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We can take 6x 3f" v^  6((j   S A ??"?j    0J 9  }A latin square of order 3x$P  C (Alatin01  H  0޽h ? 33___PPT10i.@+D=' = @B +  3+P h(  hP h C (Alatin026 h 0ةK >A latin square of order n defines a quasigroup on 3 elements. >?0x2$ 3f6 Q h 0;w  Y is a quasigroup if is a groupoid and have unique solution .>Z02+-6 .^ h 6d h C A ?"? y  ^  h 611d  h C A ?"? cg  H h 0޽h ? fff̙___PPT10i.ׯ+D=' = @B +   p l(  l l 0Uiw @Two latin squares and of order n are orthogonal if 0n02n$cd l C A ?"? K ^ l 6--d l C A ?"?  / ^  l 6LL^  l 6LLd  l C A  ?"?   l 0xf7  'Two orthogonal latin squares of order 4( 2($P l C (Alatin03S ^H l 0޽h ? fff̙___PPT10i.@X+D=' = @B +K   bZ p(  p p 0ҭ7 Z are mutually orthogonal latin squares of order n (MOLS(n)), if for 0 >b0283f#6!^ p 6((d p C A   ?"?d"  ^ p 6%%d p C A   ?"?l m  ^  p 611d  p C A   ?"?jo  P  p C (A latin04   p 0w2` ZMOLS(4)  2 h  p 02  LTheorem Let n be a prime power and n`"2 . Then there exists n-1 latin squares (Best possible!) of order n which is a collection of mutually orthogonal latin squares. b 2G6@R H p 0޽h ? fff̙___PPT10i.0RN+D=' = @B +J  aY t(  t t 0|2d< kOrthogonal Latin Square Graph 2 t 02M9 L00bPSNbbNecvuZPޞ  NvWv vubܕOO[N PW OLSG0"'02'g! t 0Ti,  LVNNbNec@bbvƖT0 EiQ Pޞv#N/U \avbNecNvWv0 ''g$> t 0H3i6 :"  jON000 pk PNvWvvbNec GRN [QOvOLSGp 0"6026g. !^  t 6((d t C A   ?"? g   ^  t 6((d  t C A  ?"? /    ^  t 6((d  t C A ?"?   H t 0޽h ? fff̙___PPT10i.RP+D=' = @B +Y  ph x(  x^ x 6d x C A ?"?zC @ x 0iF  )R(uL"MbPS[Qxef fp vN PPermutation bfp1u \ f v1-1, onto Qxe0 Of((1,1))=(0,3) f((2,3))=(1,0)pc2gg3fgg3f(g   ^ x 6%%d x C A ?"?2"2 j  x S A ??"? `  j  x S A ??"? H x 0޽h ? fff̙___PPT10i.tQ+D=' = @B +  |@(  |^ | 0i9  ( ) NpvN PcR _N/fcR GR vQ-N 0 001u NvP}g bPSN|vsvuL"MBf LWvvnbNec\ gn P0@bN $Rekey/f1uTiQ Pec@bb_b&N N[f4ggE ^ | 6%%d | C A ?"?m3#  ^ | 6%%j | S A ??"?  H | 0޽h ? fff̙___PPT10i.R+D=' = @B +      * (  b  0 `9Zdw Theorem [Lindonear et al,(1979)JGT] NGpNaW GR_X[(WP=|V(G)| PnbNec n Y'Y O_1up PbNec@b[QOvOLSG H 00004agZg%  ^  677d  C A ?"?    <oD___PPT10$n___PPT9PH Cryptosystem from MOLS(n) O(unbNec0 g PMessages(Plaintext)0 NnbNec^iN PW(OLSG) G GR g |E(G)| Pkeys0 (geO(uk PMOLS(n) GR g Pkeys.)J ccg g  g 3f g  g 3f g ggb  8Xj  S A ??"?m s  d  C A ?"?I~ u  H  0޽h ? fff̙___PPT10i.ЍR+D=' = @B +5  LD0(    0tw  D֊ 1.vunvvu'YBf MOLS(n)v Pxe_Ng_'Y0 2.YgSnaMOLS(n) key space\0 3.Orthogonal mate0 N[f~b(hQ~bQO)HY;gg3fg  8XH  0޽h ? fff̙___PPT10i.&8+D=' = @B +  P6(  Z  0$wMw  L00bPSNx n pPrime Power, YdkN(n) NvWvvnbNecxe(max) 6q/fg'YN(m)=n-10 00 00 \e N/fPrime Powervn 0000 ffx0W [T. Beth, Abh. Math. Sem. Hamburg 53(1983)] @bN g YYvMOLS(n)a0@@gg$gg>g    ^  6d  C A ?"? 3 ^  6d  C A ?"?]  H  0޽h ? fff̙___PPT10i.9+D=' = @B +  pD(     0wF  Rce!j_ (Sharing Scheme) 00 00(WяN g_Y͑'Yvzl[ pNxOzlV{N zl gfu 8^gc(u1uY PN Tav`l NMbWLOYё^ |v\8hH_& 0@bN ^zN P|q}O_\vNxe!ql_U/f g[v_'`0xrcccBg g3f g.<H  0޽h ? fff̙___PPT10i.g:+D=' = @B +s   (    0dwI,  F@A critical set C in a latin square 00 is a set (partial latin square) with the following two properties 1. L is the only latin square of order n which has symbol k in (i,j)-cell for each and 2. no proper subset of c has property (1).\! 3f;**/d  C A ?"?&/ ^  6%%d  C A ?"?  ^  6d   C A  ?"?p Z     0+wf   lA critical set(P  C (Alatin05 w H  0޽h ? fff̙___PPT10i.G:+D=' = @B +  <(    0Hw<w>  LO 00 NbWQv N PMOn NiQ Pgb_bN PLuƖ(Critical Set)0\N bYN N/fLu6q YN(3 P)_NSN|~~kXb/UNvbNec0HQg g3f&g@P  C (Alatin06/ H  0޽h ? fff̙___PPT10i.p)J;+D=' = @B +  (    0,id ?aFact 1 A critical set C of a latin square L provides minimal infos from which L can be reconstructed. Fact 2 Deciding whether a partial latin square is a critical set is NP-complete. (From completion point of view.) Fact 3 Denote the minimum size of a critical set of order n by M(n). [D. Curran & G.H.J.van Rees, Cong. Numer. 1979]<bx3f?~ H  ^  6d  C A !?"? w Y !H  0޽h ? fff̙___PPT10i.@;+D=' = @B +c  <(  P  C (Alatin07* 6 P  C (Alatin08 V6   C ,A latin07-2* 7 ,$D 0  0$aV v  dCritical sets, n=5 2H  0޽h ? fff̙_W___PPT107.++D ' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DM' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B-randombar(horizontal)*<3<*+!   8 0   (    0LBad <Conjecture 00 2 cg>  0daz  @Theorem (ؚ, Fu, C.A. Rodger, JSPI 1997) For n>20, 0 M*(n)Maxmim size of a critical set of order n. Theorem (D.R.Stison, G.H.J. van Rees, Cong. Numer 1980) For 0 Theorem (ؚ, Fu, ^, Cong, Numer, 1995) For Jg$0-A g g  .  8 ^  6d  C A! "?"?@ # "^  6d  C A" #?"?j Y #^  6d   C A# $?"?   $^  6((d   C A$ %?"? -u  %H  0޽h ? fff̙___PPT10i.ȅ+D=' = @B +  0b(  *  0awH  Rce!j_ 1.Key ! L (bNec) n public 2.xN PƖT[/fL-NY PLuƖvoƖS0 3.bvQ-N t f"|S|0(SN[1ؚc gYNN(i,j ; k), uN PCritical Set) 4. YYvNsSS_0R Key 0Xc@gg3f&gL      H  0޽h ? fff̙___PPT10i. +D=' = @B +  P(  S  0(adj OA (s,t)-secret sharing scheme is a system where k pieces of information called shares or shadows of a secret key K are distributed so that each participant has a share such that: 1. the key K can be reconstructed from knowledge of any t or more shares; and 2. the key K can not be reconstructed from knowledge of fewer than t shares. \P3fX3f $IH  0޽h ? fff̙___PPT10i.PА+D=' = @B +  pL(    0@a*  @LuƖvxd g_Y ! \eƖvN NY0 ! XR4x㖦^"!!g .  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