An introduction to binary sequences, combinatorial designs and how they are related to communication theory and computer security is given. An exhaustive search algorithm for normal sequences is presented. This is the first time that the lengths n = 24 and n = 25 have been searched through completely. No sequences of length 24 are found. It turns out that all the normal sequences of length 25 can be derived from Turyn sequences. This construction is subject to a new theorem that is given here

Gysin, Marc

Seberry, Jennifer

English

Research Online

University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 1995 New normal sequences of length 25 Marc Gysin Jennifer Seberry University of Wollongong, jennie@uow.edu.au Follow this and additional works at: https://ro.uow.edu.au/infopapers  Part of the Physical Sciences and Mathematics Commons Recommended Citation Gysin, Marc and Seberry, Jennifer: New normal sequences of length 25 1995. https://ro.uow.edu.au/infopapers/1120 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: research-pubs@uow.edu.au New normal sequences of length 25 Abstract An introduction to binary sequences, combinatorial designs and how they are related to communication theory and computer security is given. An exhaustive search algorithm for normal sequences is presented. This is the first time that the lengths n = 24 and n = 25 have been searched through completely. No sequences of length 24 are found. It turns out that all the normal sequences of length 25 can be derived from Turyn sequences. This construction is subject to a new theorem that is given here. Disciplines Physical Sciences and Mathematics Publication Details Marc Gysin and Jennifer Seberry, New normal sequences of length 25, ACSC'95 - Australian Computer Science Conference, Glenelg, January, 1995. This conference paper is available at Research Online: https://ro.uow.edu.au/infopapers/1120 [245] Rep Marc Gysin and Jennifer Seberry, New normal sequences of length 25, ACSC'95 - Australian Computer Science Conference, Glenelg, January, 1995. New Normal Sequences of Length 25 Marc Gysin and Jennifer Seberry * The University of Wollongong, Wollongong, NSW 2500 Australia e-mail: marc@cs.uow.edu.au.j.seberry@cs.uow.edu.au 10 July 1994 Abstract An introduction to binary sequences, combi-natorial designs and how they are related to communication theory and computer security is given. An exhaustive search algorithm for normal sequences is presented. This is the first time that the lengths n = 24 and n = 25 have been searched through completely. No sequences of length 24 are found. It turns out that all the normal sequences of length 25 can be derived from Turyn sequences. This con-struction is subject to a new theorem that is given here. Key words: Hadamard matrices, normal sequences, near-Yang sequences, Turyn se-quences, exhaustive search algorithm. 1 Introduction We start with a definition: Definition 1 (Nonperiodic Autocorrelation Function) Let X = {{Xll"",Xln},{X21"",X2n}, . . . , ... , { Xml , ... , xmn}} be a family of m se-quences of elements 1, 0 and -1 and length n. The nonperiodic autocorrelation function of the family of sequences X, denoted by N x, is a function defined by Nx(s) = 'Supported by the ARC grants A49131885 and A9130102, The University of Wollongong and the Cen-tre for Computer Security Research 1 n-, L(XliXl,i+, + X2iX2.i+s + ... + XmiXm.i+.), i=l where s can range from 0 to n - 1. Initially people were interested in single se-quence (that is m would be equal to 1 in the above definition) such as Barker sequences. A crucial criterion was the autocorrelation func-tion being zero or one for each possible shift (s 2: 1) and being a comparatively large num-ber when not shifted (s = 0). We observe that for s = 0 the autocorrelation function always returns the total number of nonzero elements in the sequence(s). Barker sequences were use-ful to measure long distances such as from the earth to the moon or to an aircraft. A single sequence was sent out to the remote object and overlapped with the reflected sequence. The autocorrelation function was then calculated. When the returned value increased from one or zero to a large number it was known that the sequences were no longer shifted against each other. Therefore, assuming that the ve-locity of the signal sent out was known, the distance to the object could be easily worked out . Later the search turned to sets of sequences (m > 1) such as Golay, base and normal sequences. Golay sequences are two binary sequences with zero autocorrelation function (from now on we shall denote the nonperiodic zero autocorrelation function being zero for s = 1, ... , n - 1 by zero autocorrelation func-tion). Golay sequences could be used in spec-trometry to cancel out all but one frequency of for s = 1, ... , n - 1. Furthermore, their struc-light and in sonar using a distance, say a sub- ture has to satisfy certain symmetry condi-marine, between transmitters. Base sequences tions. are four ternary sequences Xl, X 2, X 3, X 4 with X 1 ,X2 having entries 1, -1 and X 3 , X 4 both Ifn is odd their structure is: starting with 1, -1 and both ending with p ze-ros (p = 0, ... , n - 1). Turyn sequences are a special set of base sequences with certain sym- A {I, al, a2,.··, am, -am, ... , -a2, -al, -I} metries imposed on the sequences. A formal B = {1,bl,b2, ... ,bm,-bm, ... ,-b2,-bl,1} definition is given in the text below. Near- C {CO,CI, ... ,Cm-I,Cm,Cm_I, ... ,CI,CO} Yang sequences are a more generalized form D = {d d d d d d d} 0, I,···, m-I, m, m-I,···, I, ° of normal sequences. We give the formal definitions of normal se- where n = 2m + 1. quences and of Turyn sequences. If n is even their structure is: Definition 2 (Normal Sequences) A triple (Fj G, H) of sequences is said to be a set of normal sequences of length n, denoted by N Sen), if the following conditions are sat-isfied: (i) F = Uk) is a sequence of length n with entries 1, -1. (ii) G = (gk) and H = (hk) are sequences of length n with entries 0,1, -1, such that G + H = (gk + hk) is a (1, -1) sequence of length n. (iii) 9j + 9n-j+l o (mod 2) j = 1, ... ,[%] ° (mod 2) (iv) NF{s) + No{s) + NH(s) = 0, s=l, ... ,n-1. It can be shown that the quasi-symmetry of the sequences, that is, condition (iii) is implied by all the other conditions. A proof is given in [Gysin93] or [lKKSYY91]. The second definition arose from the work of C.H. Yang [lYang82], [2Yang83], [3Yang83] and [4Yang89]. Definition 3 (Turyn Sequences) A quadruple A, B, C and D of sequences with entries {1, -1} oflength n + 1, n + 1, n, n are called Turyn sequences, denoted by TS{2n + 1), if they have zero autocorrelation function. That is, if NA(s)+NB(s)+Nc(s)+ND(s) = 0 2 A = B = C = D = {I, aI, a2, ... ,am, am+l, am, ... , a2, at, I} {I, bl , b2 , ••• , bm , bm +l , bm , ... , b2 , bl , -I} {CQ,Cl, ... ,Cm,-Cm, ... ,-Cl,-CO} {do, d l , •.• , dm , -dm , ... , -d l , -do} where n = 2m. Many sequences can be constructed from each other. Golay sequences, for example, are al-ways normal sequences by definition. But there is also a special method concerning the symmetric and skew-symmetric part of the Golay sequences which also leads to normal sequences. We can get base sequences directly from normal sequences. Turyn sequences lead to special sets of normal sequences as shown below. On the other hand, the sequences are also strongly related to other combinatorial designs such as orthogonal designs, Hadamard matri-ces, difference sets and bent functions. Some combinatorial designs can be gained from the sequences in a straightforward manner, oth-ers are subject to quite sophisticated theo-rems. Hadamard matrices are studied widely and they can also be used in communica-tion theory. Bent functions play an impor-tant role in the design of Substitution Boxes which are of great importance to many cryp-tographic algorithms such as DES (Data En-cryption Standard), LOKI and GOST. These functions are also useful in hashing algorithms such as HAVAL which produce digital finger-prints of messages. 2 New Normal Sequences Derived from Turyn Se-quences The following theorem is well known and a proof is given in [Gysin93]. Theorem 1 Let A, B be sequences of length n with entries 1,0, -1, where A is skew (ak = -an-k+d and B is symmetric (bk = bn-k+d and a[~l = 0 for odd n. Let A + B and A - B be 1,0,-1 sequences of length n and let C = A + B. Then Nc(s) = NA(s) + NB(S), S = 1, ... ,n-1. [IKKSYY91] construct normal sequences from Turyn sequences: Theorem 2 [lKKSYY91] {o, co, bl , ... , Cm, bm+l , -Cm, ... , bl , -co, o} Hz = {I, 0, ... ,0, -I}. We define G- Hz = {O,D,bl, ... ,O,bm+I,D, ... ,bl,O,O}. We note that G2 = G+H. N ow by using Theorem 1: NG2 (s) = N(;(s) + NH(S), S = 1, ... , n - 1 (1) and NG(s) = N(;(s) + NH2(S), S = 1, ... , n - 1. (2) Let A, B, C, D be Turyn sequences TS(2n + 1). Let We write (2) as F = Ale= {al,cI,az,cZ, ... ,an,cn,an+l} G BIOn = {b l , 0, bz , D, ... , bn , 0, bn+l } H = On+dD = {D,dl,O,dz , ... ,0,dn,0} where On and On+l are sequences of nand n + 1 zeros. Then F, G, H are normal sequences of length 2n + 1. NH2(S) = NG(s) - N(;(s), s = 1, ... , n - 1 (3) and add (1) and (3) to obtain NG2 (s) + NH2(S) = NG(s) + NH(S). For odd n, the proof works exactly the same except that sequences which were skew are now symmetric and vice versa. 0 to Therefore, if there is any triple F2 , G2 and H 2 derived from Turyn sequences according to Theorem 3 is an NS(2n+ 1), there is always a We found the following new method construct normal sequences from Turyn se-quences: Theorem 3 Let F, G, H be normal sequences NS(2n + 1) derived from Turyn sequences according to Theorem 2. Then the following sequences F2 = F Gz {0,g2 + hZ,g3 + h3, ... ,g2n + h2n ,D} Hz = {gl,D, ... ,D,gZn+d are normal sequences of length 2n + 1. Proof. We prove that G and H have the same autocorrelation function as G2 and H 2 . We have to distinguish two cases, one for even n and one for odd n. For even n, the sequences involved have a structure as follows: G {l,O,bl, ... ,O,bm+I,D, ... ,bl,O,-l} H = {O, Co, D, ... , Cm, D, -Cm, ... , D, -co, DJ 3 triple satisfying the structure of F, G and H as in Theorem 2 and vice versa. 3 The Algorithm We briefly outline the problem and the al-gorithm. A full description of the algorithm together with the program-code is given in [Gysin93]. The Problem Given a set of sequences, it is easy to test if they are normal sequences or not. The prob-lem is to search for normal sequences. We note that the search-space grows exponentially in some manner according to the definition of the search-space itself. While the search for smaller lengths n is only a matter of seconds, finding longer sequences can take months of CPU-Time. This is because one always faces the problem of the combinatorial explosion. A first simple algorithm may be implemented in the following manner. Treat each sequence as a binary or ternary number. Given a length n run through all the 2n x 3n x 3n possible com-binations of the numbers. Decode each com-bination of numbers into sequences and check if normal sequences are obtained. Needless to say that this first algorithm runs out of CPU-Time very soon. For lengths n 2: 12 CPU-Time invested exceeded one day. A better Definition of the Search-Space By looking at Condition (ii) and (iii) from Def-inition 2 we first observe that we can redefine the search-space. Let us examine triples of se-quences which "look" like normal sequences. That is, they fulfill Condition (i) to (iii) from Definition 2, but they may not have zero au-tocorrelation function. This drastically cuts down the search-space. Moving through the Search-Space By looking at the autocorrelation function we see that for s = n - 1 the equation is fdn + glgn + h1hn = 0, and for s = n - 2 Note that these equations are only influenced by the outermost pairs of elements of the se-quences. The innermost pairs make no contri-bution to these equations at all. The search is now performed in the following way. 1. First, try to find all possible combinations iI, In, gl, gn, hI and hn which fulfill the "last" equation, that is, for s = n - 1. 2. Move to the next equation, s = n - 2, and try for each previous successful combina-tion iI, In, gl, gn, hI, hn to find all the new combinations 12, In-I, g2, gn-I, h2, hn - 1 which satisfy this new equation. 4 3. Continue until all the elements of each se-quence are determined. That is, the par-tial sequences F part , G part and Hpart be-come complete sequences F, G and H. 4. Test the remaining equations from the au-tocorrelation function for each combina-tion in order to find all possible normal sequences F, G and H. We observe: • The search is performed from the outer-most to the innermost triple of pairs of elements. • If we add a new triple of pairs, we have 32 possibilities to do so for normal sequences (in accordance with the new search-space described above). • The combinations of the first triple of pairs, that is, iI, In, gI, gn, hI and hn which satisfy the last equation do not de-pend on n. For a large enough n, the same could be said for the next equation and so on. (Consider for example the last two equations from the autocorrelation func-tion for n = 20 and n = 21: they are ex-actly the same, and therefore the triples of pairs which fulfill these equations are also the same.) The last statement is very important. It tells us that we can store triples of pairs and use them again for larger lengths n, and therefore the time-consuming testing of the autocorre-lation function only has to be performed once. Storing, Compressing and Reusing Triples Each triple of pairs of elements is assigned a number (from 0 to 31) and consecutive triples of pairs of elements that passed the corre-sponding equations from the autocorrelation functions are stored in a file. Further com-pression can be done by the observation that due to the nature of the tree-search algorithm, many successful configurations start with the same triple of pairs of elements. Therefore, we only store the triples of elements which have changed since the last configuration. Example: Uncompressed triples Compressed triples 17832 17834 17 8 3 10 17842 178439 1923 1 17832 "end" 4 "end" 10 "end" 42 "end" 29 "end" 192 3 1 "end" These triples the can be reused to start the search for longer lengths n as the equations from the autocorrelation function are the same assuming that the elements do not overlap each other. The Search-Tree and Branches of the Tree Cutting The triples of pairs of elements represent a search-tree with branching factor 32. Each node of the search tree corresponds to one triple of pairs of elements of the sequences. At each node one equation from the autocor-relation function is considered. Downwards steps are performed when the corresponding equation from the autocorrelation function is fulfilled for this particular configuration. If a certain triple of pairs fails its correspond-ing equation from the autocorrelation func-tion, the next node on the same level of the search-tree is considered. If there are no more nodes on the same level, an upwards step is performed. If the algorithm reaches the bot-tom of the tree (that is a leaf of the tree is examined), all the remaining equations from 4 Computational Results We were able to carry out the search up to length n = 25. The lengths 24 and 25 have been searched through exhaustively for the first time. For length n = 24 no sequences were found. The algorithm found 24 sets of inequivalent normal sequences of length 25 in approxi-mately 2500 hours CPU-time on a Sun Sparc-4 computer. We considered two triples of se-quences to be equivalent if one triple of se-quences can be changed into the other triple of sequences by reversing and/or negating one or more sequences of the triple. We found that all of the normal sequences of length 25 can be derived from 12 inequivalent sets of Turyn se-quences using Theorem 2 and 3. An example is given in the following table: +-+-++-++++--++-+---+++++ G1 = 0+0+0+0+0-0+0-0+0-0-0-0-0 H1 = +0+0+0-0-0+0-0+0-0-0+0+0-F2 = +-+-++-++++--++-+---+++++ 0+++++-+--++--++----+-+-0 H2 = +00000000000000000000000 - . The Turyn sequences constructing these nor-mal sequences would be: A +++-++-++-+++ B +++--+-+--++-C = --+++-+---++ D = ++++-+-+----. the autocorrelation function are tested. Ifpos-itive, the configuration under consideration is a triple of normal sequences. 5 Summary and Conclu-Testing an equation from the autocorrelation function at each step means cutting branches of the tree when appropriate. In fact, there are further cutting mechanisms. These are de-scribed in [Gysin93]. 5 . Slons We presented a tree-search algorithm for searching for normal sequences. The algo-rithm is quite simple and straightforward to implement. Previously calculated results can be reused for the search for longer lengths n. We were able to carry out the search up to length n = 25. The last two lengths were new and results were previously unknown. Length 25 leaded to a new theorem. The tree-search algorithm is not only limited to normal se-quences: it can be adapted for many groups of sequences. Although this algorithm is very fast we still have to deal with the problem of the combinatorial explosion. Acknowledgement We wish to thank the Center for Communi-cation and Information Science, University of Nebraska, for letting us use thousands of CPU-hours on their machine "ramoth". References [Edmondson91] Genet M. Edmondson, More non-existent Turyn sequences, M.Sc. Thesis, The University of New South Wales, Australian Defence Force Academy, Canberra, 1991. [EdmSebAnd92] Genet M. Edmondson, Jen-nifer Seberry, Malcolm R. Anderson, On the existence of Turyn sequences of lengths less than 43, Mathematics of Computation, 62,205, 351-362, 1994. [Gysin93] M. Gysin, Algorithms for Searching for Normal and Near-Yang Sequences, M.Sc. Thesis, The University of Wollon-gong, 1993. [GysSeb93] M. Gysin and J. Seberry, New Re-sults with Near-Yang Sequences,. Utilitas M athematica, to appear. [IKKSYY91] C. Koukouvinos, S. Kounias, J. Seberry, C.H. Yang and J. Yang, On se-quences with zero autocorrelation, Codes and Cryptography, to appear. [2KKSYY91] C. Koukouvinos, S. Kounias, J. Seberry, C.H. Yang and J. Yang, Multi-plication of sequences with zero autocor-relation, Australasian Journal of Combi-natorics, to appear. [KouKouSeb90] C. Koukouvinos, S. Kounias and J. Seberry, Further results on base sequences, disjoint complementary se-quences, OD(4t;t,t,i,i) and the excess 6 of Hadamard matrices, A rs Combinato-ria, 30, 241-256, 1990. [KouSeb91] C. Koukouvinos and J. Seberry, Addendum to Further results on base sequences, disjoint complementary se-quences, OD( 4i; i, i, i, t) and the excess of Hadamard matrices, Congressus Numer-antium, 82, 97-103, 1991. [IYang82] C.H. Yang, Hadamard matrices and b-codes of length 3n, Proc. Amer. Math. Soc., 85, 480-482, 1982. [2Yang83] C.H. Yang, A composition theorem for b-codes, Proc. Amer. Math. Soc., 89, 375-378, 1983. [3Yang83] C.H. Yang, Lagrange identity for polynomials and b-codes of lengths 7t and 13i, Proc. Amer. Maih. Soc., 88, 746-750, 1983. [4Yang89] C.H. Yang, On composition of four-symbol b-codes and Hadamard matrices, Proc. Amer. Math. Soc., 107, 763-776, 1989. 

New normal sequences of length 25

New normal sequences of length 25 An introduction to binary sequences, combinatorial designs and how they are related to communication theory and computer security is given. An exhaustive search algorithm for normal sequences is presented. This is the first time that the lengths n = 24 and n = 25 have been searched through completely. No sequences of length 24 are found. It turns out that all the normal sequences of length 25 can be derived from Turyn sequences. This construction is subject to a new theorem that is given here

Marc Gysin

Jennifer Seberry

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New normal sequences of length 25

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