e¯twlmn ¨ twhPtw ifNnr¹ - mimuw.edu.plagnese/mikro/MIKROSKRYPT3.pdf · 12$ 6 "-6
Transcript of e¯twlmn ¨ twhPtw ifNnr¹ - mimuw.edu.plagnese/mikro/MIKROSKRYPT3.pdf · 12$ 6 "-6
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ò�tw|&��tw�&tw~^}r� ¢Oj�y�lon¤}�l�{u|fNj�vxjOhS}�lu¨ ªovxfNªm�Nj��&p��&t�{uh&fijw�¯¾���}�S}�vOjxhPtXq&smp��&�.|{o�«S}���tXnOj+�smp�yLt.��h&fijXprq ��twvOt$��prS}�lofi©X�rp�yÞpr�r�r�ij�q&smp��.�&|wp4yLtw£w�;�r�.}$S},hPt�{upw~�fNtwlm{�vOjxhPt¡|S{u�rsmjO�rpwª>nvxnx}$h&h&fi|w�4y¸S}P��t>nxjOsmp�yLt.��~�pr�w�NfNS}$ªo~�}��rp^nx�&¢x}yLtw£_hPtw�&~�fNjxsoh&fij�î�~�twsoh&p�{usut�y�fi£w�hPt^vxp>q�pR�n¤y§tw�Nt¯hPtw~Cn�t��pr¢xjOh&fijLp9~�pr¢O�ifiy?prªovxf~Xtwsmh&pw{osut�y�lm{IyLtSð¤�w�r�.}$¢§h&fivLS}�{op9hPtwl�h&fNj?|wprlmnx{op�yLt$�Ep&���tw|�kmjxlm{�n�q�p4y�fNj¤{usmnOjO~ * q prh&fijxyLtw¢�h&fiv�h&fNj�|�prlon¤{u�Rkmj���~�pr¢Ojx~^}G�wpÛnx�&¢x}yLtw£�yÜq&smp�vxjOlofijq&smp��.�&|wv¤}4kmhS}$~ fN�ij^v��&vxjO~^}"î�h&q��>nOtwh&fNjxvOn¤}�lmnOvxn�tw£�ð¤�6y�fN©Ov>{utw|¡h&twq&sut�y��&©�h&fij�&fNjxsonxjO~^}¡�rpq p��a�SyLtw�r©>y°prq&fNlmfNj^{ojOv��&h.pr�Npr�wfNf���Î�p4y�lm{�t4ko¨'{utw|�n¤y§t�h&jGÄN��Ä�áI%±É�ìRĤë�Å0ê)%�ÂoìOÅ0Ä * |S{uprª9{osutwvxfhPt�hPtwlmnx}$~�hPtw�&~�fNjxsohS}$~ón�twh&fijOvxnx}$lonxvOn�t�h&fN����y�fN©xv�q pRkot�y�fEt�lofi©�q.sojOl�kot.��S}�nx~>�&lmfN£�hPtwlÌ�&pq���twvOjxh&fEtÌn�t�{ujLlmnO|�p��.} * h.q�� qPt�ü&lm{Iy?p¯y�q&smp�yLtw�&nxf.prq ��t�{o©?nOt¯jO~�fNl�km©cnOtwh&fijOvOn¤}$lonOvxnOjxü��q&sonxjOnvxp^�&pw{I}$v��&vxn�twlL�jxnOq���t�{uhS}�vOn¤}$h&h&fN|��&n¤}$lo|Rt�vxjOh&©gh&fNjxnOjOsmp4y§¨.����tw|^y�fi©Ov�yø|wpwü&vO��y�lonx}$l�{u|fNjvxjOhS}Ûlu¨��&p$�Pt�{uh.fNjw�
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plu¨��.p��PtR{uh&fijw�6~Xt�|$lmfN~>�.~ßkmjOl�{^q&smnx}�km~�p�yLtwh&j�hPt'&sonxjO�r�¼î�p
fi�Njckmjxlm{gprlofN¨$�St��Nh&j�ð¤�$vOnx}$�if�{�tw~Û�.�r�.nOfNjF (y) = 0
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Lc(2, 2, 105) = 200
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min z≥0,f(z)≥y
zi=zi �L��� i∈Φ
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m�aiAZ�q]>[4YEkD(ÃcS : IntRn−1
+ ×R+ → R ∪ {+∞} mya@;IY_lg�IJ c(w, y) = min
z≥0, f(z)≥yzi=zi ����� i∈Φ
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2
Lpl = 1
�~Ú�]>[4YEkD);b��A)�tS�a�]aiAd(p) = 400− 100p
�³>;>Yf§:;Va@P�Ah��A/=�LNEMFvS��!m�P6Y�W)sa:Y�L�^áA/Y�W)JG\�L:J`u
K = KmLF(^gL:JG[4[�J w �q]_[4Y�EOD);�YA$\�F,ais>PÎ^�;��A/\ra#;��
c(y) = 16
+ miny=K
16 L
13
K=K
12K + L = 1
6+ 1
2K + y3K−
12m°;ÁP®eV§q]>njL�^ A/Y�W)JM\XL:J
c(y) =
16
+ miny=K
16L
13
12K + L = 1
6+ 3
2y2 �
� W)s>P�[�A>P�;je�FIJbe4§i]>nVAA/Y�W)JM\vA>PyJkDbE!JM[p;c[p;�WiS4[VY�]¨W)s>P�[p;ZD(JG\raminy≥0
c(y)y
= 1m8Y_;I�Iet;
Fc�VW�^ ��W)Ae�]_Y�]�D(Jy = 1
3
m���WXFvSB[p;>Y4§:;jet;�E|{L = 1
9
LK = 1
9
��»yA$[4L:JrP�;>�Q��A)�tS�a�PTSV[�A/\XLd(1) = 300
mUP�LNÃ(Ex[p;QWqSV[4Y�]�D(JM\ra900
;>Y�a�S�P�[_S�E|{f�4WX^��
�������������� �������� Ñw´
·�JGWM;>FhFIAt?_;�EMF!^~S,mTE(Ab\XL:Ã}\(a@;>[VL:J(m_D(JMKX=�L�e4A�^gLO;I\ra@;�\@��W)A>P�;je�F$l�\XL:Ã}[�A_P�LT^�LNJM\XF(Y>;>Â�E|S,m�E(AF>P�L:ÃMY�\XFvSc��Av�VS�a�e4A
d′(p) = 750− 150p�
� ?,;IW)e�F/A}Y�W)sa:Y�L�^ A/Y�WvJM\�LNJ���Aet;>�UD(JM\(a300
m�P�L:Ã!E300 = 750−150p
m�P�L:Ã(E�E(JG[p;�]>\ra@;I=�L\XL:Ã2[p;
p′ = 3���ìY�W)sa:YL�^ÇA/Y�W)JG\�LNJ�^`A/�([�;�F>P�L:ÃMY�\XFvS�����WvAte�]>YEOD(Ã�F_P�LNÃMY�\�Fr;!D)l�E�Fr;Va¯W(]e�[4LNJM[4LNJ
��WG;�EXS��º}W)sa:YAA/Y�W)JM\vA>P�;��q]>[4YEkD);�YA/\XF,ais_P½^�;b��A/\(a@;��
c(y) = 29
+ 3y3,; P�LNÃ(E�Y_;I�Iet;¨�4W�^�;
D(JG\ra8\�Y4§#A$[4[p;�PTSj��W)Ae�]_YA>P2;��Argmaxy≥0 py−c(y)
m4EGFvSV=�Ly = 1
3p
12���a#l�eh�q]_[4YEkD);���Aet;>�vS
n;j§#ÃMF!L~aqAs(p) = 300p
12�Ç� Wvs>P�[�A>P�;�ejFIJ`Y�Wvsa¯Y�AtA/Y�WvJM\)A>P�JOD
750 − 150p = 300p12 ,E(A
PTSVF([�;�EMFr;¨E(JG[�Ãp′′ = 7− 2
√6�
� K�WvJ(e�[4L�^ A$YWvJM\XL:J`[�;�WqSV[4Y�]cD(JM\(a�[p;�et;I=]¢jÌ_Ì<;>Y�a�S�P�[_S�E|{Î�4W�^ A´�q]>[4YEkDML°YA/\XF,ais_Pc(y) = 1
6+ 3y2 LV��W)Ae�]_Y�EOD);�Y>;>�Ie4JkDxWvs>P�[p;Q\�L:à Argmaxy≥0 py − c(y)
m�EMFMS4=�Ly = 1
3p�y�a@l�e
s(p) = 300p.� W)s>P�[�A>P�;je�FIJdKXW)J!e�[4L:AA/Y�W)JG\)A>P�JOD
750− 150p = 300pmUE(AcPTS4F![p;�EGF$;¨E!JM[�Ã
p′′′ = 53
�� eV§q]>njL�^ A/Y�WvJM\�LNJ�E(JG[p;dF![�s>PcD(JM\(a
1mY>;>�Ie;}F��4WX^�WvAtep]_Y�]�D(J
y = 13
m>��W�FMS�[p;>Y4§:;jet;�E|{L = 1
9
LK = 1
9
L�^c;�F/JMWvA_PTS¨FvSV\�Y���»yA/[4LNJ$P2;>����Av�VS�a�PTSV[�A/\XLd(1) = 600
m8P�L:Ã!E�[p;QWqSV[4Y�]D(JG\ra
1800;>Y�a�S�P�[_S�E|{<�4WX^��
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