Classification of doubly resolvable designs and orthogonal resolutions Mutually orthogonal resolutions ROR of 2-(9,3,3) 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 . 2 In such a way we obtain a ROR – resolution with an orthogonal partner. 1 2 … … … 12.m m

Classification of doubly resolvable designs and orthogonal resolutions Svetlana Topalova, Stela Zhelezova Institute of Mathematics and Informatics, BAS, Bulgaria Classification of doubly resolvable designs and orthogonal resolutions is the topic of this talk. It is joint work of Stela Zhelezova and me, SvT

Classification of doubly resolvable designs and orthogonal resolutions Definitions Design resolutions and equidistant constant weight codes Definition and one-to-one correspondence between them Some cryptographic applications Mutually orthogonal resolutions Definition – what makes them usable in cryptography History of the study of resolutions and orthogonal resolutions Classification of doubly resolvable designs Classification of the sets of mutually orthogonal resolutions Correctness of the results, parameter range and bounds for higher parameters The talk starts with definitions concerning combinatorial designs, design resolutions and constant weight codes and the correspondence between them. Next some cryptographic applications of design resolutions are mentioned. Mutually orthogonal resolutions are then defined and their usability in cryptography is discussed. Historical notes about the resolution classification problem, and the construction of mutually orthogonal resolutions are presented. The last three items are a summary of our results on the classification of doubly resolvable designs and sets of mutually orthogonal resolutions, the parameter range we cover, and bounds for some higher parameters. in particu

Classification of doubly resolvable designs and orthogonal resolutions Definitions 2-(v,k,λ) design (BIBD); V – finite set of v points B – finite collection of b blocks : k-element subsets of V D = (V, B ) – 2-(v,k,λ) design if any 2-subset of V is in λ blocks of B Isomorphic designs – exists a one-to-one correspondence between the point and block sets of both designs, which does not change the incidence. Automorphism – isomorphism of the design to itself. Let V={ Pi } vi=1 be a finite set of points, and B ={ B_j } bj=1- a finite collection of k-element subsets of V, called blocks. We say that D=(V, B) is a design with parameters 2-(v,k,) or balanced incomplete block design (BIBD), if any 2-subset of V is contained in exactly  blocks of B. Two designs are isomorphic if there exists a one-to-one correspondence between the point and block sets of the first design and respectively, the point and block sets of the second design, and if this one-to-one correspondence does not change the incidence. An automorphism of the design is an isomorphism into itself.(permutation of the points that transforms the blocks into blocks)

Classification of doubly resolvable designs and orthogonal resolutions Design resolutions and equidistant constant weight codes Resolvability – at least one resolution. Resolution – partition of the blocks into parallel classes - each point is in exactly one block of each parallel class. Isomorphic resolutions - exists an automorphism of the design transforming each parallel class of the first resolution into a parallel class of the second one. One of the most important properties of a design is its resolvability. The design is resolvable if it has at least one resolution. A resolution is a partition of the blocks into subsets called parallel classes such that each point is in exactly one block of each parallel class.

Classification of doubly resolvable designs and orthogonal resolutions Design resolutions and equidistant constant weight codes q-ary code Cq(n,M,d) of length n with M words and minimal distance d is a set of M elements of Znq (words), any two words are at Hamming distance at least d. any two words are at the same distance d -> equidistant code all codewords have the same weight -> constant weight code. One-to-one correspondence between resolutions of 2-(qk; k;) designs and the maximal distance (r; qk; r- )q equidistant codes, r = (qk-1)/(k-1), q > 1 (Semakov N.V., Zinoviev V.A., Equidistant q-ary codes with maximal distance and resolvable balanced incomplete block designs, 1968)

Classification of doubly resolvable designs and orthogonal resolutions Design resolutions and equidistant constant weight codes Equidistant ternary constant weight (4,9,3) code Incidence matrix of 2-(9,3,1) 1 1 2 3 points V = 9 That’s a small example of this correspondence. As each point is in exactly one block of each parallel class, the parallel class is defined by the numbers of the blocks within the class, in which each point is. As this matrix is of larger dimension, it is clear why people who have constructed resolutions, actually construct their corresponding constant weight codes. So do we. The cryptographic applications of resolutions, often use the fact that each resolution class is uniquely defined by any of its blocks, and that for some designs the block itself is uniquely defined by a proper subset of its points. In this example a STS(9) is shown. In this case a block is defined by any 2 of its points. And if you construct a threshold scheme for instance, you can have 4 secrets corresponding to the 4 parallel classes, each of them is uniquely determined by two of the 3 shares corresponding to one of its blocks. parallel class blocks b = 12 words

Classification of doubly resolvable designs and orthogonal resolutions Design resolutions and equidistant constant weight codes Some cryptographic applications: Anonimous (2,k)-threshold schemes from resolvable 2-(v,k,1) designs (D.R. Stinson, Combinatorial Designs: Constructions and Analysis, 2004); Resolvable 2-(v,3,1) designs for synchronous multiple access to channels, (C.J. Colbourn, J.H. Dinitz, D.R. Stinson, Applications of Combinatorial Designs to Communications, Cryptography, and Networking,1999) Unconditionally secure commitment schemes (C.Blundo, B.Masucci, D.R.Stinson, R.Wei, Constructions and bounds for unconditionally secure commitment schemes, 2000) We’ll only mention 3 cryptographic applications here: Anonimous (2,k)-threshold schemes from resolvable 2-(v,k,1) designs , Resolvable 2-(v,3,1) designs for synchronous multiple access to channels, and resolvable designs for Unconditionally secure commitment schemes

Classification of doubly resolvable designs and orthogonal resolutions Mutually orthogonal resolutions Definitions Parallel class, orthogonal to a resolution – intersects each parallel class of the resolution in at most one block (blocks are labelled). Orthogonal resolutions – all classes of the first resolution are orthogonal to the parallel classes of the second one. Doubly resolvable design (DRD) – has at least two orthogonal resolutions ROR – resolution, orthogonal to at least one other resolution. A parallel class is orthogonal to a resolution if it intersects each parallel class of the resolution in at most one block. Two resolutions are orthogonal if all classes of the first resolution are orthogonal to the parallel classes of the second one. Orthogonal resolutions may or may not be isomorphic to each other. A doubly resolvable design (DRD) is a design which has at least two orthogonal resolutions. We denote by ROR a resolution which is orthogonal to at least one other resolution.

Classification of doubly resolvable designs and orthogonal resolutions Mutually orthogonal resolutions Definitions m-MOR – set of m mutually orthogonal resolutions. m-MORs – sets of m mutually orthogonal resolutions. Isomorphic m-MORs – if there is an automorphism of the design transforming the first one into the second one. Maximal m-MOR – if no more resolutions can be added to it. We denote by m-MOR a set of m mutually orthogonal resolutions, and by m-MORs sets of m mutually orthogonal resolutions. Two m-MORs are isomorphic if there is an automorphism of the design transforming the first one into the second one. The m-MOR is maximal if no more resolutions can be added to it.

Classification of doubly resolvable designs and orthogonal resolutions Mutually orthogonal resolutions 2-MOR of 2-(9,3,3) One resolution has r=12 parallel classes All 2r=24 parallel classes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 This is an example of a 2-MOR. The blocks are referred to by their numbers. While a resolution parallel class is uniquely defined by any of its blocks, a parallel class of the m-MOR is uniquely defined by any pair of its blocks, and that is why m-MORs can be used in ways similar to those in which resolutions are used. If the resolution has r classes (in this case r=12) and if you construct a threshold scheme for instance, you can have 24 secrets corresponding to the 24 parallel classes. Each secret is uniquely determined by two of its blocks. And if the 2-(9,3,3) design has in addition the property that each two blocks have at most 2 common points, then a block is uniquely defined by two of the 3 shares corresponding to it.

Classification of doubly resolvable designs and orthogonal resolutions Design resolutions and equidistant constant weight codes - History Colbourn C.J. and Dinitz J.H. (Eds.), The CRC Handbook of Combinatorial Designs, 2007 Kaski P. and Östergärd P., Classification algorithms for codes and designs, 2006. Kaski P. and Östergärd P., Classification of resolvable balanced incomplete block designs - the unitals on 28 points, 2008. Kaski P., Östergärd P., Enumeration of 2-(9,3,) designs and their resolutions, 2002 Kaski P., Morales L., Östergärd P., Rosenblueth D., Velarde C., Classification of resolvable 2-(14,7,12) and 3-(14,7,5) designs, 2003 Morales L., Velarde C., A complete classification of (12,4,3)-RBIBDs, 2001 Morales L., Velarde C., Enumeration of resolvable 2-(10,5,16) and 3-(10,5,6) designs, 2005 Östergärd P., Enumeration of 2-(12,3,2) designs, 2000 Resolvable designs have been intensively studied. A summary of results can be found in the CRC Handbook of Combinatorial Designs. The resolutions classification methods are well described in these two works of Kaski and Ostergard. Yet full classification is only possible for very small parameters. You see by the next several recent publications what parameters are considered, and considered by means of parameter specific approaches, and final results are often achieved after the corresponding programs run for quite a long time.

Classification of doubly resolvable designs and orthogonal resolutions Mutually orthogonal resolutions History Abel R.J.R., Lamken E.R., Wang J., A few more Kirkman squares and doubly near resolvable BIBDS with block size 3, 2008 C.J. Colbourn, A. Rosa, Orthogonal resolutions of triple systems, 1995 J.H. Dinitz, D.R. Stinson, Room squares and related designs, 1992 E.R. Lamken, S.A. Vanstone, Designs with mutually orthogonal resolutions, 1986. E.R. Lamken, The existence of doubly resolvable - ???BIBDs, 1995 E.R. Lamken, Constructions for resolvable and near resolvable (v,k,k-1)-BIBDs, 1990. The newest results and an extended bibliography and summary of previous works on the existence of DRDs can be found in the handbook of combinatorial designs and in the recent work of Abel, Lamken and Wang

Classification of doubly resolvable designs and orthogonal resolutions Mutually orthogonal resolutions History Colbourn C.J., Lamken E., Ling A., Mills W., The existence of Kirkman squares - doubly resolvable (v,3,1)-BIBDs, 2002 Cohen M., Colbourn Ch., Ives Lee A., A.Ling, Kirkman triple systems of order 21 with nontrivial automorphism group, 2002 Kaski P., Östergräd P., Topalova S., Zlatarski R., Steiner Triple Systems of Order 19 and 21 with Subsystems of Order 7, 2008 Topalova S., STS(21) with Automorphisms of order 3 with 3 fixed points and 7 fixed blocks, 2004 The newest results and an extended bibliography and summary of previous works on the existence of DRDs can be found in the handbook of combinatorial designs and in the recent work of Abel, Lamken and Wang, while a method for construction and classification of RORs and DRDs is presented in our previous works, in which the corresponding equidistant code is constructed by backtrack search word by word and from some word on an orthogonal resolution existence test is applied.

Classification of doubly resolvable designs and orthogonal resolutions Incident matrix of 2-(9,3,1) Corresponding ternary equidistant code – (4,9,3) backtrack search word by word; E test; ORE test; 1 1 2 3 points V = 9 Since RORs are resolutions with some additional properties, we use the most popular way of constructing design resolutions, i.e. we construct the corresponding equidistant code by backtrack search word by word, so that each word is lexicographically greater than the previous one. By permuting the words of the equidistant code, the symbols in each coordinate and the coordinates themselves, we obtain equivalent codes. That is why without loss of generality, we can fix the first symbol of each word (corresponding to the first parallel class of the resolution) and we can also construct only codes for which the coordinate entries in each word are in lexicographic order too (see [10], sect.5.2 for proof). We check if there exists a partial solution equivalent to initial and lexicographically smaller than it. If we find such a solution, we skip the current one. parallel class blocks b = 12 words

Classification of doubly resolvable designs and orthogonal resolutions partial solution R n empty block ROR – each of two orthogonal resolutions (different partitions of the blocks with at most one common block) RORn – partial solution of ROR words < n e - E test after each one words > n e - ORE test followed by E test words > n o – ORE test n e ≈ 10, V/2 ≤ n o ≤ 2V/3 1 Vn V Let us consider the set Vn of n points of a design resolution R and all the design blocks redefined only by the incidence with Vn. We call this incidence structure partial solution, and denote it by Rn. If Tn is another partition of the blocks into parallel classes and two parallel classes of Rn and Tn respectively, have at most one common block, Rn and Tn are orthogonal, and we call each of them a RORn. Rn might have blocks, which are incident with no point of Vn. We shall call them empty blocks. If the words are less than ne, we apply E test after each word. For more than ne words the E test already costs too much computational time, so we either drop the E test at all and only apply it after constructing a whole ROR (suitable if a small number of RORs is expected), or we apply ORE test before the E test. Contrary to the E test, the ORE test works faster if the number of words is greater, and we do not apply it to less than no words. The efficiency of the algorithm is very sensitive to these two tests. Unfortunately, we have not found a way to calculate the right values of ne and no. For each parameter set our software needs some preliminary manual "tuning" in order to run fast enough. Usually ne is about 10, and no is between v=2 and 2v=3. resolution R

Classification of of RORs and DRDs q v k  Nr DRDs* RORs * No 2 3 4 6 8 10 12 16 20 9 5 n.4 15 18 21 24 27 25 14 1 31 82 240 650 1803 4763 27121734  73534 545  128284  546  1  1895  5  4 426 149041 203047732 74700 339592 √ * 13 44 70 546 38  27269 83  76992 1  n  13 101 278 524 819 - 891 319 743 618 1007 66 145 55 Резултатите са представени в таблицата. В първата колона е показан номерът на дизайна според таблицата на Mathon R., Rosa A., 2-(v,k, ) designs of small order, във втората параметрите му. В третата колона е представен броят на неизоморфните резолюции, последван от , ако сме го получили и с нашата програма, и с * ако резултатът не е цитиран в таблицата на Mathon R., Rosa A. В колоната drNr е представен броят на резолюциите с ортогонална и в последната – броя на двойно разрешимите дизайни.

Classification of doubly resolvable designs and orthogonal resolutions Classification of the sets of mutually orthogonal resolutions (m-MORs) Start with a DRD. Block by block construction of the m resolutions. Construction of a resolution Rm– lexicographically greater and orthogonal to the resolutions R1, R2, …, Rm-1. Isomorphism test Output a new m-MOR – if it is maximal. We start with a DRD and construct its resolutions block by block. For each resolution we check if it is isomorphic to a lexicographically smaller one, and if not, we try to construct another resolution, which is lexicographically greater and orthogonal to it. We next repeat the same for R1, R2, etc, constructing at each step a resolution Rm orthogonal to all the resolutions R1, …, Rm-1, and checking if this m-MOR is isomorphic to a lexicographically smaller one. We output a new m-MOR if it is maximal.

Classification of doubly resolvable designs and orthogonal resolutions Classification of the sets of mutually orthogonal resolutions (m-MORs) q v k  b r DRDs RORs 2-MORs 3-MORs 4-MORs No 2 3 4 6 8 10 12 16 20 9 5 24 15 14 21 18 40 60 80 28 42 56 72 108 44 66 88 90 76 36 48 30 54 22 33 45 38 11 1 546 83 70 1/1 0/1 0 / ≥485 1 / 1 0 / 1 7 / 17 5 / 5 2 / 7 691 / ≥718 0 / 5 3 / 3 388 / 495 319 / 321 - 0 / 60 0 / ≥27 333 / 334 1 / 2 ≥485 / ≥485 60 /60 ≥27 / ≥27 236 596 1078 101 278 524 891 319 743 618 1007 145 55 The results are summarized in this table, where the last column shows the number of the design in the tables of Mathon and Rosa, “2-(v,k,) designs of small order”, and a/b means that the number of nonisomorphic MORs is b, a of them maximal.

Classification of doubly resolvable designs and orthogonal resolutions Correctness of the results for the numbers of RORs and DRDs The resolution generating part of our software – for smallest parameters - the same number of nonisomorphic resolutions as in R.Mathon, A.Rosa, 2-(v,k,) designs of small order, 2007. In cases of known number of resolutions, we obtained the number of RORs in two different ways. For most design parameters we obtained the results for RORs using two different algorithms: block by block construction and class by class construction. If the whole resolution R is a ROR, the partial solution obviously has an orthogonal partial solution too, and is a RORn. Experimenting how the ORE test works on partial solutions, we found out that if a resolution is not a ROR, and we take partial solutions on n < 2/3 of its points, they are usually RORns,and thus no pruning can be done by the ORE test for a small number of words. The reason for this is that some of the blocks of partial solutions may contain less points, and even no points at all and therefore there are much more ways for these blocks to participate in di®erent orthogonal parallel classes. We implemented two different algorithms for the ORE test. They both use backtrack search to partition the blocks of the initial resolution into the second resolution, orthogonal to the first one. The search stops if one such partition is constructed, or if all possibilities have been tested and no orthogonal resolution can be found. The first algorithm tries to construct orthogonal resolution block by block (BB), while the second one class by class (CC)

Classification of doubly resolvable designs and orthogonal resolutions Correctness of the results for the numbers of RORs and DRDs An affine design has a unique resolution and can not be doubly resolvable. Designs with parameters 2-(6,3,4m) are unique and have a unique resolution (Kaski P. and Östergärd P., Classification algorithms for codes and designs, Springer, Berlin, 2006). Using Theorem 1 we verified the number of DRDs of some multiple designs with v = 2k in another way. For some parameters we obtained in parallel the maximum m for which m-MORs exist. If the whole resolution R is a ROR, the partial solution obviously has an orthogonal partial solution too, and is a RORn. Experimenting how the ORE test works on partial solutions, we found out that if a resolution is not a ROR, and we take partial solutions on n < 2/3 of its points, they are usually RORns,and thus no pruning can be done by the ORE test for a small number of words. The reason for this is that some of the blocks of partial solutions may contain less points, and even no points at all and therefore there are much more ways for these blocks to participate in di®erent orthogonal parallel classes. We implemented two different algorithms for the ORE test. They both use backtrack search to partition the blocks of the initial resolution into the second resolution, orthogonal to the first one. The search stops if one such partition is constructed, or if all possibilities have been tested and no orthogonal resolution can be found. The first algorithm tries to construct orthogonal resolution block by block (BB), while the second one class by class (CC)

Classification of doubly resolvable designs and orthogonal resolutions Bounds for higher parameters - Definitions Equal blocks – incident with the same set of points. Equal designs – if each block of the first design is equal to a block of the second one. 2-(v,k,m) design – m-fold multiple of 2-(v,k,) designs if there is a partition of its blocks into m subcollections, which form m 2-(v,k,) designs. True m-fold multiple of 2-(v,k,) design – if the 2-(v,k,) designs are equal. We shall call two blocks equal if they are incident with the same set of points, and we shall call two designs equal if each block of D1 is equal to a block of D2 . and two parallel classes of the same resolution equal if each block of the first resolution is equal to a block of the second one. A 2-(v,k,m) design is called an m-fold multiple of a 2-(v,k, ) design if there is a partition of its blocks into m subcollections, each of which form 2-(v,k,) designs. If they are equal we call the design true m-fold multiple.

Classification of doubly resolvable designs and orthogonal resolutions Bounds for higher parameters m-MORs of multiple designs with v/k = 2 Proposition 1: Let D be a 2-(v,k,) design and v = 2k. 1) D is doubly resolvable iff it is resolvable and each set of k points is either incident with no block, or with at least two blocks of the design. 2) If D is doubly resolvable and at least one set of k points is in m blocks, and the rest in 0 or more than m blocks, then D has at least one maximal m-MOR, no i-MORs for i > m and no maximal i-MORs for i < m.

Classification of doubly resolvable designs and orthogonal resolutions Bounds for higher parameters m-MORs of true m-fold multiple designs with v/k > 2 Proposition 2: Let lq-1,m be the number of main class inequivalent sets of q - 1 MOLS of side m. Let q = v/k and m≥q. Let the 2-(v,k,m) design D be a true m-fold multiple of a resolvable 2-(v,k, ) design d. If lq-1,m > 0, then D is doubly resolvable and has at least m-MORs. Unfortunately we do not know any results on the exact values of the number of main class inequivalent sets of q-1 MOLS of side m for q > 2. It is on the one hand a complicated problem itself, and on the other classifications of some sets of MOLS have been done, but with respect to an equivalence defined in a different way. For many parameters it is known whether lq-1,m > 0. We use this data to establish the existence of m-MORs for some parameters. R.Julian, R.Abel, C.J.Colbourn, J.H.Dinitz, Mutually orthogonal latin squares (MOLS), The CRC Handbook of Combinatorial Designs,

Classification of doubly resolvable designs and orthogonal resolutions Bounds for higher parameters m-MORs of true m-fold multiple designs with v/k > 2 Corollary 3: Let lm be the number of main class inequivalent Latin squares of side m. Let v/k = 2 and m ≥ 2. Let the 2- (v,k,m) design D be a true m-fold multiple of a resolvable 2- (v,k,) design d. Then D is doubly resolvable and has at least m-MORs, no maximal i-MORs for i < m, and if d is not doubly resolvable, no i-MORs for i > m. Let lm be the number of main class inequivalent latin squares of side m. Let v/k = 2 and m≥ 2. Let the 2-(v,k,m ) design D be a true m-fold multiple of a resolvable 2-(v,k, ) design d. Then D is doubly resolvable and has at least …… m-MORs, no maximal i-MORs for i < m, and if d is not doubly resolvable, no i-MORs for i > m. The number of main class inequivalent latin squares of side m is known for many values of m, and for v = 2k much better bounds can be set using the this corollary, which follows directly from Proposition 1 and 2.

Classification of doubly resolvable designs and orthogonal resolutions Lower bounds on the number of m-MORs v k  Nr ROR s DRDs 2-MORs 3-MORs 4-MORs m sm 6 8 10 12 3 4 5 20 24 28 32 36 15 18 21 27 40 25 1 82 240 650 1803 4763 ≥27.106 ≥5 ≥1 13 16 44 70 0/≥ sm /≥8 /≥ sm /≥13.106 ≥8 /≥95 ≥95 7 9 - 11 352 716 2.1015 3.1042 2.1096 31824 33.1010 29.1033 19.1067 95 Using this corollary we extend our computer results for some parameters. The first 3 columns give the parameters of the designs, with Nr we denote the number of nonisomorphic resolutions. Next in the table are the number of RORs and DRDs. Sm is the lower bound on inequivalent m-MORs by Corollary 3. The designs on 6 points are true m-fold multiples of the 2(6,3,4) design which is resolvable and not doubly resolvable and therefore they have no maximal i-MORs for i less than m and they have no i-MORs for i greater than m. You see that the lower bound grows up very quickly because of the big number of main class inequivalent latin squares of the corresponding side. For the next designs we can say nothing for the i-MORs with i less than m, because there are not only true m-fold multiples.