Crosscorrelation of m-sequences, Exponential sums and Dickson

Transcription

1 Cosscoelatio o m-equeces, Epoetial sums ad Dicso polyomials To Helleseth Uiesity o Bege NORWAY Joit wo with Aia Johase ad Aleade Kholosha

2 Itoductio Outlie m-sequeces Coelatio o sequeces Popeties o m-sequeces Two-leel ideal autocoelatio uey Thee-alued coss coelatio Fou-alued coss coelatio New ie-alued alued coss coelatios Dicso polyomials Ope poblems

3 m-sequece Eample Liea ecusio s t : s t+4 = s t++ s t Pimitie polyomial = 4 ++ Popeties o m-sequeces Peiod =, pimitie polyomial o degee Good pseudoadom popeties Balaced Ru popety Two-leel autocoelatio s t -s t+ = s t+ ad s t = s t+ Decimatio by d, d, -= gies a m-sequece ace epesetatio s = t t, whee - =0 ad : GF GF is = i i=0

4 Coelatio o biay sequeces Let a t ad b t be biay sequeces o peiod The cosscoelatio betwee a t ad b t at shit is - a,b = - t=0 a t+ - b t The autocoelatio o a t at shit is - a,a = - t=0 a t+ -a t

5 Two-leel autocoelatio o m-sequeces Let s t be a m-sequece o peiod = - The the autocoelatio o the m-sequece is s,s = - i =0 mod - = - i 0 mod - Poo: Let 0 mod -. The s,s = t - s t+ -s t s t+ = t - =- sice m-sequece is balaced

6 Coss coelatio o m-sequeces Let s t be a m-sequece Let s dt be decimated m-sequece i.e., d, -= The coss coelatio betwee the two m-sequeces is deied by C d = t - s t+ -sdt I the case d i mod - the s dt =s t+ ad C d has oly two-alues autocoelatio I all othe cases, at least thee alues occu t

7 ome popeties p o C d d C d ad C d has the same distibutio whe d d mod - o whe d d i mod - = C d + C d + = C d = a whee a is the umbe o solutios o = 0 d + d d + = 0 with i GF * = GF {0}

8 Biay -alued coss coelatio C d has eactly dieet alues i the cases: - Gold : d= + whee /, is odd - Kasami : d= - + whee /, is odd - Welch s cojectue: Cateau, Chapi, Dobbeti 000 d= m + whee =m+ is odd - Niho s cojectue: Dobbeti & Hollma, Xiag d= -/ + -/4 - whe mod 4 = -/ + -/4 - whe mod 4 - Cusic ad Dobbeti d= / + +/ + whe mod 4 d= +/ + whe mod 4

9 Biay 4-alued coss coelatio Theoem Dobbeti, Fele, Helleseth, Rosedal 006 Let < be gie such that - - ad + - eist mod d +. Lt Let < < ad d = -s+ with s = - - d = -s + with s = + - The C d τ taes o 4 alues ad distibutio is ow. Cojectue: All 4-alued decimatios o the om d= -s+ is coeed by the Theoem

10 Two Cojectues Cojectue Helleseth I the peiod is -ad = i the C d has at least 4 alues Cojectue Helleseth Fo ay d, -=, the C d = - o some Rema. The - cojectue is equialet with C d +=0 Calculatios show that the cojectue is equialet to poig: The system o equatios is a pimitie elemet q- q- = 0 d d 0 + d + d + + d q- =0 has eactly q q- solutios i GF, whee q=

11 Decimatios d= +/ l + d = + / + = - + Kasami-Welch -Valued Cojectue Niho 97 d= t +/ +, t > odd, gies at most 5 alued coelatio Couteeample o t=7 Lagei, Leade, McGuie 007 ome cases ow with 5-alued coelatio - Kasami d= 5 + / +,=, odd - Bace d= 5 + / +,=, odd Coelatio alues -, -± +/, -± +/ Eact coelatio distibutio is uow Theoem Johase, Helleseth 008 d= + / + =, odd i.e., d=5/ gies 5-alued coss coelatio ad distibutio is completely detemied

12 etch o poo d= +/ +, =, odd. The coss coelatio is 5-alued with coelatio alues -, -± +/, -± +/ odd. The distibutio depeds o the umbe o solutios o + y + = y 5 + = 0. The distibutio o the coelatio alues depeds o the umbe o solutios A = N,0,0 o + y + u + = = a + y + u + = 0 = b u =0=c = 4. Chapi, Helleseth, Zioie 005 showed that Na,b,c ca be epessed as a uctio o thee epoetial sums 5. N,0,0 ca be detemied eplicitly

13 . The coss coelatio is 5-alued The coss coelatio whe d= l +/ + ca be epessed by C d = 0 - a + d = 0 - a + + l + quaig the coelatio C d + = K a o 0 whee K a is the eos i GF o L a = l +a l +l +a l- l- + Fo l= L a = 4 +a +a + Fo odd, the possible umbe o solutios is, e, e, e, 4e o e =, Hece, the coss coelatio is 5-alued with coelatio alues -, -± +e/, -± +e/ odd ad e=,= l

14 . Detemiatio o thid powes Theoem Let d= l +/ + the C d + = b whee, y GF * = GF {0} + + y + + = 0 l + + y l + + = 0 The b = +l, + l-, - +l,l-,. Poo Elimiatig y gies + l + l - + = 0 Coollay Fo l= the b =, -

15 /4. olutios o equatio system Theoem Chapi, Helleseth, Zioe 005 Let Na,b,c be the umbe o solutios,y,,u i GF o + y + u + = = a +y +u + = 0 = b u = 0 = c I is odd the Na,b,c ca be epessed by thee epoetial sums, especially A =N00= N,0,0 + + G - K - C whee C = - + Gold sum K = Kloostema sum G = Iese Gold sum ad tace is om GF to GF

16 5. O the umbe o solutios A =N,0,0, A =N,0,0 = + + G -K -C Fidig C C = εgf - + = - η - η C C =, =0 ad η, η ae eos o ++ ad C = / + whee / is the Jacobi symbol Fidig K K = = - η - η K =, K = ad η, η ae eos o +++ Fidig G + - G = 0 - = -η -η - η -η 4 G =, G = -, G =7 ad G 4 =7 η, η, η, η 4 ae eos o

17 Coelatio distibutio o d=5/ Let A =N,0,0 = + + G -K -C Theoem Distibutio o C d + I the case,= ± +/ occus A /96 times - +/ occus + - +/ -A /4 times + +/ occus + + +/ -A /4 times 0 occus m- - + A /6 times I the case,= - +/ occus - +5/ + A /96 times + +/ occus +5/ + A /96 times ± +/ occus + -A /4 times 0 occus A /6 times

18 Geeal case d= +/ +, odd All peious steps wo ecept we eed to id A Coside the umbe o solutios A o + y + + u = a = + + y u + =0 + + y u + = 0 The complete 5-alued coelatio distibutio ca be detemied om A How to id A o geeal??

19 A = N,0,0, ad epoetial sums Kloostema sum: K = Σ Gold sum: C = Σ Iese cubic: G +- = Σ 0 - Ge. sum: K = Σ 0 - whee Theoem Let be odd ad,= the A ++G C K = + Cojectue Fo ay,= the K = K ad G = G

20 Itoductio to Dicso polyomials Dicso polyomial D, u D +, = + i0 i Let u= ad D =D, D = - D + = i u i i D = D = D = + D = 4 = 5 = 6 4 D D 6 + D 7 = D 8 = 8 D 9 = D is a pemutatio polyomial i GF i, -= I, -= ad D 0 the / = /D

21 Dicso ad Kloostema Dicso ad Kloostema pemutatio polyomial is a the Let D. ad pemutatio polyomial is a the, Let D D K LEMMA D 0 : PROOF 0 K 0 D D D 0 D

22 The case l= I Coside the umbe o solutios A o Let + y + + u = a= + + y u + = y u + = 0 + y = ad y = w = ie i.e., =0 + u = +a ad u = +a s i.e., s=0 The system becomes o D +, + D + +a,+a s = 0 D +a +, + D ++a,+a s = = =

23 The case l= II Let = + ad s = s + s The A is the umbe o solutios o = + + s + s = s +s + ad A /4 is the umbe o solutios o + = δ y + + = δ + y + y + whee m = m y = ad δ = +/ Elimiatig leads to the equatio y u y C 0 o some u ad whee C u K

24 A Key Theoem A Key Theoem Theoem Theoem Let A be the umbe o solutios i GF m o + y + + u =a= + y + + u = a = + + y u + = y u + = 0 The A = 8N whee N is the umbe o solutios o T T T 0 m = m = m =0 whee 0, is i GF m ad,,

25 Desciptio o i + j s Desciptio o i j s

26 A ad epoetial ums p solutios o be the umbe o Let N The 0 {0,} 8 N A {0,} {0,}... {0,} ' sice uthe calculatios gie C K G ' -,, - C g K G

27 G = G o odd LEMMA. G Let be odd. GF whee ' s ae eos o L LEMMA. i G 4 m Let be odd. GF whee ' s ae eos o Hece, i L G G 6 m 5 m 4 L i m i - - G o odd - 5-6

28 K =K o = K =K o = Let 4 4 whee g ad g Note 5 t g t D t LEMMA whee 5 g K K K ' : Poo ' the Let g g , 0 K K 0

29 Coectio to Dillo-Dobbeti D Let Δ = Δ = Δ m- = = Δ is a -to- map ImΔ leads to Dillo - Dobbeti dieece sets Coectio: Dicso Kloostema Cojectue. Let be ee ad,=. The K GF ** D

30 Coclusios Oeiewo coss coelatio o m-sequeces Complete coelatio distibutio o ew amilies with 5-alued coelatio i A ca be calculated d= +/ +, odd Complete coelatio distibutio ib ti o = ad = Cojectued the coelatio distibutio to be the same o ay wheee,= Two ew cojectues o epoetial sums Coectios to Dicso polyomials ad Dillo- Dobbeti dieece sets

31 Appedi Computig ad = - C + Computig =-G, ad + Coectio to Dillo-Dobbeti dieece sets howig that =

32 Computig ad Co put g a d The., be odd ad Let LEMMA., GF C the Note that sice PROOF : Hece, T T 0,} { } {0, C

33 Computig, ad. The, be odd ad LEMMA Let G. the Let PROOF : {0,} G. Note. the Let PROOF : ad, is oe - to - oe sice The t t t Hece, t t t t t t t t, , m 0 G

34 Coectio to Dillo-Dobbeti D Let Δ = The Δ = Δ m- = = Δ is a -to- map ImΔ leads to Dillo - Dobbeti dieece sets Let g i0 i The g is a -to- map Img=ImΔ ad o ee ad 7 g g

35 Lemma howig that = Let be ee, odd ad,= the = = - K + Poo. ice is ee ad g -to- the Im =Im 7 7 0, 0, 0, 0 g