted to difference sets, IEEE Trans. IT 37(3) (1991) In 1998, Ding determined the autocorrelation values under the condition that gcd(p 1,q 1) = 2 in: C. Ding, Autocorrelation values of generalized cyclotomic sequences of order two, IEEE Trans. Inf. Theory 44 (1998) Independently 1998, Mertens-Bessenrodt, the autocorrelation values in: S. Mertens and C. Bessenrodt, On the ground states of the Bernasconi model, J. Phys. A: Math. Gen. 31 (1998) So exactly 200 years after Legendre, the 2nd class of such sequences was discovered. Page 21 September 2008

23 The Ding-Helleseth-Lam Construction Let q = 1 (mod 4), and let D (4,q) i be the cyclotomic classes of order 4. ( ) For all i, the set D (4,q) i D (4,q) i+1 is a q, q 1 2, q 5 4, q 1 2 ADS, if q = x and x 1 (mod 4). Their characteristic sequences have optimal autocorrelation { 3,1}. In terms of equivalence, only one sequence is obtained. Their linear complexity is known (see the reference below). C. Ding, T. Helleseth, and K. Y. Lam, Several classes of sequences with three-level autocorrelation, IEEE Trans. Inform. Theory 45 (1999) Page 22 September 2008

24 Open Problems for the Case N 1 (mod 4) There are sequences with optimal autocorrelation that do not below to the known classes: E.g., the sequence defined by the (45,22,10,22) ADS of Z 45 : {0,1,2,3,4,5,6,7,9,11,12,15,16, 19,23,24,29,30,32,35,37,39}. E.g., the sequence defined by the (33,16,7,16) ADS of Z 33 : {0,1,2,3,4,5,6,8,13,14,18,20,22,25,28,29}. Open Problem 4: Are there other classes of binary sequences of period N 1 (mod 4) with optimal autocorrelation? Page 23 September 2008

25 Part V: Construction of Optimal Sequences the Case N 2 (mod 4) Sidelnikov-Lempel-Cohn-Eastman (1969,1977) Ding-Helleseth-Martinsen (2001) No-Chung-Song-Yang-Lee-Helleseth (2001) Page 24 September 2008

26 The Sidelnikov-Lempel-Cohn-Eastman Construction Let q 3 (mod 4) be a prime power. Define C q = log α (D (2,q) 1 1). Then C q is a ( ) q 1, q 1 2, q 3 4, 3q 5 4 almost difference set, and its characteristic sequence has optimal autocorrelation values { 2,2}. A. Lempel, M. Cohn, and W. L. Eastman, A class of binary sequences with optimal autocorrelation properties, IEEE Trans. Inform. Theory 23 (1977) V. M. Sidelnikov, Some k-valued pseudo-random sequences and nearly equidistant codes, Probl. Inform. Trans. 5 (1969) Page 25 September 2008

27 The Ding-Helleseth-Martinsen Constructions: Part I Let q 5 (mod 8) be a prime. It is known that q = s 2 + 4t 2 for some s and t with s ±1 (mod 4). Set n = 2q. Let i, j,l {0,1,2,3} be three pairwise distinct integers, and define [ ] [ ] C = {0} (D (4,q) i D (4,q) j ) {1} (D (4,q) l D (4,q) j ). Then C is an ( n, n 2 2, n 6 4, 3n 6 ) 4 almost difference set of A = Z2 Z q if (1) t = 1 and (i, j,l) = (0,1,3) or (0,2,1); or (2) s = 1 and (i, j,l) = (1,0,3) or (0,1,2) Sequence: almost balanced, optimal autocorrelation values { 2,2}. C. Ding, T. Helleseth, H.M. Martinsen, New families of binary sequences with optimal three-level autocorrelation, IEEE Trans. Inf. Theory 47 (2001) Page 26 September 2008

28 The Ding-Helleseth-Martinsen Constructions: Part II Let q 5 (mod 8) be a prime. It is known that q = s 2 + 4t 2 for some s and t with s ±1 (mod 4). Set n = 2q. Let i, j,l {0,1,2,3} be three pairwise distinct integers, and define [ ( )] [ ( )] C = {0} D (4,q) i D (4,q) j {1} D (4,q) l D (4,q) j {0,0}. Then C is an ( n, 2 n, n 2 4, 3n 2 ) 4 almost difference set of A = Z2 Z q if (1) t = 1 and (i, j,l) {(0,1,3),(0,2,3),(1,2,0),(1,3,0)}; or (2) s = 1 and (i, j,l) {(0,1,2),(0,3,2),(1,0,3),(1,2,3)}. Sequence: balanced, optimal autocorrelation { 2,2}, large linear complexity. Page 27 September 2008

29 The No-Chung-Song-Yang-Lee-Helleseth Construction Let q 3 (mod 4) be a prime power. Define C q = {(q 1)/2} log α (D (2,q) 1 1). ( ) Then C q is a q 1, q+1 2, q+1 4, 3(q 3) 4 almost difference set. Its characteristic sequence has optimal autocorrelation values { 2,2}. Remark: This sequence is almost balanced, and is the 1-bit modification of the Sidelnikov-Lempel-Cohn-Eastman sequence in the (q+1)/2-th position. J.S. No, H. Chung, H.Y. Song, K. Yang, J.D. Lee, T. Helleseth, New construction for binary sequences of period p m 1 with optimal autocorrelation using (z+1) d + az d + b, IEEE Trans. Inform. Theory 47 (2001) Page 28 September 2008

30 Open Problems for the Case N 2 (mod 4) For N = 26, computer search has found the following five sequences with optimal autocorrelation: The sequence marked with * is the Sedelnikov-Lempel-Cohn-Eastman sequence. Open Problem 5: Are there other classes of binary sequences of period N 2 (mod 4) with optimal autocorrelation? Page 29 September 2008

31 Part VI: Concluding remarks Page 30 September 2008

32 Concluding Remarks Among the four cases, the two cases that N = 3 (mod 4) and N = 0 (mod 4) seem easier than the remaining cases. For the two cases that N = 1 (mod 4) and N = 2 (mod 4), there are only a few constructions. Most of the constructions are based directly on or related to cyclotomy or generalized cyclotomies. Every binary sequence with period N and optimal autocorrelation is equivalent to a function from (Z N,+) to (Z 2,+) with optimal nonlinearity. Page 31 September 2008