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K ^ l 6--j l S A ??"?  / ^  l 6LL^  l 6LLj  l S A  ??"?   l 0X37  'Two orthogonal latin squares of order 4( 2(P l C (Alatin03S ^H l 0޽h ? fff̙___PPT10i.@X+D=' = @B +   0  p(  p p 0D37 N are mutually orthogonal latin squares of order n (MOLS(n)), if for 0 >b0283f#*<^ p 6((\ p S A   ??d"  ^ p 6%%\ p S A   ??l m  ^  p 611\  p S A   ??jo  P  p C (A latin04   p 03` @MOLS(4)  2 ^  p 043  LTheorem Let n be a prime power and n`"2 . 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H x 0޽h ? fff̙___PPT10i.tQ+D=' = @B +  0 |F(  |^ | 079  ( ) NpvN PcR _N/fcR GR vQ-N 0 001u NvP}g bPSN|vsvuL"MBf LWvvnbNec\ gn P0@bN $Rekey/f1uTiQ Pec@bb_b&N N[f4ggE ^ | 6%%j | S A ??"?m3#  ^ | 6%%j | S A ??"?  H | 0޽h ? fff̙___PPT10i.R+D=' = @B +O  0 f^ (  ^  677  <A7zd~! @___PPT10 b___PPT9D< bCryptosystem from MOLS(n) O(unbNec0 g P Messages (Plaintexts)0 NnbNec^iN PW(OLSG) G GR g |E(G)| Pkeys0 (geO(uk PMOLS(n) GR g Pkeys.)M cg"g g3fgg3fg g L  8X\  S A2 ??WD \  S A ?? #  f  H  0޽h ? fff̙___PPT10i.ЍR+D=' = @B ++  0 B:0(    0\V7  :֊ 1.vunvvu'YBf MOLS(n)v Pxe_Ng_'Y0 2.YgSnaMOLS(n) key space\0 3.Orthogonal mate0 N[f~b(hQ~bQO)HY;gg3fg  8XH  0޽h ? fff̙___PPT10i.&8+D=' = @B +  0 pD(     0x7F  Rce!j_ (Sharing Scheme) 00 00(WяN g_Y͑'Yvzl[ pNxOzlV{N zl gfu 8^gc(u1uY PN Tav`l NMbWLOYё^ |v\8hH_& 0@bN ^zN P|q}O_\vNxe!ql_U/f g[v_'`0xrcccBg g3f g.<H  0޽h ? fff̙___PPT10i.g:+D=' = @B +k  0 z (    0І7I,  F@A critical set C in a latin square 00 is a set (partial latin square) with the following two properties 1. L is the only latin square of order n which has symbol k in (i,j)-cell for each and 2. no proper subset of c has property (1).\! 3f;**/j  S A ??"?&/ ^  6%%j  S A ??"?  ^  6j   S A  ??"?p Z     0d7f   RA critical set(P  C (Alatin05 w H  0޽h ? fff̙___PPT10i.G:+D=' = @B +  0 <(    0T7<w>  LO 00 NbWQv N PMOn NiQ Pgb_bN PLuƖ(Critical Set)0\N bYN N/fLu6q YN(3 P)_NSN|~~kXb/UNvbNec0HQg g3f&g@P  C (Alatin06/ H  0޽h ? fff̙___PPT10i.p)J;+D=' = @B +$   0 $(  r  S |V-`A`}   r  S T.` `  H  0޽h ? fff̙80___PPT10.w`  0  (    0ħ7d 5aFact 1 A critical set C of a latin square L provides minimal infos from which L can be reconstructed. Fact 2 Deciding whether a partial latin square is a critical set is NP-complete. (From completion point of view.) Fact 3 Denote the minimum size of a critical set of order n by M(n). [D. Curran & G.H.J.van Rees, Cong. Numer. 1979]<bx3f?t H  ^  6j  S A !??"? w Y !H  0޽h ? fff̙___PPT10i.@;+D=' = @B +I  0 "(  P  C (Alatin07* 6 P  C (Alatin08 V6   C ,A latin07-2* 7 ,$D 0  07V v  JCritical 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<*+  0 0b(  *  07wH  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 +  0 P(  I  0\Jdj 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 JH  0޽h ? fff̙___PPT10i.PА+D=' = @B +  0 pL(    07*  @LuƖvxd g_Y ! \eƖvN NY0 ! XR4x㖦^"!!g .  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