Binary Sequences with Optimal Autocorrelation

Transcription

1 Cunsheng DING, HKUST, Kowloon, HONG KONG, CHINA July 2004

2 The Autocorrelation Function The autocorrelation of a binary sequence {s(t)} of period N at shift w is C s (w) = ( 1) s(t+w) s(t). N 1 t=0 The set C = {0 i N 1 : s(i) = 1} is the characteristic set of {s(t)}; and {s(t)} is the characteristic sequence of C Z N. This is a one-to-one correspondence. Studying binary sequences of period N is equivalent to that of subsets of Z N. Page 1 July 2004

3 The Autocorrelation and Difference Functions Let {s(t)} be a binary sequence of period N. Define C = {0 i N 1 : s(i) = 1} and d C (w) = (w + C) C, which is called the difference function of C Z N. Then C s (w) = N 4(k d C (w)), where k = C. It is a bridge between binary sequences and combinatorial designs! Page 2 July 2004

4 The Optimal Autocorrelation Values (1) C s (w) = 1 for all w 0 (mod N) if N 3 (mod 4); (2) C s (w) {1, 3} for all w 0 (mod N) if N 1 (mod 4); (3) C s (w) {2, 2} for all w 0 (mod N) if N 2 (mod 4); (4) C s (w) {0, 4} or C s (w) {0, 4} for all w 0 (mod N) if N 0 (mod 4). A sequence {s(t)} of period N is said to have ideal autocorrelation if C s (w) = 1 for all w 0 (mod N), where N 3 (mod 4). Problem: Find binary sequences with optimal autocorrelation. Page 3 July 2004

5 Difference Sets Definition: Let (A, +) be an abelian group of order n. Let C be a k-subset of A. The set C is an (n, k, λ) difference set of A if d C (w) = λ for every nonzero element of A, where d C (w) is the difference function defined earlier. Necessary condition: k(k 1) = (n 1)λ. Remark: Difference sets do not exist for many parameters n, k, λ. Remark: This is a topic with a long history and many people have worked on it. Reference: D. Jungnickel and 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 4 July 2004

6 Almost Difference Sets Definition: Let (A, +) be an abelian group of order n. A k-subset C of A is an (n, k, λ, t) almost difference set 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) = (n 1)(λ + 1) t. Comment: Difference sets are just special almost difference sets, i.e., (n, k, λ, n 1) almost difference sets! Remark: Introduced by Davis and Ding independently. Survey: 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 5 July 2004

7 Optimal Autocorrelation and Combinatorial Designs (1) Let N 3 (mod 4). Then C s (w) = 1 for all w 0 (mod N) 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 C s (w) {1, 3} for all w 0 (mod N) 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 C s (w) {2, 2} for all w 0 (mod N) 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 C s (w) {0, 4} for all w 0 (mod N) iff C is an (N, k, k (N + 4)/4, Nk k 2 (N 1)N/4) ADS. Remark: The first case has been studied for a long time. The other cases are studied only recently. Page 6 July 2004

8 Binary Sequences with Ideal Autocorrelation The ( l, l 1 2, ) ( l 3 4 or l, l+1 2, ) l+1 4 difference sets of Zl are called Paley-Hadamard difference sets, which include those with parameters: ), where p 3 (mod 4) is prime, and the difference set just ( p, p 1 2, p 3 4 consists of all the quadratic residues in Z p. (2 t 1, 2 t 1 1, 2 t 2 1) Singer DS (m-sequences), Gordon-Mills-Welch DS (GMW sequences), Maschietti DS (Maschietti sequences), with projective geometry Power function constructions (Dillon, Dillon and Dobbertin) Other recent developments Page 7 July 2004

9 Binary Sequences with Ideal Autocorrelation ( l, l 1 2, ) l 3 4, where l = p(p + 2) and both p and p + 2 are primes. These twin-prime difference sets may be defined as {(g, h) Z p Z p+2 : g, h 0 and χ(g)χ(h) = 1} {(g, 0) : g Z p }, where χ(x) = +1 if x is a nonzero square in the corresponding field, and χ(x) = 1 otherwise; ( p, p 1 2, p 3 ) 4, where p is a prime of the form p = 4s They are cyclotomic difference sets defined by D (6,p) i D = D (6,p) 0 D (6,p) 1 D (6,p) 3, where D (6,p) 0 denotes the multiplicative group generated by α 6, = α i D (6,p) 0 denotes the cosets, and α is a primitive element of Z q. Sequence: balanced, autocorrelation { 1}, linear complexity known. Page 8 July 2004

10 Cyclotomy Let q = df + 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 = (D (d,q) l + 1) D (d,q) m. These constants (l, m) d are called cyclotomic numbers of order d with respect to GF(q). Applications: Sequences, coding theory, cryptography, combinatorics. Page 9 July 2004

11 Legendre Sequences Let p 1 (mod 4) be a prime. The Legendre sequence 1, if t mod p is a quadratic residue; s(t) = 0, otherwise. has optimal autocorrelations { 3, 1}. Linear complexity, pattern distributions, etc., see reference below. C. Ding, Pattern distribution of Legendre sequences, IEEE Trans. Information Theory 44 (1998) C. Ding, T. Helleseth, W. Shan, On the linear complexity of Legendre sequences, IEEE Trans. Information Theory 44 (1998) T. Cusick, C. Ding, A. Renvall, Stream Ciphers and Number Theory, North-Holland Mathematical Library 55, Elsevier/North-Holland, Page 10 July 2004

12 Ding-Helleseth-Lam s 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). The characteristic sequences have optimal autocorrelation { 3, 1}. Their linear complexity is known. C. Ding, T. Helleseth, and K. Y. Lam, Several classes of sequences with three-level autocorrelation, IEEE Trans. Inform. Theory 45 (1999) Page 11 July 2004

13 A Construction with Generalized Cyclotomy Let g be a fixed common primitive root of both primes p and q. Define d = gcd(p 1, q 1), and let de = (p 1)(q 1). Then there exists an integer x such that Z pq = {g s x i : s = 0, 1,..., e 1; i = 0, 1,..., d 1}. Whiteman s generalized cyclotomy: the cyclotomic class D i is D i = {g s x i : s = 0, 1,..., e 1}i = 0, 1,..., d 1. The generalized cyclotomic numbers are defined by (i, j) d = (D i + 1) D j. It was used by Whiteman to find the two-prime difference sets. Page 12 July 2004

14 A Construction with Generalized Cyclotomy Suppose that gcd(p 1, q 1) = 2. Let D 0 and D 1 be the generalized cyclotomic classes of order 2. Define C = D 1 {p, 2p,, (q 1)p}. If q p = 4 and (p 1)(q 1)/4 is odd, then C is a (p(p + 4), (p + 3)(p + 1)/2, (p + 3)(p + 1)/4, (p 1)(p + 5)/4) almost difference set of Z p(p+4). Its constructuion is related to that of the twin-prime difference sets, but different, because the former is balanced while the later is not. Its characteristic sequence has optimal autocorrelation { 3, 1}. Reference: C. Ding, Autocorrelation values of the generalized cyclotomic sequences of order 2, IEEE Trans. Inform. Theory 44 (1998) Page 13 July 2004

15 Ding-Helleseth-Martinsen s 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 and H. M. Martinsen, New families of binary sequences with optimal three-level autocorrelation, IEEE Trans. Inform. Theory 47 (2001) Page 14 July 2004

16 Ding-Helleseth-Martinsen s 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, n 2, 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 15 July 2004

17 Lempel-Cohn-Eastman s Construction Let q be old. Define C q = log α (D (2,q) 1 1). Then the set C q is a ( q 1, q 1 2, q 3 4, 3q 5 ) 4 almost difference set if q 3 (mod 4), and a ( q 1, q 1 2, q 5 4, q 1 ) 4 almost difference set if q 1 (mod 4). Sequence: Balanced, optimal autocorrelation { 2, 2} and { 4, 0} resp., linear complexity known. Remark: Someone said that the construction was given by Sidelnikov earlier. 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 16 July 2004

18 A Generic Construction with Difference Sets Building block: C an ( l, l 1 2, ) ( l 3 4 or l, l+1 2, ) l+1 4 difference set of Zl, where l 3 (mod 4); i.e., sequence with ideal autocorrelation. Construction: Define a subset of Z 4l by U = [(l + 1)C mod 4l] [(l + 1)(C δ) + 3l mod 4l] [(l + 1)C + 2l mod 4l] [(l + 1)(C δ) + 3l mod 4l] (1) where C and (C δ) denote the complement of C and C δ in Z l respectively. Conclusion: U is a (4l, 2l 1, l 2, l 1) or (4l, 2l + 1, l, l 1) ADS of Z 4l. Sequence: almost balanced, optimal autocorrelation { 4, 0}. 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 17 July 2004

19 Another Generic Construction with Difference Sets Let D 1 be an ordinary ( l, l 1 2, ) ( l 3 4 (respectively, l, l+1 2, ) l+1 4 ) difference set in Z l, let D 2 be a trivial difference set in Z 4 with parameters (4, 1, 0). Then D := (D 2 D 1) (D 2 D 1 ) is (4l, 2l 1, l 2, l 1) (respectively, (4l, 2l + 1, l, l 1)) almost difference set of Z 4 Z l. Sequence: almost balanced, optimal autocorrelation { 4, 0}. 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 18 July 2004

20 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. Some constructions are based on perfect or almost perfect nonlinear functions. Some constructions are based on interleaving. Constructing almost difference sets seems more difficult than constructing difference sets! Page 19 July 2004

21 Open Problems 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 : Open Problems: {0, 1, 2, 3, 4, 5, 6, 8, 13, 14, 18, 20, 22, 25, 28, 29}. Construct new ADSs (especially for the case A mod 4 = 2). Does a ( v, v 1 2, λ, t) almost difference set exist for all odd v? Page 20 July 2004

22 Further References J.S. No, H. Chung, M.S. Yun, Binary pseudorandom sequences of period 2 m 1 with ideal autocorrelation generated by the polynomial z d + (z + 1) d, IEEE Trans. Inform. Theory 44 (1998) J.S. No, S.W. Golomb, G. Gong, H.K. Lee, P. Gaal, Binary pseudorandom sequences of period 2 n 1 with ideal autocorrelation, IEEE Trans. Inform. Theory 44 (1998) A. Maschietti, Difference sets and hyperovals, Des. Codes Cryptography 14 (1998) J.F. Dillon, Multiplicative Difference Sets via Additive Characters, Des. Codes Cryptography 17(1-3) (1999) Page 21 July 2004