MFNWtKUb<ob<R=MDLCdNVZZJ:tN>H:xXVErps:;BNSDOETlMXlgwgiW;mD[UUUWUsKitUf]Wfv_ivmixoYKEVcsIyuyvayvUIv_ioixoOWkgxwiywOveCHwgIxiIxmyqAYs]IwgYtUiuIXpCIFiSIaBAAsa;GbYyvcixqyxeYweyuYyuWdMWTuUYuyyyyA;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::ZjifDqEtk]`N\\@Nd\\QgqxH`jwhSWDQVyPxPLAIXU`wyyySUun`r[DNZ]WmUjPuJZ]Y_lSLqqWioSxwwtLEQl@UNGiOC@XyQjXLYbIvN<xwaLnAt=uOZdQnAtE<SIdQnQJLYRIdq:`xJYryqJBhyNFvL?^^YoOA[yYelofiGbt?w[w[PhdK?gSO^DGpLYeJp]t?fjHo\\I_:yo;H]\\`\\:GoDF]`hqEht=w[F_alS=wUToTtOHPwCborY[w:=EpYdRYrYMChKdE?BDmidKG=QsC_YRmHnQBLYr?QeE_X_krige:[iBYcf_DDaGeSs\\eTPOb_wYrwsXirdIviGbNwG];TYeTKmgywvJGBsyCy]VlmFeyEQwcX=jjyx:`sQMP^\\YPho_Tk>xMsmtsIPMhKmYLwMXwIWXqMxqPIUkEQT?moDhtHEo_lY@mQHQpZDyLUrYHpn<yRutnHUv<lpxKYPWwIXR_p`I`pXfWOyy>eMy_JWu=qaR>ppVxO^funr?G`Hv^Qia]vuuocJpwUQdTgd`_mex]Tvf\\xfrhdbXvpIe_Hs[IiH>nUonv@bKpiZHtX`ibhfKO`JFdPPkIqvy^q<?m@vuvA[k`fDhbkYdNqxj_c>_fOfv_wdx^_E?uYXyQ@olFqYIf;_e]IyPVqnosfPyJA^=asuq[j`ZR?kE^yjHtHQgOHxSn\\wYoIh`TY\\Fg`Rx`Iq[Vwq:@]TyybQxv@]k>kivdaY\\ui\\dWirn[PqrTgpPYbx^tvFfkWZbihlYa>^bK@wTwsQhvOyb@?]gqhwomng_>og=>wpGarAc]hibAyX@eLogQnhlykD?s<_c\\>b@QuvA^kxm^ppAXvjVZsF^AFo^_nVVflixrifhaqi?bHI\\Jf_]O]s^`lyssAsp_b=IZ]akdPmJniAv^PnaNAw:Gi>VqmfvRIuyF_[NmpQjc?pIq^PWjiFdUYrc>glPqhP[B?jLNqKAwyxnVhq\\ajYQ^ZFVQxk?e;_f@UbISs??T<aBw=fK]UyYy[oRAMyR=HwiwEUHfmRPSty]TsStbAHxSuYMs^yGKUu=IB=QxemUA=rrwI;aIX=BJ?b^ss[_TXEYTCeEkuGgCNgeEKY:yxEKBLWbmuBHkvjOgvacI_W=_dGktRegYwr]WFQ?yTKBBUwI[HTYrByGjyF\\Wbwgvw]SxawaaWs;yAwTCAS^yxd?Xd=sBgyRaDW=DjsT:=h\\KgmMG[av\\Kd]sTJEcv[dV;fvch;wS:_DkYu]QwOCdO;sg=yoeytSG`kImsFyog^?xEOBLCFViDIgI@Gy]ot^irP;HK?hZOsjgS\\oH?EUSuDGMUAuFJIHi_FKSWUwRT[ho=Succ^;Is_VTUE=ICoSIswCWqRZQG<_iUacrCehOcaIRWuspqRfYT@ccfMuhsyCWrYmIPKIbQhdCehqx\\st?]DG]EqMIFYfW]rgUCbqvIGSgofLWg`aHJKdluEqEeu=ixkwQStrSWtWgcgwJSIGku^oxgKVyQWZEt^gBeKGZKxced=IdTOhJEfR[xrMBkKg^mGJ]Hc[trOT:_R?eFd_FVCXZCD?QCqSX]YetGF<EuQeUfcCLMhjGvVKs_STkUw^]CEUEl[f<?hNEwdoH?MWf]FbesPKU]kgH]bSES;QVV?hHqdT=ce?bp_h=GGuqGD[y@SU=IFKExEeUWAhNMX^wdRYFIMevKeHYWSsCl[HGau[AEZiR_iTJUDS]YckXsoV>]GBqb@;VM=DluVHgVuQeDqxLUE]]WSAR_oB?oxLgr==vqkR?McPAEG]WBKVP[HVOI>IrEuBkUcqSckCwpsFo_Rc?eB[hhCXYSrFChVASt[UUWWs]ceYBhyD>aUTMWZ;vDoR<MigQDu_TtCUuQeTqXLOI>QV_CiI]w_CruEHosRwoFf?EcQiJ?bh<rTuX[Xm>QN?YtNdpPQMSxUM<Lq=q@INBAKETJBhxStLsEq:\\VmYMcEJvLM`\\joAWKlvL`oTExbqR``uRqK;=PX<LAusChO?@mNEjeaP]ISWhp@yWl\\Wc=y<QlPXJQuSwlW=xtYyvJHOTtK;TW>lOIDODTJZyNoUPRLxHlwPelKxT;toREv\\Alc]kbppf`yolyvPvOMkxDK>]u\\EVC]NAanAYc=F^K_udgd[Q^Vi^Dr>[tR=H\\aG_?GT[rtSru[XBuGDsUKag?QUEGEKCigcGMeYoGB<URBIb[ebvYFAKbGyGK=CMQCQ]C^[UkUTFcXVEh=]g<[VDoBAIgOyXCgsQsd`CFc=ujQHK]Yc;xOOi@YxlOFXEbxOGeCs<khaIRVIgOms=eTOIyPyrfyBtqtVuyREy:orPce\\IgqkbVMUZAX>sHsUuOkYqAgC=syoYAsv`KChAX^WR]_xvcF_kRgAc^IcP[SI_D[Uf=MerofcQGoYBfcca[TiETvae;=HGctaqvuWHd;IbAiJEYdytG?hBordUTXWC;ebcisL[UxYDC[gkQrHMgHqebmrikvj?HrSiPyrckxkCwQADCoIeEvbUbGAboQhXEh[;d\\KHHmb_OFtWu_yb_UcROtnwbQUHjEuL=Up_Rb=UYAgUME>gCAgCiySEosEQUGqdWMWq=c?ErKMWIGFwOCeGw_?c@YBM=s`Qes?U`GvDIGu]Dh_U\\aECQCkig`KY^Id<UrFSGdidCQd?wvjsgjoc`av?ABUcCqkDbgUQmYdWTyUHIEI^?vO=xrIY@_IXKxyey:Wy]YRruxDiGiSv?uiHGbVQl:DK[HmrpPHPq^PlE\\kAMkvmLLylFAokljcev=lqi<YWtRLewqIQP]nuTjSqvo]xgtr:`TDeos`qsXoUts^<QVAKd=lHEwRQOVYqEyTo`Y:aYNPKh\\V>AsNQx;TxrdW^YJ^tja\\vHdnlUkRekoYJvXOVesOqlUMN@mPnUPoXmT@jtmUdpKoHxhmuD=QHewk\\nBlOhuqWXowys<\\VPdkZAJgERo`R@ev[evpTq`aSx<NUAvyUra]nvtRiHuBTQITs\\qV\\yLl]raXt\\@PCHS@tr;\\XmDS;XmFpVRyXuHjIMKB`mZivW=NHTSA\\srptgMmIANqeuY@qJMOFhrxELi]vomrP@kg]TEMSNEXrelmyroxkU\\YyMlm=K`AvvaXWmKQqmA<QTIU^IQhmw^IYHQq^\\sWllE]skls=QYwARtpUPHVWin>TKyeq`DLkYPD=VYxOUiu<QTo=u=PUcYXLykEMmBHYwuOSdsmuu_dRm]WlpLI\\xKlqy@K^AO:IJ\\ao;YsdHxRHpO@yD@L?IpLxrdUp_Hvcpvg]uEQVKXwvdnp`VNqVV@t[lL_io;qOIUNwLSfiJ:mt>yVTqNgMVoaoP]RNiVDQO`@VDisdHywtY;@VftLqYlstrE<vhmrBimUMr>EkJAuGxvYiYJmWxxYqdjGxKl]T@QPVYOY`LJ`m`ajN\\MBMVQmysLNDYsq<opYySDm`AvP@qBHlPiO\\Ax<qo\\@YeXrSHPR@VeYVGASrxQZYPGttsPk]eLEhWo<PGAP?QxZLXX<ucMS\\lJydSSMwG@kQLvjAMWTyUtoxULkUPXTu<PQ\\hsaPkdPKNhuHdkAtuCQPZQMKDSvQVPYypLRTxyMTPVMUUhqsmmDpncYlX=NqlqkxRpdPOekRxMp@kSlU]HW;xt?=S_Lm=Atn`LqUQEEVBAWBUnn=tBLXxptF`NSATdUNGHTE<WNINPxWNIRGewJTNwHu=MYV@uE@K<MM_eSGEk_DP@aV@ml@=L\\EuCPvcywSpka@u[tQhDp<eS@Avm@U\\Mv=AQZDW;MMwUkQ=m=aTMAY]LouuS:MN\\yPs\\QXmPVIwvqJoPMTIprAs^QRvlsS]tNdKCEl?xTwmn[prj]WMxKODNIIL^@sn<YfEkXHQNdQMtWLaPMQLqUT?MyZtWTaRCUlk@X;@mWdK?mnVDlF@xvtLVQQsIm^TWs<oX=KaEuEhYELt`]Qr@yTHXRtxBXuvDrZdt^MnVHXIaRxqLKLSGASMHw]@jdyrNTM==se]r`]oG<K\\=VP`YKDjXuTuIjE@wCQSxdM^@wPPS=Msb=k>LO[\\o;tsm]UCILdEVj<S;PTEiNUMVMmoPuJGLTHTNGpXKPKgDpJlUTHKsuo<PcJn_cxp@FwagZNY_WpeM`qAWg]h\\fIsA`bZ^atituw`>Aiayh[PrEQigpbMOwyvaJvx:HgbYg;Xm<OrMogdPw\\>^??kNFaVXqGP^dyZwFrWGxKn[kfgL>`GYnPYkdwbKqbYXpphhOGs>y[[FsuV]Av:GuVKGX?rtmbU=UAyBXQVIOwDqEKIEsoe:ad]kXJavRGdd[BwMcuEY\\eEDex][dxOe]AuRIdBKvS[D>CXgAVH]BwUUGsBYixByfVwvrkSa]BmGWgcfq_il;hgig<ARuarHuhNQdqkHkKWqAdpEcGoGCMI^iwaWcWyFSmSlqsI_WmgGcqeVismqboWhYWGdWSFGCnwvkOH?kFPCsjohaaDoygB;cesRR=HPAcA[ivarNgrGQyf;ce[cSab>oF<eXJ]bA?rgIUJKetmscqCqEULmU<MfeQwrytKsXtsvgwfR=ssee]eVr=iXgTusREKIrawVaesoD[]yqAgT=VE;XwgTX;CsicmshZIyh]tf;D>agYkSZ;cEeeOUbpgBK]C^cch=X[Wcw?TK]D_kwZcr^aIMaiv]gC?uR[iAeGn=gWIRDYxaAgLyiU[cosfTMs`WHECeWihZSi[SSH?vD]CCSyJSeLawp=dB_xOOxKqg@cy^?bb_wJSC`UYT[bNEGd]HZiIOAbVOrJeEcKIj?FnaWgAelSCA_WK]t\\AUAqSMMDhSVRgep?BFMu]kRDkvOKw`]iy]SsAxnUt>Sfi;IWuuuGev=rhWd@qggYFHucr=ffovLQB_[e>=ieAwjAd\\ihDUtOSHFeW^wiAeWnMCcsXsmh]oxC?g_?fYgrnsSvguuMb[If?Cf`mRPEvYusl;XQ=SqmDRAsuctPGvsgxHgBKui\\;Fv]GLAD^[CoME>?SOWUigTKUYTYDTcIVCGt;`FO]UfkWV^GwkaW`ufkKPZt?\\Dw^oO_\\^iw@bn_mvN^]?vKqc`v`tVs<@yfN_e`\\DvsFg`iYcLHtw_bKwZk^hKVnhVfmg\\sIwQPfAG`Pnhuikjww=atXPbdweN@jNWuG`agib^ViYfeaPir?g;_[=Or:x\\vqi=iqZiZTvk@afXvpdqlSW\\ipydarYG\\JFm=Y_xge?_ZBqcDH_NwrNXxkPn=W[lYn=a_=HZ[in?PnAayT?yxviixbhXr;igdHiooQ]eDWfbiF@Sx=ctvwu[QWIiI^sE]WrP[EbMugWRnsBd[yR[d?YU=MybchadqR=loMvwDSWYPWxR\\=LOdWj<LImuL]w<XNAEROhwKHLxmKftSxYUf<KEMxxtLNlYKxr>uQtpTVYUt\\mudTa\\t]enZDQp`YkqSI@o]qOp`lo]PCxxxAkrxx_DsTImQHSayThaRE\\JLtptqRAuQyXONdUyUN;ax[MWsxxN=t?=J_yU?PK?Qx:Hn_YNk<SLIy\\UUDYOhhv:MufmMkIOrUJqXslDrU@wYqQEhLKAk[Xnkqv_]xcUKl=TaEnUENPmvHipVpSHYtcPQKxTIUw`XXeAq[iXyhqXdW]quSlkuUNTQw[hyWalRESQhKudMPuRNHQVTpZ<VAHnsHlYxv^mmVHJtDYoTw]at\\XX^Yy<LjJ`kV@o]tS=XJqMQ]Epq`sVIJ^pWtPKcLmkdNIhuZ=WYyvhTNrpTPiMwTQaqqZERnXje=nI`Ux=pPutt]X^@wGUKLxj[=qJQnY]VMLUFMrq`wIDsXhPc@Q\\AyrtTuHN=IkfdUeqO[lLqQpHtv=qJ>qNNmRUUL^]wRTLD@q]`sWhxkTK<hN^drxQmkARFQrVApJXYrlLWhVKqs`pxRLTwuQjdqf<wR=lOmu_]N\\aXCisgmJtDXsdUtUL<Pk`Xp^<URPuE\\Ty@UGDKXMKlQM:pt<`nHyvg<kEyn^lVFIQ>qPnuwBeyruTmHYmXJ_urDpKqIRpPLLxRV]JtLSkujxmokElxMuxAXNYWchP=hRXxUjpvqqnGEnv=YjQrZTP^epTqja\\PFDYnucSPdiflVg[jw[Hw\\j^_owuVPdfg_CVgdXnHhhkQwMVshgZTxolYbh`ojHqw_`eXZ\\>wXOne`m?goL_wOn`Bw`a_vfyyXGuJGugfso`mgivtHmX^cSpmQaf\\^]nyh=oZx_wPXnOitrib:XwYOhpWy]qdlWvu`cX>]Jgsm>tdqssn_F?anfyNhZKgg\\NgDyp;Ah:_lhAs[vtDF\\Mwh\\gwBAl[ybMX^?We]YdEnZwhy:Qw>aut@_lOl:>hfgaoxuFQbKnbYHpHQobw^C_nW@qDnpcQqEGawV\\`@rnpclhck>^XGdN@qdAu[FfUI^u_\\:`qfvq?_soG\\UguAA\\An]kPlFNdB@sKVpdNtH^gAfoipdaGdEGlPwbJPt[OsQn^UN`mFZZvlnob>ygL^wWYm\\VheVeMGjPhrJHenIbp@x\\we]Xoc``hpe`xp:vuXweMYg[PqTpniH`oo[Jg]t?si@`pvofItsn\\^Id`ovVagAqlaIxV@]jV]dvaQFal_hbowAOxD`_aYjJhloqkWYlJ^fAfbi>lMP`QNf[grX>r@_nH_j_a^TNvoxiJVrs^euPco@\\QO[O_pE>gYPm@_moP^UQ_BpfENcH`jMnZiYtmx`VOgxv\\fOqhod@yoWAoHNk^WbCYdsOhrygJndKvqVXbR@]i>jAHyW^]h?]fxgCIcNn]Io^lNwHFf>@gYAkQVcD@iB?\\UGrTV^hfjDifg^ytAyIv\\Q`myVx`v_DQZ\\Hxt`^Qq_sQm@hdCntT@c=xfg@`UYo\\YxxfpgYjHI`dggYo_q`thI^W`a=GrBheUVoPwkhxydGZS^np?yF^mGhhvh]TI^<qhwq^HF\\sQpVGtoo[GabIV\\f@fBywC?jOwoGF\\cFyqnmmNhewn:wkfxoaOipho:w_^w]GXi@^xiQeqFiOn_gA^oVpUYn<NxEgl?Iigi`ZQhlGuWovA_xna\\XNs:yb_PprX^Giv;Fhqxg<Ite^dDFajHfSvoQYi?WxZPdcI_NGm=iZ^Iv`>dY?p=qhmPp=>]O`bIQwNgelQd?VbY@i_O`\\IbDIeZfrmpblvlZfZy^svnsnIhmNh[apjVbmVfUfZ[At=`fBgvKfgWxkb?cfojdGvrhiLfv`Y_C_dipgXwoCXtsHl\\n]NPZmO`yW\\e^hT_xDFlh`[PI^ZnpWpmDgZ=_cGfccVvZnnJYkVofg^hlWw>pa\\`lMpfHPjCPj>GfnO`T>icv][fj@vktPronymPTgdbPp>yNdTnpdorQmTay:DoCxr`iP>QLchN@DXDTryXyI]jG<uEhL@TuA=leyOf\\XrPtZpOimYiqQgQrqXvrlWquqtLPvQyp]r?QYNur`uNWeR`xq^HjFipgUYXDsAYxNTrm@OJqPIYn:eoFMXYEtcLPRqLGHwKlnaUMpHocMwN=yZVfE^^_Iq\\FnC_cTHhnWsW?oN@nbP]]hrvGbs?oqnmB_a[xvn>fc@_EOi>XhNfpuVa@xhNIc^ormIqffoF_mfHcgydgN]__v:H[p@_R^^K`]eaoMijRW\\ZOy`hhApeUpmBh`@@mQ`n[fhMqbT_oOfhEphtAl`?lnidb@vhwbq^xVGcmo`nhuJavkNZI^`<gghHoD?`h_mIQyIHqFVg[q_T_n]Xb^H\\Sf^]nt`wfWo_VnvLHfSnbsYyui`AQgcq]D>sK@fqf[ChiS_Zvff[HtBRlgF`?sOSSrqvdAIH?xtmU]uW]GGDaRiSF?MTpKvokhbYF>QhhKUN?Dy=Vfwv=EVGWivoeK_uLagD]rVCtM?d?=s=GRQoYfIY;=UqoerMDsgILoS>EX<mCk_GFAY\\iEvIDggEAqehqgdWhSaSOUdAUBqeyMMco;gf]wO_slsGEaIlUtoegg?EaITkSu`QrwovxKy^keL]r]Gvj[WJshO_igADBwg]`vSmLLIsRHoCdLlHMFPxfpkLPr`]jmPOFQxvIkdLn>xVQysDtsEMptxvKaN``thlRl`qI`XiimSeJ:\\SiLU<mR]Hrb]xAmudTkWePQiuQ<lourv]tMtWUajiyo:pnW=PE=oLDQ^yL>`oG`jXtLplxs`mj`t[HODPnkDOtettHqVHKU=TglNadR^xUKYR;Xn;<YG`PtQYXTOlPtSDQNEliMw^dsvIpftPUikHal>LVqxkXxRShUC`TipvaAN;pssPOEQptlL^mrK=MKyTC`uc=mEdpR`u=Aqexjr@RkUPqXUGALn]l>HvcXKD@ycIv;QWKtSUmU_URB`kR`MBDlXTnbLnq=YXUnVPtutyO`Qx`JfmUbmQGxKlQmTQlLUMi=RZXmFeN`lXU<uQatxDYXqQLaVUXrdyUKMyAhkMQqTDjuqSTxpJTKBqv^QynlK]dXl@sXpogxR^qOXdvGxmYYWIMnf\\OYuueLM?iRneRwywcQRrPnFLPG=VRPO^iK]<RqpMYlquAxWpYwqKmufyfiq?^c@_DNnnYw@o]k>[^iuBfhQIyrNu@inlp[_Wfu^_Yp]EAal_y^ve_`bP^agga=AeRwaHAjQoeDOyV^e]oqUG`y?chw]=NxxO\\wVwZndk?bV>pPVjPYjDng@xc=qpQ_cOH\\Q>\\_f`Gfm^odnQ`>XZdWe\\GmFFvsPuapp=`lFVthi[Bya=NqqFxYYcmq\\pfsaGx;A\\DQldPwBPqMf\\UonsVuhHhX^fcNoQoceOkhIsti^;qtZGtOnxX?^vqdYPpjo]hPl<qc@VkTheGOwEokn@[Naiy^mBQj\\IrcOmOnk<ok]qt=qr=VoIV_d_jXvuqiowXw[ncxH_LY]tgpundPNeGpik>v[ivRFt_GuKnqFIw\\n[Dpht`vgIkZav_Q]NqvbAkmHqAqgUOq_H^ZXjE>c@ObIXjtnofGo?qq^gek>ut@iu`tp?l\\IpOhwCas\\aonfiIvm?PdqQgMNhVQjFgeDAdSFf\\VeDAghqlRG\\UFqMYepp^xvlPPv^G^e@dF?^L@mKIurQ^^yxKFiKnmXGj:IvRymv>lgOlr?q<htJI]lw[^quL@_\\Gfugp<?xGw[l_w?ieg_ijn\\D^\\eInBw[<gZnH\\oQ_FAbVnaFaaR`_>G`rHna^skP[@Fjhih>`bBpbgvvrAaBxp=Aqoqb@w_j^q;qa\\VuFq[@x[V@[?AgJ^[kxerPenhqfWafWy@qfmqvcwZkng[awq@ona^EG]KgrbY]_`]Nx^fGZrOxeA`RFh:@wFNjfhZ]Ncgiq=YddW\\b^m@@erFcgq[LQmRApeA[qAxA>rTGrBHmJpxoxayqqiagVO`cIn[>[^Xr@IhK>b:Ywm^koNxqg[X`g<QyC_gM`[x_fy>t\\@fbxb;A`h^hdQiKXufFnFH]Lhqchj^fyOeC[fcowYsvRqI;kW[;teGV^?TIUwW[etGDOmhlUf[ubfcyi]IMogiwH^Uh:owAwfWaRpmtCCu=IFdAe^OhsKrueeZESZUB^cinMIfwE_mfskxJsEn?fOac@?GJ=uXkDIiB:qYOWBUKR?=u?qgNath?vdGFP?u`Uyw]fCccUOtvUGYci?Cu=gXp]xIcIyOrAmyxYxIcUNKFpQDYYR;CwB=sPqiF?eF?C>UTYef>=RJMT\\EBCGUDihhKUR]vgAenUvZiCeibWkCcUBoqdoocwyCC_r@if;Ssqksw_DVoBuyY;yrdqfuSchAYAUgqSySkgZuWmkD=QcvYwY[bksrJsEaKWiWuM=u_gc:Eg_YfMYtnqHogRisEi=xpsXq]grIhrWFiECySU:mRe_xgEBs[C\\yRo?c:sifCgn[XHeDUgWKoiicEPYIqqHxkWwAgYchqgDjgI;aFpKDWQv_OBZYt>WwdAUYuRL;coyqNuVh]Uqptdik;pXdykxTQGlTQtvX`YV=q:@Vm@rfTrITlTyjYtxo@jnUSSHSs]q_uyq]sCeWNMuHmK>\\jW=lwdNZdoB`WspoPDLoImBUJaaXXmXPaLSeon=UuamQqmpEUWHsLTyvEXTeOiiYyiqy=qq]XHARH`M_IQc<YAenHyQTInyMlRYKKxkfEXyAmYyQT=pthy[duxAmCAmAyYwUosdyA`PkQkUtOyxr`IR^tmMLvePL;DYZenryvu<YGuNr`xkPqPxMrmmC=S@\\vayJkMxnutcav>@jVDv`lqdmL;iwfTkBeLqlUWQPfUXQXyZuLgmsOyQ=HV@uNxeQuumr@QOMykDMcLms=OH@sQIU=HQm]rxujluyGqpxYuNqUD]Yxar;avC`mH\\U;tP;lWkuQsIJmxMGMLvMNaYPS`mrYOhqYJ@SreQx]TbHOrHlcIm?EypIjnhlkEwgYnFtXohJSUslLrw<kq`XcXuHmS:IPPMW^\\neLNrEYTELkhyWDW:pv@PRNASAqoq`YLxUytXZMvr`TjlmkUq=hQPQRvdtSEXTYUuULq@YxAT?ySLmUVdNoQNV\\oKQwoUX=myJ@v^yrfIsJdOR=X>=WX@URHjyIVIuXILLS`nk\\LsdLJxVXpmjPrpyJ:YqgMSUurFXqeaWDxpW`Y:=RpPLLao?TQMDNl`PdYp<mX>yXXELE`wQmT>]QetyZEUExkr=R?yraIXTIYjuufinriK\\@ySqKExX^]x]`rX=R:LVhpPBlPeYY\\dLbUNT@PXeS^awZiwAukjXW<<medjwP`qweXNrkoh_Yo=hbkW]w`wE@eNWblwkhX\\h^rNobMQrywiKYkYy`yWt]ifhylt^hyAx\\x\\yHkTYkBxsk_yjAh=hv[_^yfua_d\\?ktxar>jowbYyZlQaPVmj^iwPq?`[AH^^Q`pAy`wdZ`cMpyvi^;FrmxqywtPpwWN`qqa[QdlXxJ_hwgZL_rHxkiGaZWq\\xqMHw`xvNAhI_pjwsL@pMWrx^ohqiZ?ohyw^xx<OtQW[cqlrFcZoa\\N]SWbKwq?hykwj=y]`_feNsf_Z[@i^xvxX`iv^w^fXWdQv`t?bqYvfV^qphkgnm_hOIlfn\\fynmpxP?yfvpuiepnhyfruyi?>qYo^rxuxAu;hxeia_wyYf[Iv]`Hiq?nHWxDy^IPZyhm?QaVojWapPgmnI_x`n_yim@`jyr?OyyW\\`xqA@uXNaHW\\?pdAykuveI@c;AqmIyCw\\eAf?OoWapPGxIHpxyjRfjyakN`gUFxaOcSvyYAl?qefnu;fvI`xtowHHrpWh^icYOitqy<pdyOmS>yin`\\ya=hntvo[gtmpyYVjkyiO?btFfuaxe_etyhlXZyyZ?xpAxy\\>kYFv;P[to`bIvl@n_ifMIa]P`Iyy`Pcw^xjyqEnks_b:ojdIi<fxpVyqQtKYryyabHsKv\\E?lYgvjouu>dw^etHyjX`s?yTqZUYuR^^X`Z?ykxoit?qn?sL`_;hlj?]^xdshgsyqmAoNw[w^yyN[Bv_<AgOftih`SIa=h_fpbx@uIAdIvlHV^f?bdYf@hwS__:vh^Iw_IoyPoVYbIv\\=V_J>p;FmhYeJ>xan]bYoxoitCV[bBKeqqroeGBCI=MHlQycIw[Qv;?cAaxJ=Rxay_EfXKYryy:oxvcdr=TVYCCuCw?X:IX;CIrIiAoEhSEtiWkqEt=w?tKw\\x<vjXniu^yAv]AYcNiedPgjD:;j^PNaLNQENjD5BUser-Friendly ElGamal Public-Key Encryption Scheme\251 Czeslaw Ko\305\223cielny 2003 University of Zielona G\363ra, Poland Institute of Control and Computation Engineering Academy of Management of Legnica, Poland Faculty of Computer Science c.koscielny@issi.uz.zgora.pl

This worksheet demonstrates the use of Maple to create an ElGamal encryption schemeThe following Maple techniques are highlighted:Developing a cryptosystemEncrypting a messageDecrypting a messageDemonstrating the power of the scheme

The progress on the discrete logarithm problem induces the users of basic ElGamal public key cryptosystem, working in a multiplicative group of NiMtJSNHRkc2IyUicEc=, to permanent increasing a prime modulus p in order to ensure the desired security. For long-term security, at least 2000-bit moduli should be used at present. Common system-wide parameters even larger key sizes needs, since computing the database of discrete logarithms for one particular p will discredit the secrecy of all private keys computed using p. Yet the task of finding necessary for encryption generator of a multiplicative group of NiMtJSNHRkc2IyUicEc= is unfeasible for an ordinary user if p > NiMqJCIiIyIlKz8= (p > 0NiMqJi0lJkZsb2F0RzYkIiQ6IiEiJCIiIiokIiM1IiQuJ0Yp), so, basic ElGamal cryptosystem cannot be named user-friendly. It is possible to relieve this inconvenience, forming such an ElGamal public key cryptosystem, which works in a multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw==, providing that to construct NiMtJSNHRkc2IyklInBHJSJtRw==, primitive polynomial NiMtJSJwRzYjJSJ4Rw== of degree m over NiMtJSNHRkc2IyUicEc= will be used. Since the root of primitive polynomial NiMtJSJwRzYjJSJ4Rw== is known, one can easily determine any generator of a multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw== and without any problem compute public and private key. With this end in view, the routines for multiplying and exponentiating in the multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw== and for key generation, encryption and decryption, have been implemented, as well. Next these procedures have been used for testing ElGamal cryptosystem and it then appeared that the cipher works even in such algebraic system, which awkwardly simulates the multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw==. Thus, the latter algebraic system has been shortly described, named spurious multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw==, and denoted as NiMtJSRTTUdHNiMpJSJwRyUibUc=. ElGamal encryption scheme, presented here, fully deserves the name uses-friendly, since it has an extended keyspace and its key-generation and encryption/decryption algorithms are exceedingly simple. Therefore, from this point of view, ElGamal cryptosystem seems to be much safer and more efficient than RSA encryption scheme. But this question should be carefully still examined.

It is known that basic ElGamal encryption scheme can be generalized to work in any finite cyclic group G. As is rightly stated in [1], the operations on elements of the multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw==, namely, multiplication and exponentiation, are easy to implement and the discrete logarithm problem in this group should be computationally infeasible. Therefore, ElGamal encryption scheme, based on the multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw==, is considered in this contribution. A concise description of a little modified algorithms, characterizing this scheme, is given below.Key generation: Each entity creates its public key and a corresponding private key. So each entity A ought to do the following:Choose a primitive polynomial NiMtJSJwRzYjJSJ4Rw== of degree m over GF(p) in order to construct a group, isomorphic to the multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw==. The group, having an order n = NiMsJiklInBHJSJtRyIiIkYnISIi, consists of the set G = {1, ..., NiMsJiklInBHJSJtRyIiIkYnISIi}, and of an operation of multiplication of elements from this set, which is performed by means of function P(x, y) , NiQlInhHLSUjaW5HNiQlInlHJSJHRw== . The function Pow(x, k), carrying out the operation of rising any element from G to a k-th power, NiMtJSNpbkc2JCUia0c3JCwmJSJuRyEiIiIiIkYrLCZGKUYrRitGKg==, is also defined. The generator of this group is p, since the polynomial NiMtJSJwRzYjJSJ4Rw== is primitive. Select a random integer NiMtJSNpbkc2JCUickc3JCIiIiwmJSJuR0YoIiIjISIi, such that NiMvNiQlInJHLCYlIm5HIiIiRighIiJGKA== , and compute the other generator NiMlJmFscGhhRw== = Pow(p, r).Choose a random integer NiMtJSNpbkc2JCUiYUclIkdH , and compute NiMlJWJldGFH = Pow(NiMlJmFscGhhRw== , a).A\342\200\231s public key is NiMlJmFscGhhRw== and NiMlJWJldGFH, together with NiMtJSJwRzYjJSJ4Rw== and functions P and Pow, if these three last parameters are not common to all entities. A\342\200\231s private key is a.Encryption: Entity B encrypts a message m for A, which A decrypts. Thus B should make the following steps:Obtain A\342\200\231s authentic public key NiMlJmFscGhhRw== , NiMlJWJldGFH , and NiMtJSJwRzYjJSJ4Rw== together with functions P and Pow, if these parameters are not common. Represent the message m as a number from the set G.Choose a random integer NiMtJSNpbkc2JCUia0c3JCIiIiwmJSJuR0YoRighIiI=.Compute NiMmJSJjRzYjIiIi = Pow(NiMlJmFscGhhRw==, k) and NiMmJSJjRzYjIiIj = P(m, Pow(NiMlJWJldGFH, k)).Send the ciphertext c = NiMmJSJjRzYjIiIi, NiMmJSJjRzYjIiIj to A.Decryption: To find plaintext m from the ciphertext c = NiMmJSJjRzYjIiIi, NiMmJSJjRzYjIiIj , A should perform the following operations: Use the private key a to compute g = Pow(NiMmJSJjRzYjIiIi, a) and then compute NiMpJSJnRywkIiIiISIi = Pow(NiQlImdHLCQiIiIhIiI=).Retrieve the plaintext by computing m = P( NiMpJSJnRywkIiIiISIi , NiMmJSJjRzYjIiIj) .Remarks on implementation: It is supposed that we will use primitive polynomials, but to determine an arbitrary primitive polynomial of higher degree is not so simple, even for Maple. There exist a few useful tables, for example [2], where about 600 primitive polynomials of degree from 311 to 3604 are listed. It is a pity that is difficult to find similar tables for p odd. Nevertheless, one may easily determine, using Maple, random irreducible polynomials of degree m over NiMtJSNHRkc2IyUicEc=, with p odd, satisfying NiMtJSNpbkc2JCklInBHJSJtRzckKiQiIiMiJSs1KiRGKyIlK10= , and work with ElGamal encryption scheme based on the subgroup of the multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw==. Furthermore, Maple provides very efficient arithmetic in the domain of univariate polynomials over the integers modulo p, allowing without difficulty, to implement functions P and Pow for multiplying and exponentiating elements belonging to the multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw==.

To use ElGamal encryption scheme based on a multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw==, the following procedures have been implemented in Maple interpreter and written into the file ufelgp.m: ElGinit := proc(pn::prime, ip::polynom) ... end proc: returns two global variables of small size, namely, a prime p, equal to the actual parameter pn, and a NiMlJ3pwcG9seUc=-type irreducible polynomial ext (converted actual parameter ip), needed for proper operation of the procedures, listed below, P := proc(a, b::nonnegint) ... end proc: returns the product of two arbitrary elements a and b of the multiplicative group of GF(NiMpJSJwRyUibUc=), Pow := proc(a::nonnegint, k::integer) ... end proc: returns k-th power of an element a belonging to the multiplicative group of GF(NiMpJSJwRyUibUc=), ElGmggfkey := proc(p, m::posint) ... end proc: by means of this procedure the user can compute its private and public key, ElGmggfE := proc(m::posint) ... end proc: enciphering procedure, converting any integer m, representing a plaintext, to a, cryptogram, consisting of two integers the NiQmJSJjRzYjIiIiJkYkNiMiIiM=, ElGmggfD := proc(c1, c2::posint) ... end proc: deciphering procedure, which regains the original plaintext m contained in the cryptogram NiQmJSJjRzYjIiIiJkYkNiMiIiM=, ElGs2blE := proc(m::string) ... end proc: more application-oriented procedure than ElGmggfE; it converts an arbitrary message m, in form of a string of ASCII characters, to two lists of bytes NiQmJSJjRzYjIiIiJkYkNiMiIiM=, ElGbl2sD := proc(c1, c2::list) ... end proc: deciphers cryptogram, generated by the procedure ElGs2blE, and returns the string of plaintext. Two last procedures may immediately be used by Maple community in transmitting rather short e-mail messages (up to several kBytes) with fine privacy. It is obvious that above procedures can easily be adapted to enciphering/deciphering files of arbitrary sizes. otinsmg := proc(p::prime, pm::polynom) ... end proc:if one wants to construct ElGamal cryptosystem working in NiMtJSRTTUdHNiMpJSJwRyUibUc=, one ought to call the procedure ElGinit with actual parameters p and pm, where pm denotes an arbitrary polynomial of degree m over NiMtJSNHRkc2IyUicEc=, and to realize key generation, encryption and decryption using all procedures previously listed. To familiarize the reader with the spurious multiplicative group of GF(NiMpJSJwRyUibUc=), procedure otinsmg, which returns a square matrix, representing the interior of table of operation in an NiMtJSRTTUdHNiMpJSJwRyUibUc=, and a set of its invertible elements as well, has been implemented. edtest := proc(dg, ks::posint, pp::prime, px::polynom) ... end proc: an auxiliary procedure, making experimenting with ElGamal cipher easier.

For all prime p, for any positive integer m and for any polynomial NiMtJSJmRzYjJSJ4Rw== of degree NiMlIm1H over GF(p) there exists an algebraic system <Gx, .>, consisting of the set Gx of all NiMsJiklInBHJSJtRyIiIkYnISIi non-zero polynomials of degree NiMxJSNkZ0csJiUibUciIiJGJyEiIg== over GF(p) and of an operation of multiplication of these polynomials modulo polynomial NiMtJSJmRzYjJSJ4Rw==. Such an algebraic system is a generalization of the multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw==, therefore, will be called the spurious multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw== and denoted as NiMtJSRTTUdHNiMpJSJwRyUibUc=. The NiMtJSRTTUdHNiMpJSJwRyUibUc=, applied in the presented application, is obtained using the mapping NiMlJnNpZ21hRw==, defined by the function NiMvLSUmc2lnbWFHNiMtJSJ2RzYjJSJ4Ry1GKDYjJSJwRw== , converting a polynomial NiMtJSJ2RzYjJSJ4Rw==, belonging to the set Gx, to a number from the set G (see second section of this worksheet). It is then clear that multiplication in NiMtJSRTTUdHNiMpJSJwRyUibUc= can be performed by the same routines as in the multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw==, but the interior of multiplication table of NiMtJSRTTUdHNiMpJSJwRyUibUc= is not a Latin square. In principle, NiMtJSRTTUdHNiMpJSJwRyUibUc= is a commutative groupoid, in which the operation may not be closed, nor fully associative, since divisors of 0 exist if NiMtJSJmRzYjJSJ4Rw== is not irreducible. The existence of such groupoids, which seem to be important for cryptography, have been noticed by the author recently (maybe other cryptographers experimented with similar algebraic systems), thus, all their properties are not yet known to him. However, using ElGamal cipher based on NiMtJSRTTUdHNiMpJSJwRyUibUc=, we must remember, that there exist non-zero elements belonging to this groupoid, which have not their multiplicative inverses. The set of all invertible elements form an abelian group under multiplication (function P), generally not cyclic. It is then easy to observe that any message can be encrypted by means of ElGamal cipher based on NiMtJSRTTUdHNiMpJSJwRyUibUc=, but the sender, after the cryptogram NiQvJSJjRyZGJDYjIiIiJkYkNiMiIiM= has been generated, should verify if NiMmJSJjRzYjIiIi is invertible, that is, if Pow(NiQmJSJjRzYjIiIiLCRGJiEiIg==) exists and can be computed. It is condition sine qua non of successful deciphering. In any case, NiMtJSRTTUdHNiMpJSJwRyUibUc= will work in this application and the reader may generate and observe its table of operation and see the set of invertible elements using the procedure otinsmg. These are several examples of small NiMtJSRTTUdHNiMpJSJwRyUibUc='s:

restart; read "ufelgp.m";otinsmg(2, x^3 + 1), otinsmg(2, x^3), otinsmg(2, x^3 + x + 1); otinsmg(2, x^3 + x^2 + x + 1);

otinsmg(5, x^2 + 2*x + 1); otinsmg(5, x^2);

otinsmg(2, x^4 + x^2 + x + 1); otinsmg(2, x^4 + x^3 + x^2 + x + 1);

If either a set of invertible elements or a table of operation in a tested NiMtJSRTTUdHNiMpJSJwRyUibUc= is needed, it suffices to execute, for exampleotinsmg(2,x^8 + x^2 + x + 1)[2]; otinsmg(3,2*x^3 + x^2)[1]; Further experiments with NiMtJSRTTUdHNiMpJSJwRyUibUc= are left to the reader.

Now we are ready to test ElGamal cryptosystem, based on a multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw==, on its subgroups, or on a NiMtJSRTTUdHNiMpJSJwRyUibUc=. restart; read "ufelgp.m": Taking from [2] the primitive polynomial of degree 3036 over GF(2) one can construct and test ElGamal cryptosystem working in the multiplicative group of GF(NiMqJCIiIyIlT0k=), by executing the following statements:dg := 3036: ks := dg: pp := 2: px := x^3036 + x^1087 + 1: edtest(dg, ks, pp, px);It is also possible to determine, using Maple, random irreducible polynomial px of degree dg over GF(pp) with pp odd, of much smaller degree than in the first example, such that NiMpJSNwcEclI2RnRw== will be equal to about NiMqJCIiIyIlM0k= . In this case the order of a root of px is not known and ElGamal cryptosystem will probably work in a subgroup of the multiplicative group of GF(NiMpJSNwcEclI2RnRw==). The test, of course, will be satisfactory.dg := 188: pp := 46327: ks := dg*length(convert(pp, binary)): px := Randprime(dg, x) mod pp: edtest(dg, ks, pp, px); The system works faster than in previous example, therefore, by increasing pp and keeping roughly the same key size we will fobtain the shorter times of enciphering/deciphering:dg := 29: pp := 162259276829213363391578010288127: ks := dg*length(convert(pp, binary)): px := Randprime(dg, x) mod pp: edtest(dg, ks, pp, px);Finally, an arbitrary polynomial NiMvJSNweEcsLiokJSJ4RyIiJiIiIiokRiciIiVGKSomIiM3RikqJEYnIiIkRilGKSomIiIjRikqJEYnRjFGKUYpKiZGLUYpRidGKUYpIiM2Rik= of degree 5 over NiMtJSNHRkc2IyUjcHBH with NiMqJCUjcHBHIiIm approximately equal to NiMqJCIiIyIlTkk= has been chosen. The execution of the following statements shows that px is not irreducible pp := 531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127: px := x^5 + x^4 + 12*x^3 + 2*x^2 + 12*x + 11:Factor(x^5 + x^4 + 12*x^3 + 2*x^2 + 12*x + 11) mod pp;Under the current circumstances ElGamal cryptosystem working in SMG(NiMqJCUjcHBHIiIm) will be tested:dg := 5: ks := dg*length(convert(pp, binary)): edtest(dg, ks, pp, px); If you find this worksheet interesting, drop me a line, please.This message contains 63 bytes and to encipher it using ElGamal cipher we need a key of size ks in bits, satisfying NiMxIiQvJiUja3NH. Thus, we can take from table [2], e.g. the primitive pentagonal of degree 515 over NiMtJSNHRkc2IyIiIw==, and execute the following statements:restart; read "ufelgp.m": pp := 2: dg := 515: mess := "If you find this worksheet interesting, drop me a line, please.": px:=x^515 +x^418 + x^328 + x^123 + 1: ElGinit(pp, px): k := ElGmggfkey(pp, dg): n := k[1]; alpha := k[2]; beta := k[3]; a := k[4]; c := ElGs2blE(mess); cryptogram:=convert(subs(0=1,c[1]),bytes),convert(subs(0=1,c[2]),bytes); md:=ElGbl2sD(c[1], c[2]);If tables of primitive polynomials are inaccessible, we may randomly select a polynomial and use ElGamal cryptosytem, working, for example in NiMtJSRTTUdHNiMqJCIiIyIkOiY=, executing statements:restart; mess := "If you find this worksheet interesting, drop me a line, please.": read "ufelgp.m": pp := 2: dg:=515: px :=x^515 + x^2 + x + 1: ElGinit(pp, px): k := ElGmggfkey(pp, dg): n := k[1]; alpha := k[2]; beta := k[3]; a := k[4]; c := ElGs2blE(mess); cryptogram:=convert(subs(0=1,c[1]),bytes),convert(subs(0=1,c[2]),bytes); md := ElGbl2sD(c[1], c[2]);Let us see yet, the result of executing the statement:Factor(px) mod 2;At last, we may use Maple to find random irreducible polynomial px of degree 83 over GF(83) and use it to construct a cryptosystem as follows:restart; mess := "If you find this worksheet interesting, drop me a line, please.": read "ufelgp.m": pp := 83: dg := 83: px := Randprime(dg, x) mod pp: ElGinit(pp, px): k := ElGmggfkey(pp, dg): n := k[1]; alpha := k[2]; beta := k[3]; a := k[4]; c := ElGs2blE(mess); cryptogram:=convert(subs(0=1,c[1]),bytes),convert(subs(0=1,c[2]),bytes); md := ElGbl2sD(c[1], c[2]); We can, of course, easily construct ElGamal cryptosystem for practical use with satisfactory key size, much greater than about 500 bits, in many different ways, making use of the approach presented here. In the end of cryptographic experiments let us suppose that the user A, who has his private key a, saved on the disk in the file named "a", has received the cryptogram c, generated by means of the routine ElGs2blE and saved in the file "c". Besides, he knows that his private key of length 5208 bits has been calculated using the prime pp and the polynomial px, stored in files "pp" and "px", respectively. Let the reader decipher this cryptogram using the procedure ElGbl2sD.

It has been shown that the ElGamal encryption scheme, based on a multiplicative group of NiMtJSNHRkc2IyklInBHJSJtRw==, on its subgroups and on NiMtJSRTTUdHNiMpJSJwRyUibUc= are user-friendly, because in such scheme the key-generation and encryption/decryption algorithms are simple and effective, even if a key size of order up to 10000 bits or more is needed. Particularly interesting seems to be the use of NiMtJSRTTUdHNiMpJSJwRyUibUc= which makes the cryptosytem partly unpredictable and expands its keyspace. However, the problem, presented here, is quite new and far from being fully discussed.

[1] Menezes A. J., van Oorschot P. C., Vanstone S. A.: Handbook af Applied Cryptography, CRC Press, 1997 [2] Zivkovic M.: Table of Primitive Binary Polynomials. II, Mathematics of Computation, Vol. 63, No. 207, July1994, pp. 301 - 306

Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material.. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities. MFNWtKUb<ob<R=MDLCdNVZZJ:@L>H:TKGxMkJ:<O`Lo\\lQxlQWdMWpsHqShmWhYoeXOPmTPmV`mvqyxq=Xj=xXquXaxnaXcEWc=UR=UweYwELKDLqtPq<R:=r^av^uRAurZ@nZtVauVb=WbMYtMyvayvYyuYYxmYxqyxqYyuYyEYsEYpmXpyyyyypqxp=J:>::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::dy<TypC>qULCTJcDXoXusT<aupkcfWMX@JCeU`dNuTmWxyyyppuPCDSSuLClu><xTpQmlsb]MihUO`qTeXSQO;@JxV]wOl:@syFv<w\\t@tsNnQn\\V?w<w\\?FqJijXynZVvnyHErmiB__tWit[MyxYRIIXvWgtSS=;gQMwAIC]IYrGXRogc[EpqYtsxn=BVSUGuEA[WxKrWaSHssoYBPkynKctqgmyUKAYQYUw_rs=wboYTWXI?IQKyo[X@wydqytYRGAy`ixs[SlyXaSyquy:mel=dXqydIfvgRIeSUkUmUBGwuZitS;eQ?S>AdMasnkySGbDSuimbSabjytNAyMuXlaTWaCp;y?at;_txaTwath?cj=GbgYVGCA[eAkh^ihyaIGoVdGxyWeQatamVHYx:SEIewyacmcSBAvgOyyssEyBVWCwQFtYWxYdMgcY_y^Uy?gce[WXQCDcwGuwHMw?qwx[gacscGrwOtuKFXKsc[FZIBOqIrII]kuICfRosM_yTSEWWcKQs_qGHeIiaWBsvaAXWoFsYTyuIYSdWCet[fZpOYtv[\\XSMvN=Xhluxel]ylvUn;PYsqvkmmCxSEQPsMOeUpQEKN`yVAqcqRQpYxHr[xU\\AtgPVexmHHQYDXptL;ey_\\XHxyTpLQ=qJhJklqA=wPxqOtpPmwQ=kWdSSYjxhQt=li<X=Pr\\HoxMKxppdUPGxl`<RadWsEMUhnMinaqvy\\t]pJw\\Pttt:lw_hy;PxuElWpfypiQyg<IbgHqQ?wRwvFgcQnmtI]lXZoauvw\\]Vi\\?yuIjGqyA_]j^cia\\^vaYfmXYvV_foyd_wZa?yIPfNXpOimbInwiieQyZ@[jf[p_`s?\\N@qaw[<a_=qpdIu]>gnHpUi\\^a[AGcS_y]pnHg_oIi=XkM`bK^yUWjFhhCpif?llhelhkKqk=qgCqqIokJadZ@]IOspHjgQgUv^Mp^[akXNokxcFaxMX>Efx=GJyY]=uKWXuefcYCV_DO;X]oeDwI]UrhIXhKdtYgv=sYMxyMhEAbdKdFED;MBimUYgvNsfBuDgqw^sRZoieyiYEfEAsYOcU;uf_C^;g>EIUmWy]xZ[H?UTiwhayb<EWUAhmghUee]ODLyfkYdOQDNMsleg]mHGkynUrrUhjgbvstrICsOiU?upUhtME_cVUeywWrSeSvIwHqsEUvwaS`mv_kCEgDEEVOoyfSFYGXh[xe;wfsya?Hbcu_SiHUfrStqsgICUKmR;IEGGiEUxSSewkBRcic?f[GHs]WBCeFSXMec@qwQYiOCFi;bd_epghCcrSIbrUFfKXpOE>CdGUVH_ss=GaEF\\Mh_uDJcXeWGSkIA=T`[uhOiKOy;Ido_sBQgPGbiMxZIx[=RNQHCUwlIhVAs>Mxv=t;Iekec[iToeB]YSVsI]UGkMgC=xM_cv]rCkGlOyE=wVsymoRPERGUWoKs>?dNGcqOvL=DcgUUid=SdBYtacBcyT;sC??sXsBFEIPKdwUibUUuowtCxLERxGUPOc=eeWWDJ_tBIFj[RMWXoaIniFDYyvIfFYH;EifaWAAdkQgSuIoYHS?s\\aYnkYcCRXAy;=urSsUEGXovmkdU?bIkuvIhf;hHKRmsIqkGkCIEGSQiUy?r[chy]DW?UJweo_HI;I[iRPuYCce]yIQGSR=SFcY@IHNabEyhT;H\\gC[iiEubXIY[?FhkfAaRyccQ;D<MBLksUGvM]FOSWZaFnmUVOB]Mh`gu]ew:CSX[VU[d^iWCITMkingVmcY;EuIkFZgetaSlkeD_SlUd?SU[Wh`_IHkuNaIBEY@KhQ[IbSfl_CpgV]IBgcf:CrOWWliVPSDMuEkwBYQbgKxGiWfcdg_cCoXDyFoAF<CYd_fZSUKOXmUErmvpWgaQIeWGyMiuOfheFY[UWgdGwe[;X@Yh<owskTwUgjYdvEhnTP`LJatUmyo]xlkUpgPSHmSOiSXtM?HsHhWglnu=ypMosmPWQtXmlLDR^erappAPq@Twu\\mf<ytMo_tNQDmwuUBal[TKM]UZ\\VsUPg\\OhXU]iw>lT>TtolYUeM\\`q:iNFQkMeuB<Y^yq[TqwLxyYk^mPDhUTEL[mxdYTrUwHYpp`R]tsyhm<\\rdhN\\]VGejEyTBLlXhUidSklVcImkuJA\\OFAJxXTJ\\oRpUr\\qnEUf<POaocioXxYUTRxhmKHnoUuBavvxt]@ordyqIl`tycEyg=St<V;LY`DoDElChWYdkpIkSMophnhqkeMW<QX^dogEmM<kxAYM=mpPKmTTMmXeQLnuK?HMeIU``TqMSdeNqmxHeLK=OUpx^@kiYp`xXVdoU@L=PprAPIuR[Qp@YlvPWwQToMpG`jOXyFhxAETieRADKgioVPOyXUlXT:Iwc<NgeMNup\\XWrdQFPQvlP=Toseo>qXbiWO\\yE=PUiPAASgLtxXLG=STASAxj=@WixwX`XOAtHloIeoHiLvyuouMtLtTyJsAxBXr@TqWXOsEKopuAEU<uyO\\LTyPAXm=tOUQneaND]KOYyLyXbtxuhmcYrXMkh\\ylLo_eq`tSeAOH]lqUwiPnkPwlHPgHrehY^pKhPwGPJ;<O<`qU=tMxUUEPW@RdITfYjjaowTqMQjXHJS\\M<EvappT@mWMJ@iOVhyLQKq]T=Eyc=UhqNa]PJ\\X\\Lu[DsQ@O[XRw<Rb`P`tSuejceYX@UN=rFexuHmDmk]XRLaYElRmIP]Pech`rxma?araaCxvWQ[\\aZ`yiFAj?gvVVd^@mGy[hhjxQvjIwMVwPGyXW_EpjDNnsy^EhvE_d:PnkOaDA^CnxEAoCh_ewc;pb[I[ZwcU?kpGwxvcVV\\OWaYGZWqbGG^jVkAQ]mXckfwTVfovZVnZLwfoIeS>e@HtcvsgPn<YqDOxcqbdNmPxtqwhsfag>myOedhqCFkNWqspy]@_VQrIIu]ncLIb>_xdQ^[yw^`^YqbSxeyga>OkV@fpVfeNhmxeSwn^?_GOklf`QqgK_yK?yj@pxvwbHtI`yYai?HvJ^wvQvYngAVo=XhwcReBIMflKTU_b`qrFQC<UGRWY=kVWAiv]X<CSyMycyweoE>?ttksVgBTmtGIXvKDT;D`atpaGQEVA=efoH@]TgswsCfWGEbCCLIYtSwG;tRaC?]hi[TfwSPUcSQYZCuloE[KTnOSTuDPqfpQU_Yx[?UZ=b`yCuETUectcrsaWIGhPUVdCXo[Dn;GTof=AVBcYRGgaaYbsvt=UBuVIOeZKgGmhHQr]]umsifyTPWtneyZKydmHjoWRAsSQHewDS=Hj]C>qdH[XHIgkwTGuvI_sgYDgabSsiLYrb]Ic[uZUuCeGN]InyyjiVnMuJibq]E>=sH[thQDXgT\\qhNwTVmGdoSiKsD]DD]UOksO=fX;XvIdbUwRiisCEv?tEAS?eH[EHiOy[mcE?hY;ewKCr[x;ECpUEaItRMUeMI@wF=GuqIdriXmAiHouB]UEkvboD`]bDeu^UHOsxwKSogVE_GNQbBAduMYQ;Y_]XbqBe[FFYGF=tXgxryYpAFDoidIRHgUf?uXGg]WguGig]URQrp;u=MHYIXxcIamsqEl<uR<PMwtwNMqNYMB?\\aIiqvboxhknwDOv]^r:a\\[WhExsn_cdQo@Ng]orLPnCptE?wJqi:ad`?gjX\\Bol:@dJis[vel^pK>]TpcIHhoSZoXJOhw[WgsesuBfEg]=uuUY=qXZWVYMSZECHWHqeX<Su^EuvYX;AFQQC]]Fl]SNqIO=ILQwhIwZoeqEoOqVY@TTprWANqYsuxNA@WjlpuaXytmXMRkdpI]K\\LT@=Pd\\SxHJSXNhulFYQmtwJhWI<QsuRUpwm\\rQDLyuMgMv>@pS@pftRiUniTV:uRRil<lRY<wltSViLhHKD@vViS`DOfaTvAsyMuKmQUhvqlQuLW@qlr`RddRKIm^QYAaXxdP\\TuVlktMYmyPA`xRivRUoLxKmANalL`qV`eTDIO;MY\\HoQiYnMkHLNqhylUJ\\tS^uKJIMKAY[qufMrxAXfxJyXxe`RPqxOiorlJW]XEHXw\\lJqr=XwN<T>`nFPklHv^LTd]kviu:YwlhWkTyDpLSUVUqQCAuTTliPopuoTHNSQyRts>IqKYKhTNQMseAjoalrQvbIslMp=\\ojLUMDuDQymaoiQulmPMELwhpuplnIvypP`XlCDM>LY@`rdqtoyn@MLFTUUPo\\UWR\\WMetOAoEewLIUctRw@t]ERG@XtqKuHQWqjWLqZ`LTUOTusmHPcYk?DN=uT\\aXSeLNuKrttf@kIunUTXCMtYyRUQplXw`Xv=iXppuLmRUqwTMm[]qxhLElt>lNi@qQ=Q_lRL<NgerhhXwAryAL=iw]IxYTUyhj;poqXPmUgHG\\ganfWfF>hrAwtwy[Ys<VuGXhSGxePjM^exn\\vabHNjTffFYwDNre@qoheHWmoW`]P\\gfq]Ikxx\\?vknnc\\giupovIhMaZOIkjIdVqtv?efnhe`i=OixVueVopxjJOuNY`[W\\jX\\SNkeqrQ_pUghjNiNQtpG\\CIe_IabYs@wwBw\\L`xO?r`qZi?c@WsW`^@fjogeppjkIpnXkKPndGadGidocE>m?Fjf_bYf\\\\?p]HieNqWggeIuCAnhiZwaepYnkgeFyjvOhu_[GQkpioSNa?ndiprUFjcV\\pQngw]R?]WFeWx`>i_H@tAwdbny<x__O`FyggqujAtJhaiAnSAs=xwtp^aYnloln?eYQtA^mJvwD?k\\Ql]xqMPc`_sjV]gvreOsIOkpP^Vy^[Vw`O[gwmLqi]NmZ@hBAriP]O>[@HdmYZyir[Nn<YpeNfonso^]dnfIYuXwkEAcUyn^A`]VeyYulPogAn;?\\K?mt^gp^jXGxf>ysfZsgu=`seb_aIESSJcWewtmCrECfgERaqENChB;f^IvxYL=PS]=yKXmGeMYLmrTSBpL_`UAlmXmXlUTXEn^EsSmmfyREXsDEwelvQqlQaX@@tj<pkTYkDSNqxPQjlusiTJELXQ\\Rw`sPaSUYJwPjdes_QsK`j@Ij_DuFmJmPLmllh<SSPKV<W[eOaaTN@wLltv=qd@OOHrc<K>huhPP=ApSURP]mbIVSurlDLqpKuaVliV>IoOxJxLyGXOhqt=QPBQVItRjdV?]PFPPCyvs]YB]RXAsPLysQT^MuLUODMueDP=UPpHsFUx:XJ`hNlEYKykqQLQHSEur^aX_XJH]UyxtgMRCXtjuo?EQWML[aRSikidoeLsUduWEMthYZyQ[qwxHT[tOu<VGxqb`qp<OQAWOeYIIw^Tv`HrNyP;EKhDLiTqcXLq<NXejsEKseT;MYA<osmuf@U@txUMJYaMFuvVajUelv]xX`ncuThTxB\\wxtvCiu@HsQUQ:msJyUVXLOeUALmdaY]TMouqEExW`xK=QQLyGAyiHP\\xOf]tG>cJw`gxw^f]mIdJwgXiybX]_^\\]x]wXoovfJ`vgQklWrhq`sxqThd_AuXHotauxqvVPs>fXQEG_YGyujGWqaCOyE>WX[wuEwysMHsACawYfsIiqvWiWpWGoGYmqwAeh;_XqGSy[YQUW<kFaUGmuhqeYE;xdwbDUDdWV<OYjmwc]rL?TpuwF_snWumiiaAInyB[aUbyx\\yy`cSLmHxsInwYLwf=ob_ktxgUJWTB]TtIvKkDDMICMVZCH<WWF;vXeuOGe^QeLwik]HkCfrUXu_DgoC[OIyuh_Iyb[eEhqryQ?MwTexIuNbumv<sOiwy]uO>ie?oNXpnFb]iykyv@pnM?^bQbcOp]@pM_wOIZ\\i]tVpGIu=PdbHfMxcxXat?aWPZsww>xaDvv<wqQvyk^piAr_@fdYyfoxsactW_uvgBPmqvmK_ZMArZWZyAvCPmuYd\\AbZp]ZNgXwryXaxva>wfYpcZgem>uxiu[GiYnuwQu<aiJns?\\UNpqHgjfwhq[bahb@xCGbHVkk_nTPeiobfycUf`XnaxidlwiTHjmheF?sw>qWXxTWygQbupZtYpgqpkwwfWvcHZcAw[iuMiyb^mEfyh_yyXsIIosXdJfxvq]>yaR_ZVxy\\bS?EbAws]w]wvcOFoMhwSURagyCYdiTwABuAEGWFuSIGoEkKYIGFYUY]uw`uwXoGuAFVWkGwqyfb@qrrifj?sYpu=@_]on=g[Q@ltQbQNZDf\\FWe\\yquw[<pu^>lvQx\\Yw<w\\<VxRPn=yxiN[CNgB^irOpwGnEfyyWntqw:gwEfZSpi_G\\<?`QnxV?wygm<NZ^qyaGpxxiMpk_OhqYrWx\\t@t?@vAA\\eq_rQqv>uy@tya`Wyy:xvmysXwyYf[MWxoWmIgvoE:;B:MTKWDKWgJ;eZ1: