Exhaustive Search for the Binary Sequences of Length 2047 and 4095 with Ideal Autocorrelation

Transcription

1 Exhaustve Search for the Bnary Sequences of Length 047 and 4095 wth Ideal Autocorrelaton Seok-Yong Jn and Hong-Yeop Song. Yonse Unversty

2 Contents Introducton Background theory Ideal autocorrelaton Equvalence Exhaustve search Search methodology N=047 case N=4095 case Concluson: results and analyss

3 Introducton Sets of sequences n communcaton and cryptography Random characterstc Low correlaton: Dstngushable from shfted verson of tself and others Balanced, run, span property, etc. Ease of mplementaton Lnear Feedback Shft Regster (LFSR) sequence Securty Large Complexty, Long Perod Canddates: PN-sequences Bnary sequences wth deal autocorrelaton

4 Ideal Autocorrelaton Two-level autocorrelaton θ ( τ ) s T = = 0 ( ) s( ) + s( + τ ) T, τ 0 (mod T ) = r, else (τ ) θ S γ Out-of-phase autocorrelaton as low as possble n absolute value r = 0 : T=4u, for u o such sequences (crculant Hadamard conjecture ) r = - : called have an deal autocorrelaton property

5 Hadamard Sequence Defnton Balanced perodc bnary sequence wth deal autocorrelaton (two-level autocorrelaton wth all out-of-phase correlaton value -) Propertes Length must be 4t- for some postve nteger t. Balanced: # of s) # of 0 s) = Exstence (Not Completely Known). N=4t- s a prme. N=p(p+) s a product of twn prmes 3. N= n - for n=, 3, 4,... Conjecture: These are all (verfed up to 0,000 except 3 cases) Specal nterest: case 3 How many (truly dstnct) knd?, What constructon?, etc.

6 Equvalence of Hadamard Sequences S, =0,,N- : perodc bnary sequence of length N Cyclc Shfts: U = S +d s a (cyclc) d-shfts of S Decmaton: R = S t wth gcd(t, N)= s a t-decmaton of S If for two same length sequence A and B, there exst t and d such that A =B t+d for all, then A and B are equvalent. D=d, d,, d k v,k, If td = td, td,, td k = d+d =d+d, d+d,, d+d k for some t and d wth (t,v)=, then t s called a multpler of D.

7 Equvalence: Example Equvalence classes of Hadamard Sequences of Length 7 A B C D E A C B D E

8 Exhaustve Search () # of (bnary sequences of length - =047) = 047 # of (balanced bnary sequences of length 047) 045 Reducton by cyclc shfts: 045 / = 034 How to reduce canddates by decmaton? Queston: Is a decmated verson of Hadamard sequence also Hadamard sequence? Answer: Yes, due to the followng Theorem Any ( n -, n- -, n- -)-CHDS D has always as a multpler. Moreover, there s unque d such that for D =D+d, D =D. Then the characterstc sequence of D have the constant-on-the-coset property.

9 Exhaustve Search () Only to check sequences wth constant-on-the-coset property: S = S, for all =0,, N- #(cyclotomc cosets modular 047) = canddates: stll mpossble!! Then for what else? Theorem (Baumert) If a (v, k, )-CDS exsts, then for every dvsor w of v, there exsts ntegers b (=0,,w-) satsfyng the followng three equatons. ) ) w = 0 w = 0 b = k b = ( k λ) + ( λv / w) ) j =,...,w, w = 0 b b j = λv / w

10 Procedure for Exhaustve Search. Docompose cyclotomc cosets mod n-.. Establsh & solve dophantne equatons. 3. Exclude redundant solutons usng decmaton property. 4. Construct sequences from the solutons found. Use decmaton property once agan. 5. Check deal autocorrelaton.

11 Coset Decomposton

12 Dophantne Equatons and Decmaton 047 = w=3: 4 solutons nly solutons to be examned w=89: 88 solutons only solutons (not redundant) For one soluton set (a 0, a,, a 6 ), h 6 = h= 0 N a h h 0 5, where N h = for 6 h's Such solutons are more than Total (at least) 0 30 canddates took more than fve hundred years n pentum.4ghz CPU.

13 Known Hadamard Sequences (length 047) s( t) = Tr( α e t ) Sequence m-sequence 3-term sequence 5-term sequence Segre hyperoval sequence Glynn type I hyperoval Glynn type II hyperoval Trace Representaton, 33, 49 3, 5, 7, 73, 4 5, 5, 05, 309, 469, 83, 39, 9, 3, 9, 73, 33, 9, 7, 49, 5, 9, 3, 9, 37, 43, 67, 69, 37, 63,, 93, 5, 3, 7, 9, 37, 49, 6, 69, 8, 93, 0, 3, 5, 39, 47, 5, 7, 73, 83

14 Lnear Complexty Lnear complexty of Hadamard sequences of length 047 from monomal hyperoval n projectve plane Sequence Type m HS HG HG Lnear Span

15 (4095, 047, 03)-case () Dvsor structure of

16 (4097, 045, 03)-case () Decomposton of cyclotomc cosets mod 4095

17 (4097, 045, 03)-case (3) Cyclotomc cosets mod 4095: total 35 cosets # of cosets coset sze Classes of cyclotomc cosets mod 4095: 36 classes # of such classes # of cosets n class For only one soluton set (a 0, a,, a 36 ), approxmately number of choces of choosng canddates, whch s greater than total number of possbltes for - case a a a a a a a a a

18 Summary Exhaustve search for (047, 03, 5)-CHDS Snce 047 s a product of only two dvsors (3 and 89), and s a prme, Prevous search methodology does NOT work effcently. Partal Search No more nequvalent Hadamard sequence of length 047 was found. Exhaustve Search for (4095, 047, 03)-CHDS Actually, the fnal case of the current exhaustve method may work. Stll tryng