View
2
Download
0
Category
Preview:
Citation preview
HAL Id: inria-00070569https://hal.inria.fr/inria-00070569
Submitted on 19 May 2006
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
RN-codes : algorithmes d’addition, de multiplication etd’élévation au carré
Jean-Michel Muller, Jean-Luc Beuchat
To cite this version:Jean-Michel Muller, Jean-Luc Beuchat. RN-codes : algorithmes d’addition, de multiplication etd’élévation au carré. [Rapport de recherche] RR-5438, LIP RR-2004-60, INRIA, LIP. 2004, pp.17.�inria-00070569�
ISS
N 0
249-
6399
ISR
N IN
RIA
/RR
--54
38--
FR ap por t
de r ech er ch e
Thème SYM
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
RN-codes : algorithmes d’addition, de multiplicationet d’élévation au carré
Jean-Luc Beuchat et Jean-Michel Muller
N° 5438
Décembre 2004
Unité de recherche INRIA Rhône-Alpes655, avenue de l’Europe, 38334 Montbonnot Saint Ismier (France)
Téléphone : +33 4 76 61 52 00 — Télécopie +33 4 76 61 52 52
������������ �������������������� �� !#"$��%��&������')(*��+�+,��&����-.�����/�0������'1�/�2#"$3��43657�0���4��'8��,�������93
:<;>=<?A@�B�C9DFEG;HC�DJIK=ML%;NL%:<;>=M?�@4OQPRDJI9;HSTOUC9SVSR;HW
X6Y[Z]\_^�`Hacbedf`>gihkjlZ4\_^mhnhkg>\poMqNrtsvuHw[^]hxzy{q}|$^4j�~Tyl�]���}s�y{^
�n���[�MqNykjn�i^�y{^]�&Y�^4y&�&Y[^��<�9�}�N����d��7�]�4^4\po[yl^������}��d��J��������^mh
���9�4�]�J�[�}�N� ~ �[y{q��<^4ylj$g�q�¡�j{Y[^�q�y{s���s����}r£¢/q>q}jlY.y{^]�4q>�[st�[�_svh6jlYM�Jjnj{Y[^�¤�y&h$jn��q��i¥§¦4^]ylq_�is���stjn¡¨q�r�rtqJ©�s��[��.ª0s�hT�[^]�4^]h{hl��yls�r�g*«�ª�}���_¬>sv��^n¬N^4y&hl���X6Y[svh®��rtr�qJ©nh9j{q��[y{qJ¬�^/jlY��}j®j{ylw[�M�4�Jj{st���0jlY[^c¢�q>q�jlY_yl^m��qi�is��[��q}¡A�0�>w[\poM^]y®¯%s�h®^]uHw[s�¬°��rt^]�Hjzjlq�y{q�w����is��[��¯jlqjlY[^6�[^m�}y{^]hkj]�x�}±7q�y{�[^]ylw[�p�}���p²� ¥§b³°b´w[rtr�^4yKs��>¬�^mh$j{st�H�Jjl^m�0jlY�^/�<qNhlstjls�q�����rRµJy&���ist¶�·zµ}�>w[\poM^]yKhkgihkjl^4\h9hlY��}y{st���njlY�s�h9y{q�w[�M�ist����[y{q��<^4ylj$g¸�}���.�]�}r�rt^m�¸j{Y[^4\¹�nº�¥»��qi�is��[�Hh4®X6Y�s�h�y{^]hl^]��y{�&Y¸yl^]�MqNykjnsvhn�i^]¬�q}j{^]�¸jlq_j{Y[^�h$j{w��ig%q}¡K���[�istjls�q��Aµ[\�w[rtjls��[r�s��]�Jj{stqN�¼µM�����hluHw���yls��[�_�}r���qNylstjlY[\h�¡¨q�y�y&���[s½¶%¾G�nºn¥§�4q>�[st�[�Hh�¿¨sV ^Nz¢/qHq�jlY�yl^m��qi�is��[�Nh&À�9Á)^�hkY�qJ©ÂjlY��}jnst�Hj{^4��^]y��}y{s½j{Y[\_^�jlsv�0�}���%r�q���sv�cw[�[stj{hq��<^4y&�Jj{stqN��hz�}r�r�qJ©Ãj{q����[�GqNy®\�w�r½j{st�[r�g_¢�q>q�jlYy{^]��qi�is��[�Hh4�Á�^T��r�hlq��i^mhl�4yls�oM^7�}r���q�y{stjlY[\h�j{��Ä>st�[�p���i¬J�}�Hj&�}��^6q�¡<j{Y[^n�[y{q��<^4yljls�^]hq}¡�jlY[^�qNyls���s�����rÅ¢/q>q}j{Y%y{^]�4qi�ist����j{q��^4��^4y&�Jjl^cq��[jls�\Gs�¦4^m��Y��}y&�i©6�}y{^Tq��<^4y&�Jj{q�y&h4Æ*ÇHÈÅÉlÊ�Ë �JÌ��N�ÎÍ q�\_�[w[jl^4yn��ylstjlY�\G^4jlsv�}µ[�nºn¥§�4qi�ist���Nh]µ[¢�q>q�jlY.yl^m��qi�is��[��µ>y{q�w[�M�ist���pj{qGjlY[^��[^m�}y{^]hkj
ϼаÑÓÒ£ÔVÕ»Ö4Ô�ÑÓÒA×&ØÓÒRÙ6×kÚ&×&ÑÓØÛ×&Ü°ØÓÕz×&Ò�×/ÝRÕ$ÒRÕl×&ÝßÞ§Ð�ÝßÕ$à�Ù�ݨÔ�Ù&á�ÔRаÕKâi×&Ü}Ù�ÝR×{ÔVÙ�ÑÓÝßÕ®ã°ÕzØvä åçæmá�Ù�Ýßèn×{ÔRÑÓé]ê°Õ�ã°ê0ë[×&ÝR×&ØÓØÓìkØÓÑÓÒRè�Õ�í�î�î&ï�ð¨ñ�ñlò�ò�ò�óçô{õmö4÷lø{ù4úlõ�óvû�ü4ñ&ý°þ$ÿ��
�¹� ���/�%�� ������������9�&���.� �� !)"k���%�&������'#(*%�+�+,��&����-����4���0������' �/�2)"k3���3657��������'���,�/������3
��� ����� � ��� �[^*�[y{q��[y{s���jl�*�iw y{^]�4q>���}��^.�i^´¢/q>q}j{Y ���}��hh{��¡¨q�y{\G^*qNyls���s�����rt^�^]hkj¸uHw[^.rt^*�[y{^4\_s�^4y��&Y[sÅyl^*�[qN� �>w[r7uNw�shlw[stj�w[��ª�^mh$j¡¨qNy{�4�4\_^4�Hj*«�ª�^�j�y{�]��s��[y{qiuNw�^4\_^4�Hj]��»r�^mh$j¡ß�N��s�rt^*�i^*\_q��Hj{yl^]y_uHw[^Nµ/���}��h_w[� j{^4rny{^]�4q>���}��^NµKjly{q��MuNw�^4y_^]hkj�muNw�st¬J�}r�^4�Hj� ���yly{q��M�istyK��wG��rtw�h®�[y{Z]h]�x9N±7q�y{�[^]ylw[�G^�j/²� ¥§b³�b*w[r�rt^]yzq��Hjzy{�]�4^4\_\_^4�Hjz��j{w��is��T�is�¬�^4y&hzhlgih$j{Z4\_^]hz�[^��Hw�\G�]y{�}jls�q�� *�&Y[sÅyl^mh�hks����[�mh0���}ylj{����^m�}�Hjc�4^�jkj{^�[y{q��[y{st�4jl�_^�j�r�^]h0qN�Hj��}�[�<^4r��]h��nºn¥§�4qi�i^]h]��7����h���^4j��}yljlsv��r�^�µ¼��q�w�h���q�w�h0s��Hjl�4y{^]h{hlq���h� �i^mhT��rt�Nq�y{s½j{Y[\_^]hn��� �����is½j{stqN�*^�j0�i^�\�w�r½j{st�[r�sv�4�Jj{stqN�*�MqNw[y7�[^]hT�Tº�¥»��qi�i^]hT^4�´oM��hl^�¾*¿ß��� ^mh$jl¥� }¥§�is�y{^��i^]hTyl^m��qi�[�}�N^]h��i^G¢/qHq�jlYMÀ�ºTq�w�h�\_qN�Nj{ylqN��h0uHw�� s�rz^mh$j��<qNh{hks�o[r�^_��� wij{str�svhk^]y�rt^mh0w[�[stjl�mh��}y{s½j{Y[\_��j{s�uHw[^mh��i^]h0�[y{qi��^mhlhl^4w[y&hc�MqNw[y������is½j{stqN�[�[^]y�q�w)\�w[rtjls��[r�st^]yr�^]h7�nº�¥»��qi�i^mhn��qN��hksv�i�]yl�mh4®ºTq�w�hT�[y{q��<qNhlq���h6�]�N�}r�^4\_^]�Njn�i^mhT��rt�Nq�y{s½j{Y[\_^]h/jls�y&�}�Hjn���}yljls��i^]h7hk�<�]�4s½¤M�4s½j{�]hn�[^]hT�Tº�¥»��qi�i^]hn�MqNw[y�N�4�[�]yl^]y6�[^]h�qN�M�]y{�}jl^4w�y{h/\�Jj{�4y{st^]r�h]��Ë �]� É ��� � �%� ~ny{s½j{Y[\_��j{s�uHw[^��i^mh�q�y&�is����Jj{^4w[y&h]µ>�Tº�¥»��qi�i^]h]µiy{^]��qi�[����^0�i^�¢/qHq�jlY¼µM�}y{ylqN���is£�}w��[r�w�h��[y{Z]h
������������ ������ ���������������� "! �������#�$��%'&(��)�)*��+�#� ,-�.�$���/�0�$�/%1���23! 45�64�7/�/�0�$�/%8��*9�:�����;4 �
< = 'p�A����,��z���4��'
ºnqNw�h���q�w�hGs��Nj{�4y{^]h{hkqN��h )�i^mhG�]��hp���}yljlsv��w�rts�^4y&hp�[^%y{^4�[y{�]hl^4�Hj&�Jjls�q��Mh F�&Y�s Åyl^mh_hks����[�mhp�[^]hG�[q�\po[y{^]h]?>�q�y{\_��rtsvhl�4^]h����}y~A¬>s�¦4s�^4�[svh�^4�´�@�A[��B���C§µH�4^]h�yl^]�[y{�]hl^4�Hj{�}jls�q���hz���[����y{��s�h{hl��st^]�HjK�[�$| �s�\_�[r�s��4s½j{^4\_^4�Hj/���}��h�rt^7yl^m��qi�[����^n�i^c¢�q>q�jlYDB �/CÅq�w��[����h®r�^]hyl�mhkw�r½j&�Jj{hK¡¨q�w[y{�[svhK���}yKr�^]h®�}r���q�y{stjlY[\_^]hK�[^T�is�¬HsvhlstqN�`[��XEB �[µ�@�CV�ºTq}j{q���hK^]����qNyl^�uHw[^nr�^]hz�[yl^]\_stZ]yl^mh9s��i�]^]hz���}��hz�4^�jljl^T�is�y{^]��j{stqN�yl^]\_q��Hjl^]�Nj Í �}wM�&YHg9B �/C§F¼^�y{^]�4q>���}��^0�i^�¢/q>q}j{Y.���}��hnh{�p¡¨q�y{\_^�q�y{st�Nst����rt^_¿ ��[^�����h��4q��i¡¨qN���iy{^��°¬�^m�7�i^mh�sv�i�4^mhn�is<�]yl^]�Nj{^]h]µ�hlq�w[¬N^4�Hj�yl^]��y{q�w[�<�4^mh
hkqNw�h�r�^�[q�\Î�i^D�;�����G��5�H�I��KJL��/�M�8���/� NO4�µ�uNw�s®hkqN�Hj�rt^mh�hk^]w[r�^]h0wij{str�s�hl�4^mh��i^���qNh/|$q�w�y{h��<q�w�y�yl�m�}r�s�hl^4y��i^mh�\�w�r½j{st�[r�s�^4w[y&h{À��qN��hksvhkjl^ �y{^4\_�[rv���4^4y��[����h/r � �]�4ylstjlw[y{^0o[s����}s�yl^0��� w[�.�[q�\po[y{^7r�^]h6hlq�w�hk¥§�&YM�QPç�[^]h/�i^0r��p¡¨qNyl\_^�R[ªNª�ª�ªNªTS5SGSlªNª7�M�}y�ªGR�R�R�R�R(SGS5SUR�V ª}µq�W9V ªc�i�mhks�����^Tr�^c�&Y[sÅyl^G«�ª��F � sv�i�4^0�i^0¢�q>q�jlY��4j{�}stjn��� ^]h{hl�°gN^4y®�i^c¡ß��sty{^7���[���}y&�QPvjly{^nr�^0�[rtwMh��<qNh{hlsto[r�^7�[^c¦]�4y{qNh��[�}��h/r � �m��y{s½j{w[y{^��� w[�.\�w[rtjls��[r�s��]�Jjl^]w[yT�}¤������ ^ <^m��jlw�^4yT�i^mh6\�w[rtjls��[r�s��]�Jj{stqN��h��[r�w�h6y&�}�[sv�i^4\_^]�Nj�^4j�X°qNw��°¬N^]�7\_q�s���hn�i^0\�Jj{�4y{st^]rR
� �[^��ylqN�[yls���j{��i^]h�y{^]�4qi�[�}�N^]hc�i^�¢/qHq�jlY#^]hkj�uHw[^_rt^��yl^]\Gs�^4y��&Y�s Åyl^��q��#�>w[r®uHw[s®hlw[stj�w[� ªG^mh$j�¡¨qNy{�4�4\_^4�Hj V ª_^�j�y{�]�4s½¥�[y{q>uHw[^]\_^4�Hj] �»rA^]hkjTy{^4rv�Jj{st¬N^4\_^4�Hj6¡ß���4str�^���� ^4�´�i�m�iw[s�yl^puNw�^�rt^��[q�\po[yl^�qNoijl^]�>w�^]�.jly{q���uHw����Hj ���}yljls�yT�i^�� � s�\_�MqNykj{^�uHw[^4r�rt^�MqHhkstjls�q��Aµ��isvhkqN��h��4^4r�rt^%�i^¸�<q�sv�[hp¾QY>µ�rt^mh�o�s½j&h��i^%�MqNs��[h�¡ß�}s�o[r�^]h���� w[�³jl^4r/y{^]��qi�[����^_^]hkj .w��[^%�isvhkj{�}�M��^¸�iwÃ�[q�\po[yl^s��[stjlsv�}rst�[¡¨�4y{st^]w[yl^ �¾QY/Z\[9]���/%^ _*�%`�U�5�2�;�����G��5�H��&��0�;�/%-a�*^�5�b�� ���4:a5*���7/�/�6�5%^�)c1�����;�/%-/���d�/*d,e�.*f �,e�;g� � Í ^�jkj{^.�[ylqN�[y{st�4jl�¸^mh$jGjly{Z]hst�Hj{�4y{^]h{hl���Nj{^I]¼^4r�rt^�]¬>s½j{^jlqNw�h0rt^mh��[y{q�o[r�Z4\_^mh��istj{h��i^9��/*^h5�6�I�/���;��%e��ç�x9�±7q�y{�[^]ylw[��^�j�²M ¥»b Ab´w[rtr�^4yiB A�C®q��Hj�y{�]��^]\_\G^]�Hj��j{w��is��Tr�^]h®�[st¬N^4y&hzhkgihkjlZ]\G^mh ��&Y[sÅyl^mh®hlst�N�[�]hzuHw[s<����ykj&�}��^]�HjK�4^�jljl^T��ylqN�[yls���j{��µ}^4j�r�^]h®q��Hj��}���M^]rt�mh®�nºn¥§�4q>�[^]hn¿ßq�W_��� ¬>s�^4�Hj��i^�L��*�%eK�j���)���/�;�� ��vÀ� �»rAqN�Njn��ylqN�MqHhk���i^mhn�}r���qNylstjlY[\_^mh/�[^���qN�H¬N^4y&hks�q����i^��[qN\�o[y{^]h6�m��y{s½j&hn�[�}�Mh���^]hThkgihkjlZ4\_^mh�¬�^]y{h6^4jT�i^4�[w�s�hrt^mhnhlg>hkjlZ]\_^]h6w�hlw[^4rvh] �»r�h6qN�Njn^]�Hjly{^0��wijly{^]h/qNoijl^]�Hw.r�^0yl�mhkw[rtj{�}jnhkw[s�¬J�}�Hjk]lkm �HË �/n � Ç1o$Æ¸Ë �/p Ç � �rqEs � ��� � Ç �ftvu^wew^x\yQz�{ ������·|�}�ih��Q 5� ���%^*��K4��;���#�$��%9*��0���.�~ 54��G�� { �£·E�� ������L,'������&O�/�6���� (�����H�'�H� Z\[ ��� Z-� �H� Ze� S5SGS����HS � Z3[ � Ze� SGS5S �� ���*�%_�L�������G��� ����5�T 5�5*��6���K�5%^�T ��
�e��� «�·i�ª¾ � �H� �·*« ª¾ S
� { �A·�� ¾f&2���6�/�� ������ � � � Z3[ � � Ze� � � Ze� SGS5S:� � S � Z\[ � Z-� S5SGS_����7Q�����H��� «�ª � R � ª��i�� ��(*�%D���k���:���) ��(���O 5�5*��6���K�5%^�2 ���6�� ����f� �(�:�� �%e��%9%�*��� � 5��%^�O��� ��+��%e�� ��/�.�j����%-4� 5�� { �n·��� ���,'�������5� �*,^45���$��*�����*E4U������c9�'&��/�6���� �¡�¢� � � � Z\[ � � Ze� � � Z-� S5SGS;� � S � Z3[ � Ze� S5S5S£�� ���*�%`�L�������G��9 �� ��� ��5*��6�5���5%^�� ��
¤ ���j�/*f ��6�� k���f� ���;�� � 5�/%��O���7/�/�}��*�����h� 5�/�+*^����%¥�4����$�5*��;� �/*94U�H�/�6� c�·§¦}¾�¨© �� ���ª � � ª��·§¦}¾f&����6�/�� ��6�(,e�:��� �$��� �:��� �(�;�)%e��%d%^*��"a5*��§ �*����Oci��;�/���j������ � �� ��2��k ��+��%-� ��,�,^�Q 54�ci���5�.*������� � �
xK�}y�^�¶i^]\G��rt^Nµi^4�.o���hl^�¾iµ¼ªR�R V ªJª V ªc^]hkjnw[�*�nº�¥»��qi�i^Nµij{�����isvh�uNw�^ªGR�R�R V ª V ª�µMuNw�s¼yl^]�[y{�]hl^4�Hjl^crt^�\�«]\G^0�[qN\�o[y{^0�[^�r � ^mh$jn���Nh4¬z��o��Nhk^��µJjlqNwijl^mhzr�^]hK�m��y{s½j{w[y{^]hK�[^7��q�\_�<q�ylj{���Nj®uHw[^�r�^]h��&Y[s<y{^]h2V ªNµ�R0^4jTªnhkqN�Hj��i^]h®�nº�¥»��qi�i^mhT¿¨^4j®q��_�<^4wij/yl^]�[y{�]hl^4�Hjl^]yz�}s���hksjlqNw�h�r�^]h6�[qN\�o[y{^]h&À��¬z�.o��Nhk^_ªR[µ''S�ª5�>¾fV�HV ªG'VfV® �^V��V ªGR'V¾HV ª5_��qN��h$j{s½j{w[^�r�^��i�4o[w[jT�i^0r��_y{^4�[y{�]hl^4�Hj&�Jjls�q��%�i^�¯��M�}y�w[�*�nºn¥§�4q>�[^��7����h�rv�hkw�s½j{^�µ[�[qNw�h��[^p��q��Mhksv�i�4y{q��Mh/uHw[^��[^]hn�nºn¥§�4qi�i^]h�^]�*o���hl^�¾.¿ß��� ^mh$jl¥� }¥§�is�y{^0�[^]h�y{^]�4q>���}��^mh6�i^�¢/q>q}jlY<À��ºTq}jly{^�o[wij
�i^]h��i^n\_q��Hj{yl^]yzuNw � s�r�^mh$j���s�hl�n�i^n\���[st��w[rt^]y®�i^mhz�[qN\�o[y{^]h9y{^4�[y{�]hl^4�Hj{�]hK�[����h®�4^nhkgihkjlZ4\_^NN��^]\_��y{uHw[qN��h�jlqNwij���� �}o<q�y&�_uHw[^nrv��[y{q��[y{st�4jl���i^mhzhks�����^]hK�}rtjl^]yl���]hK�i^mhz�&Y[s<y{^]hK��� w[��nºn¥§�4q>�[^�o[st�M�}s�yl^6�<^4y{\G^4j®�i^�r�^]hK\_�4\_qNylsvhk^]yK�i^�\�}�[s�Z4y{^���q�\_���N��j{^�¿¨s�r�hkw'°Gj�i^T\_�]\GqNylsvhl^4yKr�^7hlst�N�[^n�iw��[y{^4\_s�^4y®�&Y�s Åyl^T�[q��i¥§�>w[rVµ��[w[svhz^]��hlw[s½j{^7�i^Tyl^]�[yl�mhk^]�Hjl^4y®rt^mhO±pª�����y®r�^no[stj7ª�µN^�j®r�^]h®¦4�]ylqHhK���}y�rt^To[s½jRNÀ� Í ^�jljl^�y{^4�[y{�]hl^4�Hj{�}jls�q����4q�\_���N��jl^6^mh$jKs��Nj{�4y{^]h{hl���Hjl^ 7w[jls�rtsvhk^]yz�<q�w[yKw[�hkjlqi�&ÄJ�}�N^�^]�p\_�4\_qNsty{^�hk^m��q��M�[�}s�yl^N°x�qNw[yK\�}�[s��[w[r�^4y��^mh��4q>�[^]hzr�q�y&h{uNw�^n�[q�wMh®^ <^m��jlw�q���h/�i^]h�q��<�4y&�Jj{stqN��hz��ylstjlY[\_�4jlsvuNw�^]h]µN�[qNw�hzy{^4��yl�mhk^]�Nj{q���h®w[���[qN\�o[y{^�²��i^´³´�&Y[s<y{^]h� �r � ��s��[^�i^_�i^4wi¶�¬�^]��jl^]w[y{h7�i^�³³o[stj{h´²9µÂ^4j´²8Z³jl^]r�h�uHw[^ ²¶�·²9µ#«�²1Zz Í q��Hjly&�}s�yl^]\_^4�Hj ¸r��%yl^]�[y{�]hl^4�Hj{�}jls�q���h��/���;��¸v�U 5�/7/��B �QCVµ�[q�wMhTs�\_�MqHhkqN��hTw��[^py{^4��yl�mhk^]�Nj&�Jj{stqN�*w[�[svuNw�^G�iw)�&Y[s<y{^)R�µ �hl�°¬Nq�s�yn¯ µ� � ¯ Z� �¹R[ Í ^�jkj{^G�4q��Hjly&�}s��Hjl^��H�}y&�}�Hjlstj7uHw[^ ²9µ ^�j²1Z#�[^���qN�Nj{st^]�[�[^]�Njn��w���w[��^0�&YM�QPç�[^��i^ª��i^�rtqN�[��w�^4w[y�hlw[�M�]yls�^4w�yl^ Gw[�¼ºL»¼Ç �1q � Ç�¼ zk{ �/���?³D�¾½��\¿r�� �%e����h5�;�� �ªGR�R'V ªJªGR�R�V ªi�5�®ª�V ª�ª�V ª}ª�V ª}ª�V ªK '! 4��5����7Q�5%^�O�;�� #,^�����0��7/���K��%��
ªGR�R V ªJªR�R V ªL� ªGR�R�R[ªR�R�RÀ Á� ÃÄ(Å«DR�R�R�ªGR�R�R[ªÀ Á� ÃÄ�Æ
���fª V ª}ª V ª}ª V ª�ª V ªL� ªR[ªR[ªGR�ªGRÀ Á� ÃÄ(Å«IR�ªGR�ªGR[ªR[ªÀ Á� ÃÄLÆ
�7����hpr��)hkw�s½j{^��i^���^4j_��ykj{s��4rt^Nµ��[qNw�hG�[y{�]hl^4�HjlqN��h��i^mh_�}r���q�y{stjlY[\_^]hp��� ���[�istjls�q��Q¿ßhl^]��j{stqN��h�*^4j_�HÀ�^4j_�i^%\�w[rtjls��[r�s��]�Jj{stqN�¿ßhl^]��jls�q��.�HÀ�[ºnq�wMh6�[ylqN�MqHhkqN��h�¤�����rt^]\G^]�HjTuNw�^4rvuNw�^]h6q��[jls�\Gsvh{�Jjls�q��Mh6�MqNw[y�st\_�[rv�}�Hj{^4y�r � �]rt�]¬°�}jls�q��.�}w.�4��yly{�G¿Rhk^m��j{stqN�*�NÀ�
Ç"ǹæ�È'É�Ê�Ë�Ì
� ���:��%^�#¿v*^��J���*^���'���2�5�������/%����D�$�:�^�����I*����}���
� � %��&���4��'��T5���� �/��'G5��6�9 A����'1��'���� ��������������
FA^7�4��r��4w[rÅ��� w��¸�[qN\�o[y{^�� �i^��M³��³ª��z�&Y�s Åyl^mh®^]��h��/���;��¸v�U 5�/7/�/yl�mhkw[rtj{���Hj��i^cr � ���[�[s½j{stqN�¸�i^7�[^4wi¶¸�nºn¥§�4qi�i^]h��i^�³´�&Y�s Åyl^mh�4q���hkjlstjlw[^Tw[���4�NhK���}yljlsv��w�rts�^4y®�[^nr � �}r���qNylstjlY�\G^���� �����is½j{stqN�^4�_j{^4\_��h®�4q���hkj{���Hjz^4�9h��/���;��¸v�U 5�/7/�z^4j®�M^]yl\_^4j®w[�[^Thls�\G��rtst¤M�4�}jls�q���i^pr � q��<�4y&�Jj{^4w[y7�[y{q��<qNhl���[�}�MhkBÓ�QC§<~n¤����i^G�i��j{^4y{\_st�[^]yTr�^]h7o[stj{h7�MqHhkstjlst¡ßh7�[^ ��µ<�[q�wMhT�����is½j{stqN�[�[qN��h�²9µ ^4j�!)µ6eFA^p��qi�[����^�N�iq��ij{�F�[qNw�h��H�}y&�}�Hjlstj¸uHw[^�hlsc¯ µ� � ª ¿¨y{^]hl�M^m��jls�¬�^]\_^4�Hj#" µ� � ªmÀ�µ/��rtqNy{h�¯ µ� µr[ �¶R ¿¨y{^]hl�<^]��j{st¬N^4\_^4�Hj$" µ� µ?[ �¢RHÀ�/x9��y�4q���hl�]uHw[^]�Njmµ<hls�r � ���[�[s½j{stqN�F�i^�¯ µ� ^�j��i^%" µ� �N�4�[Z]yl^�w[�[^Gyl^4jl^4�>w[^%¿ß�} ¥ [ ¥§�£Åhls�¯ µ� " µ� � ª°À�µÅ��^4jkjl^_�i^]yl�[s�Z4y{^��[^Ghk^]y{���[ylqN���}�N�4^uHw[^���� w[�[^��<qNhlstjls�q��¼[ºTq�w�h6qNoijl^]�[q���h���st�Mhks
& µ� � ¯ µ� " µ�& µ� � ¯ µ� Z\[(' " µ� Z3[& µ� µ?[ � �߯ µ� µr[ ' " µ� µr[ �*)%¯ µ� " µ� �-�,+.- R � S5S5S � ³.« ª0/
FA^�\�«4\_^��[y{st�M��s��M^�h � ���[�[r�s�uHw[^¸�4¬>s��[^4\_\_^4�Hj��}w³�]�}rv��w[r®�[^1��ZzrF � �N�[�istjls�q�� �°¬�^m�_��q��>¬N^4y&hks�q���^4�¾h��/���;��¸v�U 5�/7/��h � ^ <^m��j{w[^ r � ��s��[^_�[^¸¾/³ �4^4r�rtw�rt^mh��'�/� ¥�����������¿ßq�w32T~cÀc^�j��i^�¾4�$³´« ª5�T�<q�yljl^mh0rtqN��svuHw[^]h76c�98�¿R�}r���q�y{stjlY[\_^��mÀ�3FA^�&Y[^4\_s��)�4ylstjlsvuHw[^�i^r � qN�M�]y{�}jl^]w[y��4q��Hjls�^4�Hj���st��hls�w��[^n�<q�yljl^nr�q��Ns�uHw[^�:96c� ^4j®w[�[^7�MqNykj{^nr�q���svuHw[^96c��¿¨¤���w[y{^��mÀz^�j®r�^7�]�}rv��w[r<h � ^ Å^]��jlw[^NµN��� w[���<q�s��Hj�i^�¬>w[^0j{Y[�4qNylsvuHw[^�µ>^]�%j{^4\_��hn�4q���hkj{���Hj]���<; Ë �>=¨� m � ¼�~��[�istjls�q��*�°¬�^m�7�4q��>¬�^]y{hlstqN��^4�8h��/���:�/¸v�U 5�/7/��º pÅ�4� �>Ç ��� �T^]wi¶��Tº�¥»��qi�i^]h2²Î^�j?! �i^�³)�&Y[s<y{^]h@ Ë �°��= Ç � `iq�\_\_^����i^�²Î^�j?! ��qi�i�]^0^4�8h��/���;��¸v�U ���7Q�T�°¬�^m�%�$³K� ª��6�&Y[sÅyl^mhACB Í ��r��4w[r�^4y�^4�)����y{��rtr�Z4r�^ED�µGF ²9µ ' !kµ6µHD)ZIF ²1Z ' !KZ®µ�J�µGF ²9µLKM!)µ ^4j7JKZNF ²1ZOKM!KZ �r � �}sv�i^�i^�¾/³�4^4r�rtw�rt^mh��'��� ¥��������5�QP
R�B Í ��r��4w[r�^4y/^4������y{��rtr�Z4r�^S�LµTFVU µ� Z3[ ¾� ��¾4�WJ µ� Z-�YX � )�D µ� Z3[ZX [ �'�O[ µ� ^�j\��ZLFVU Z� Z3[ ¾
� ��¾4�WJ Z� Z-�YX � )�D Z� Z3[ZX [ �'�O[ Z� �r � �}sv�i^�[^�¾4�M³�« ª��/�<q�yljl^]h6r�q��Ns�uHw[^mh�6 � P
0
HA HA HA HA
v+3 u+
0
y+0 x+
0y+2 x+
2 y+1 x+
1y+3 x+
3
t +0t +
1t +2t +
31
HA HA HA HA
v−3 u−
0
y−0 x−
0y−2 x−
2 y−1 x−
1y−3 x−
3
t −3 t −
2 t −1 t −
0
t +4 t −
4
0
s+0s+
1s+2s+
3s+4 s−
0s−1s−
2s−3s−
4
d0d1d2d3d4d5
Additionneur en base deux traditionnelle 1
Rec
odag
e de
Boo
thSo
ustr
actio
nA
dditi
on e
tco
nver
sion
en
borr
ow−
save
= =
]\^`_�a��?b¸~��[�[s½j{stqN�.�[^0�[^4wi¶.�nº�¥»��qi�i^mh��i^�uHw��Jj{yl^0�&Y[s<y{^]h]
ced Ù�ê°ÒAÞkÙ�æmÒVÑÓã°ì$ÝßÕkÝßÙ�æ°Ò�ÔRÙ�êYfßÙ�ê°ÝßÒ�ã°Õ$ÒAà�Ù�ݨÔRÕkÒ�ØÓÙCg�ÑÓé]êmÕkÒh�ãmÕkê]ÖcÕ$æ�ÔVÝßì$ÕkÒ �
å d ÇAåji
������������ ������ ���������������� "! �������#�$��%'&(��)�)*��+�#� ,-�.�$���/�0�$�/%1���23! 45�64�7/�/�0�$�/%8��*9�:�����;4 �
� � �%��������' ��Î���,�� �¹� ���/�%�� F � �}r���qNylstjlY[\_^p��jlwM�ist����w��M�}y&�}��y&�}��Y[^��[y{�]�4�]�i^]�Nj��4q���hkjlstjlw[^Grv�%��yl^]\Gs�Z4y{^G��j{���M^�i^r � ���[�istjls�q��#�i^�i^]wi¶��nºn¥§�4q>�[^]h0�i^�³
�&Y[s<y{^]h�² ^�jS!iºnqNw�h®�4q��>¬�^]ykj{s�h{hkqN��hK^]��hkw�s½j{^Trv��y{^4�[y{�]hl^4�Hj{�}jls�q��dh��/���;��¸v�U 5�/7/��� �i^L²¹�.! ^4�%��q�\_�[r��4\_^]�Nj ��i^]wi¶ �r � ��s��[^��� w[�[^�hlq�w�hkjly&����jls�q��¼µH�[w[svh���qN��h$j{ylw�s�hlq���h�r�^0�Tº�¥»��qi�i^���h{hkqi�4st�� p�4^0yl�mhkw[rtj{�}j] ��Z)��j&�}�Hj�w[�%\_q�j��[^��$³��ª���o[stj{h]µ��M³��þ0��o[stj{hhkqN�Hj��[�m��^mhlh{�}s�yl^mh 7r � �m��y{s½j{w[yl^6�i^��$«S�kZ�A^]�p�4q�\_�[r��4\_^4�Hj 7�[^4wi¶£°xK�}yK��qN��hk�muHw[^4�Hj]µ�� � �Lµ0«7��Z�^]hkj9�]�N��rt^]\G^]�Hj�w[�G�[q�\po[y{^�i^E�M³ �³¾0�®o[s½j&h4�`is£�[q�w�h6���[�[r�s�uHw[qN��h®r�^0yl^m��qi�[����^7�[^c¢/q>q}j{Y� G��^0yl�mhkw�r½j&�Jj]µH�[qNw�h6q�oij{^4�[qN��h�w[���Tº�¥»��qi�i^0��q�\_�<q�ylj{���Njmµi�[�}��h��^]ykj&�}s���h��4�Nh4µ��$³K� ¾0�6�&Y[sÅyl^mh4ºL»¼Ç �1q � Ç u3z�{ ���$��%���² � V ª5R�R�R8�5� ! � V ª5R�R�R£���*��b���k���:���� ���k�����f� ���;�� 5�T¿k! �/� �H�/�����M�f�K� ¤ �;�5�U��*���%e��� µ ��R�R�R�R�R£�����Z9� ªR�R�R�R����r%e�3¥��/�~ ,�Lµ �5��k«S��Z�(���G�4� L�5%i���/�L,e�645���5%^�§ck��5*���&\%e�/*f (��h5�U�5%e��%' (D � �Lµ��$�$«S�kZ�T�|R�R�R�R�R�R-�%ª�ªGR�R�R�R��¿"�k�;�����G��5�H�)��´J��G�����9���D �� ���*�%D%-�/�Kh��:�1� V ªGR�R�R�RD�� ® �:��� �(�;�� ����^��*��U�#¥��/�~ �&T�6���� �a5*^�L² ��! � ªR�R�R'&?%-�/*f ���h5�U�5%e��%' D �¢R�ªGR�R�R�R���R�R�R�R�R�R|�5� � ª V ªGR�R�R�R��(�����K�/�;a5*^�/%^ 9a�*^�����9�:4� �*��.�j�/�K������5� 40�H�/�}���K��%���*�%e�£4��������0*��;�d �*�� ® �:��� �(�;�� 1� ªR�R�R�R��b*q��Hj{ylqN��h®uHw�� strÅ^mh$j/jlqNwJ|$q�w[y&h/�MqHhlhls�o[rt^c�i^0��qi�i^4y�1��² � ! �r � �}sv�i^c�[^ �$³)� ª����&Y[s Åy{^]h]HºTq�w�h/wij{str�s�hlq��Mh®w[��^7�[y{q���yls���j{�
�i�]�4q�w[rv�}�Hj�s�\G\_�m�is��}jl^]\G^]�HjT�i^�r��_�i�4¤��[stjls�q��*�iw*�nºn¥§�4q>�[^0�4q���hlsv�i�4y{�c���}��hn�4^�jn�}yljlsv��r�^�� � Ë q��>= � � �1¼ z){ �-² �� ��O*�%_���k���:���)��L³¾�:��� �(�;�� �&O�/�6���� )ª ²`ª � ²������2� ¾ � Z\[ �F¼^¸jl^4y{\_^%�i^ �M³9�Q¾0��o�s½j&h�� � � µ « � Z ���[����ykj{st^]�Nj_�}s���hls� - «T¾ � � SGS5S � ¾ � /H?FAq�y&hluHw[^�� ^]hkjGhkjly{s���jl^4\_^]�NjG�[�4�H�Jj{s½¡$µ
�[q�wMhT�i�m�iw[svhkqN��hT�i^pr��¸�i��¤���s½j{stqN�F�iwF�4q�\_�[r��4\_^4�Hj ¸�i^4wi¶*uHw[^)� � µr[ ��� � � ª��xK��y7��qN��hl�]uHw[^4�Hjmµ�w[�[^�¡¨q�svhTr�^�y{^]��qi�[����^��i^¢/qHq�jlY^ <^m��jlw���µ�� � µr[ ��R�^�j®j{q�w�h�r�^]hz��q�\�o�yl^mhz�i^Tr � ^4��hl^4\po[rt^ - «T¾ � � S5S5S � «�ª0/Thk^ny{^4��yl�mhk^]�Nj{^4�Hjz����y®w[�¸�Tº�¥»��qi�i^T�i^ �M³��#ª���&Y[s<y{^]h][`>w[�[�<qNhlq���h/\�}s��Hjl^]���}�Hj�uNw�^�� + - R � S5SGS � ¾ � «³ª /H Í q�\_\_^´� � µr[ �|� � �|R[µ>rt^0y{^]�4q>���}��^c�i^0¢�q>q�jlY%�[^��2�[y{q>�[w[s½j F��q�w[¬N^]�}wÃw[� �nºn¥§�4qi�i^��[^3�$³9��ª5���&Y�s Åyl^mh4§F¼^.hl^4w[r��4�NhG�[ylqNo[r��4\�Jj{s�uHw[^�hkw[y{¬>st^]�Hjp����y_�4q���hl�]uHw[^]�Hjpr�q�y&hluHw[^�� �e¾ � ]�}�[��rtsvuHw[� kR�ª"R(SGS5S�RÀ Á� Ã��� r�^Ty{^]�4qi�[�}�N^n�i^7¢/q>q}j{Y���4��Z4y{^nrt^7�[q�\po[y{^T�i^%�$³���¾0�z�&Y[s Åy{^]h7ª V ª\R(S5SGS;RÀ Á� Ã��� � ��^T\_qi�is½¤<�4�Jj{stqN�¸�i^mh��4^4r�rtw[r�^]h�i^�y{^]��qi�[����^0�i^ �µ� ^4j���Z� �<^4y{\_^�j6jlqNwijl^4¡¨q�svhn�i^0yl^]�[y{�]hl^4�Hjl^]y�¾ � _r � ��s��[^0�[w.�Tº�¥»��qi�i^��i^1�M³ � ª��6�&Y[s<y{^]h0ªrR(S5SGS�RÀ Á� Ã��� ]
� µ� ��� �! ª hls3� � µr[ �|Rp^4jL� � � ª_¿R�} ¥� � ¥»�£"� �U¾ � À� � Z\[ hls3� � µr[ �|� � ��R.¿R�} ¥� � ¥»�£"� +.- R � S5SGS � ¾ � « ª /JÀR hls��[q��
� V�H� µr[ �$�H��)_�H� Z3[ �� Z� � # � � V� � Z3[ hks"� � µr[ ��ª¿ß�� ¥ [ ¥§�¼"� +.- «T¾ � � S5SGS � «�ª /°ÀR hks��[q��� �H� µr[ ��� V�H� Z3[
F¼^�yl^m��qi�[�}�N^*�i^�¢/q>q}j{YÂ�[�]�4^]h{hkstjl^F��st�Mhksc�i^4wi¶Â�4j{�}�N^]h��[^F�<q�yljl^]h¸rtqN��svuHw[^]h^4j�h � ^ <^m��j{w[^F�]�N�}r�^4\_^]�Njmµ®��� w[� �<q�s��Hj��i^´¬>w[^jlY[�]q�y{s�uHw[^NµN^]�jl^]\_��h��4q���hkj{���Hj] Í ^4j/�4j{����^T�i^c�4��r��4w[rM^mh$j6��qN��h$j{s½j{w[�7�[^%�R¾�³��³ª��z�MqNykj{^]h®rtqN��svuNw�^]h®~7ºT�pµi��� w��[^7�<q�yljl^TrtqN��svuNw�^6c��^4j0�i^¾/³ s��>¬�^]y{hl^4w[y&h]�F � �}r���qNylstjlY�\G^_�y{�]hlw[\_^pr�^]hc�is<�]yl^]�Hjl^]hc�4j{�}�<^]h0�i^Gr � �N�[�istjls�q��F�i^_�i^]wi¶F�nº�¥»��qi�i^mhc^�j0r��¸¤���w�yl^%�str�r�w�h$j{yl^r � �}y&�&Y[s½j{^]��jlw[y{^G�[^r � q��<�4y&�Jj{^4w[y��4q�y{yl^mhk�<q������}�Hj] �»rz^]hkj���qN��hkjlstjlw[����� w[� �����is½j{stqN�[�[^]w[y0jly&���istjls�q����[^4rn¿R���[�istjls�q����[^4w[y yl^4jl^]�Hw�^n�[y{q��������4^Nµ������is½j{stqN�[�[^]w[y®�[y{��¤�¶>^Nµt]4&À9^�j��[^Tr�q���svuHw[^nhlw[�[�[r��4\_^4�Hj&�}s�yl^n�<q�w�y®^ Å^]��jlw[^]y®rv���4q��>¬�^]y{hlstqN�p^]�9h��/���;��¸v�U ���7Q�4µ�i��j{^4y{\_st�[^]y7r�^p�4q�\_�[r��4\_^4�Hj� ��i^4w[¶*�[^¸«S��Z³^4jcy{^]�4q>�[^4ynr�^�y{�]hlw[rtj{�Jjm<�c�}�MhTr�^��[s�y{^p�i^mhT�4�Nh4µ<�[q�jly{^��}r���qNylstjlY[\_^��}|$q�wij{^�w[�[^�MqNykj{^�:96c��µiw[�[^��<q�yljl^ 6c��µ[�[^4wi¶��MqNykj{^]h�~Tº7� ^�jT�[^4wi¶�st�>¬�^]y{hl^4w�y{h/�}w*�&Y[^]\Gs��.��y{s½j{s�uHw[^��i^�r � �N�[�istjls�q��[��^4w[ym� � w[���MqNst�Hjc�[^p¬>w[^prtqN��sv��s�^4rVµ<�4^�jc��rt�Nq�y{s½j{Y[\_^�h � �°¬NZ4y{^��[rtwMhc�4q�$ij{^4wi¶£-FA�¸�[yl^]\_stZ]yl^��4j{���M^G�i^pr � �}r���q�y{stjlY[\_^��p�[�]�4^]h{hkstjl^
^4r�rt^�hl^4w�rt^uHw��Jj{yl^_s���hkjly{w���j{stqN��h4��»r®^]hkj��[ylqNo��}o�rt^]\G^]�Hjc�[r�w�h6|$wM�is��4st^]wi¶)�[^��4�}rv��w�rt^]y1�M²9µ « ²1Z�v� �W!�µ « !KZ �� �r � ��s��i^�i^�i^4w[¶¸hkqNw�h$j{y{�N��j{stqN��hK^�j���� w[�[^0���[�[s½j{stqN�¼µN��w[s�h6�i^7yl^m��qi�i^]y®r�^7y{�]hlw[rtj{�Jjm � �[^0�}wij{yl^Thlq�r�wij{stqN�¸hk^]y{��s½j6��� ��jlwM�ist^]y�r � st�Hjl�]��y&�Jj{stqN�¸�i^]h�i^4w[¶%��j&�}�N^]h6�i^�r�q���svuHw[^����[�[s½j{stqN�[�[^4rvh��[����h6w[�.�[y{q>�4^]h{hk^]w[y���qN\G\_^]y{�4s���r<^4jT�i^���qN\G��rt�4jl^4ynhlq��p|$^4w*��� s���h$j{ylwM��jls�q��Mh4ºL»¼Ç �1q � Ç&% z(' �/%^ ��$�4��:�/%' k��� O���k�����G��� ´�����5*�� ���f� �(�;�� ���NT%D3! �����.*f ��0�;�5���6�T¥��/%e�5�0�$�/%^%-���K�5%^����´�;! �/� �H�/�����M�f�K�)"! ���/���0�$�/%,e�;��,'�/ 54G�(¿r��,-�;�:,-���$4��j4 ¤ ��%e��$a�*^�da5*^���¡��,�,^�/���0�$��%���c - «(� � S5SGS � � / ��Nr��*��;� © �����)���/*f �6�� i%e����h5�;�� b��D���5�k��%^ ��5�Kh��6�_ 5��;�j,e�;4� 5��%��U�5%^�2cK��! ���$��K3! *�%I�����������)�������f� �(�;�� 5�
Ç"ǹæ�È'É�Ê�Ë�Ì
A ���:��%^�#¿v*^��J���*^���'���2�5�������/%����D�$�:�^�����I*����}���
���<; Ë �>=¨� m �¡u ~��[�istjls�q��*�i^��i^]wi¶%�nºn¥§�4q>�[^]h��i^�uHw��}jly{^0�&Y�s Åyl^mh4º pÅ�4� �>Ç ��� �T^]wi¶��Tº�¥»��qi�i^]h2²Î^�j?! �i^�³)�&Y[s<y{^]h@ Ë �°��= Ç � � �*�nº�¥»��qi�i^ 1��² � !!�i^E�M³i�ª5�/�&Y[s Åy{^]hACB Í ��r��4w[rA�i^�² � ! ^]�8h:�����;�/¸v�U 5��7Q��¿ß�}r���qNylstjlY�\G^G�mÀ��F ² � !NP
R�B Í ��r��4w[rA�i^ �¹^4�*�4q�\_�[r��4\_^4�Hj _�i^4wi¶#¿M³ � ¾0��o[stj{h&À� F �Lµ´« ��ZEPË B � µ� F RSP � Z� F � � PÊ B � Ë ��� � ª ³%« ª Ì ËÉ B � µ� F V���0�H� Z3[ P�� Z� F ��� V�H� Z3[ P��B Ç p�Ì�� Ë �� B � µ� F V� � µr[ �$� � )b� � Z\[ � P�� Z� F � � µr[ � � V� � Z\[ P
D positif ou nulD négatif
0000
0010
0001
0100
0011
0101
0110
01111000
1001
1010
1011
1100
1101
1110
1111
S positif ou nulS négatif
000
100
010
011
101
110
001
111
100
(a) Représentation de D en complément à deux (b) RN−code représentant S
]\^`_�a<�7bi¬K¶i^]\G��rt^���� �����is½j{stqN���[^��i^4wi¶��nº�¥»��qi�i^mhn�i^��i^4w[¶��&Y[s<y{^]h]
� � ,%������-.�����/�0������' �� %��,�� �¹���[�/���� ~n¤��*�i^�\�w[rtjls��[r�st^]yn�i^4w[¶��nºn¥§�4q>�[^]h2²Î^�j?!�µi��q�w�h6�<q�w[¬Nq���h6�M�}y�^�¶i^]\G��rt^��4��r��4w[r�^4y
².!·� �M² µ «�² Z � �W! µ «3! Z �T�|² µ ! µ «�² µ ! Z «�² Z ! µ �1² Z ! Z ¿$�°À^]� ��qN\_�[rt�]\_^4�Hj ��i^4w[¶#^4j_�4q��>¬�^]ykj{styprt^%yl�mhkw�r½j&�Jjp^]�Ãw[� �nº�¥»��qi�i^NvFA�´��ylqN�[yls���j{�´�¸�H�}y&�}�HjlstjGuHw[^%rt^��[y{qi�iw[stjG�}�����}yljls�^4�Hj - «T¾�� � Ze� � S5S5S � ¾�� � Z-�>/*^�j���q�w�h_wij{str�svhkqN��h_r��)\�«4\_^´�Nh$j{w���^*uHw[^´��st¥»�i^]h{hkwMhG�}¤��Â�i^.rt^´�4qi�i^4y�hlw[yM�R¾�³ «�ª��_�&Y[s<y{^]h]TF¼^mh�[y{qi�iw[stj{hL²��,!� ^4jL²��,!� �M^]w[¬�^]�HjT�4¬>sv�i^4\_\_^4�Hj7hl^��4��r��4w[r�^4y r � �}sv�i^��i^�\pw[rtjls��[rts�^4w�y{hT�[q��´hlst�N�[�]h]�x®rtw�hls�^4w[y&hn�[y{q���yls���j{�]h�i^mhn�nºn¥§�4q>�[^]h6�<^4y{\G^4jkj{^4�Hj�jlqNwijl^4¡¨q�svh��i^]hnhls�\G��rtst¤M�4�}jls�q���h6��q}j{��o[r�^]h]
�� �� ����������������� �"!��#�%$&���'� �(!*)ºTq�w�h6�i�m��y{st¬Nq���h/w[�[svuHw[^4\_^]�Nj�r�^��4�}rv��w�rÅ�[^´²��,!�6'FA^]h6�[y{q��[y{s���jl�mh/^�jn�}r���qNylstjlY[\_^mh®�[y{q��<qNhl�]h/h � ���[�[r�s�uHw[^]�Nj��]�N�}r�^4\_^]�Nj
²��(!6� � Ë q��>= � � � u\z){ �����r² *�%i�L�������G��G� { �£¯ �� � ª�&��/�}�/�� n¯ �� µr[ �¯ �� Z\[ ��R9�5��¯ � �|Re�Í ^4jkjl^*��ylqN�[yls���j{��hl^*�i�]�[w[s½j�s�\_\_�]�isv�Jj{^4\_^4�Hj¸�i^.rv�)�[��¤��[stjls�q��Â��� w[�Â�nºn¥§�4q>�[^�z~n���[rtsvuHw[�4^´��w �]�}rv��w[rn�[^b²��H!��6µK^]rtr�^
s����isvuHw[^7uHw[^Thls[¯ �� " �+ ��ª�µ�jlqNw�hzr�^]hzjl^4y{\_^]hz�[^nr���¡¨q�y{\G^�¯ �� Z3[ "-,Hµ�¯ �� µr[ "-,>µN¯ �, " �+ Z\[ ^4j�¯ �, " �+ µr[ hkqN�Njz�>w[rvh]µ ��.#+ - R � S5SGS � ³�«Fª /¿¨¤���w[y{^����HÀ�'¬®rtr�^��<^4y{\G^4jn�4�H�}r�^4\_^4�HjT�i^�y{�]�iw�sty{^��i^�\_q�stjls���rt^���q�\�o�yl^��i^���ylqi�iw[stj{hn���}yljls�^4rvh _���[�[s½j{stqN�[�[^4yp¿ç¤��Nw[yl^��N�]À�'¬z�^ <^4j]µ
¾ � µ + �߯ �� " �+ �³¯ �� µr[ " �+ Z\[ ��� ¾� µ + �߯ �� " �+ )%¯ �� µr[ " �+ Z\[ �
å d ÇAåji
������������ ������ ���������������� "! �������#�$��%'&(��)�)*��+�#� ,-�.�$���/�0�$�/%1���23! 45�64�7/�/�0�$�/%8��*9�:�����;4 �
ºnqNw�h�hlq�w[Y���s½j{q���h0�N�[�istjls�q��[�[^]y�rt^mh0�[y{q>�[w[s½j&h�����ykj{st^]r�h0^]� ���/�����/�U 5��7Q� %r � �}sv�i^��i^���qN\_�[yl^mhlhl^4w�y ��� � ¾0����n�}�[�<^4r�q��Mh�uNw�^���^]hq��<�4y&�Jj{^4w[y&h/���4��Z4y{^4�Hj�w[�*�&Y[sÅyl^0�i^�hkqN\_\G^���� + � � + �/^�j7���!«� �®o[stj{h��i^0y{^�jl^]�>w[^�hkqNykj&�}�Hjl^���� +� �����ykj{sty��i^�� o[stj{h��[^c\�«]\G^�MqNs���h � � +� ^4jT�i^E����«� �®o�s½j&hn�i^�y{^�jl^]�>w[^0^4�Hjly&�}�Hjl^�� � + Z\[ � B��]�QC
¾� � + ��� Z���
��� � � � +��� � � + ��� Z3[���� � � � +� ��� Z���
��� � � � + Z3[ �
�»rK^]hkj0�[q����pst�Hj{�4y{^]h{hl���Nj��i^Gyl^]��y{q�w[�<^4yTjlqNw�h0rt^mh7jl^4y{\_^]h0�i^G\�«4\_^_�MqNs��[h��[^]hc�[y{q>�[w[s½j&hc���}yljls�^4rvh0�[�}��h0�i^mhc¬�^]��jl^]w[y{h0�i^Go[stj{h� � + ��� � +, Z\[ S5S5S�� � +� uHw[sK��qN��h$j{s½j{w[^4y{q��Hjcrt^mhT^]�Nj{yl�]^]hc�i^]h0�4q�\_�[y{^]h{hk^]w[y{h�¿ß�}r���qNylstjlY�\G^G�¸^4j7¤M��w[y{^p�N�]À�źnqNw�hcq�oij{^4�[qN��hc�}s���hks�R¾�³%« ª��/\_q}j&h��i^���qN\_�MqNykj&�}�Hj6�[^ª� ! M³r¦}¾�"�o[stj{h]
y+6x+
4 y+4x+
4
y+5x+
3
y+6x+
2
y+5x+
4
y+6x+
3
y+5x+
6 y+2x+
6y+3x+
6y+4x+
6y+6x+
6
y+5x+
5 y+3x+
5y+4x+
5y+6x+
5
y+5x+
2
y+3x+
4
y+4x+
3
y+6x+
1
y+1x+
6
y+2x+
5
λ =0(7) y+
6x+1 y+
5x+2
λ =2(7) y+
2x+5 y+
1x+6
y+0x+
0
y+4x+
2
y+5x+
1
y+2x+
4
y+3x+
3
y+6x+
0
y+3x+
2
y+4x+
1
y+5x+
0
y+1x+
4
y+2x+
3
y+2x+
2
y+3x+
1
y+4x+
0
y+0x+
4
y+1x+
3
y+1x+
2
y+2x+
1
y+3x+
0
y+0x+
3
y+0x+
2
y+1x+
1
y+2x+
0
y+0x+
1
y+1x+
0
λ =0(0) y+
0x+0
λ =0(2) y+
2x+0 y+
1x+1
λ =1(2) y+
0x+2
y+0x+
6
y+0x+
5y+1x+
5
Λ(7)
(c)
+jy = 0x+
i
+2x+
2y = 1 y+5x+
0 y+4x+
0 y+3x+
0 y+2x+
0 y+1x+
0 y+0x+
0
y+3x+
1 y+2x+
1 y+1x+
1 y+0x+
1y+4x+
1
y+0x+
2y+1x+
2y+2x+
2y+3x+
2
y+0x+
3y+1x+
3y+2x+
3
y+0x+
4y+1x+
4
y+0x+
5
y+5x+
1
y+4x+
2
y+3x+
3
y+2x+
4
y+1x+
5
y+5x+
2
y+4x+
3
y+3x+
4
y+2x+
5y+3x+
5y+4x+
5y+5x+
5
y+5x+
3
y+4x+
4y+5x+
4
+jy = 0x+
i
+jx+
i y = 1 y+5x+
0 y+4x+
0 y+3x+
0 y+2x+
0 y+1x+
0 y+0x+
0
y+3x+
1 y+2x+
1 y+1x+
1 y+0x+
1y+4x+
1
y+0x+
2y+1x+
2y+2x+
2y+3x+
2
y+0x+
3y+1x+
3y+2x+
3
y+0x+
4y+1x+
4
y+0x+
5
y+5x+
1
y+4x+
2
y+3x+
3
y+2x+
4
y+1x+
5
y+5x+
2
y+4x+
3
y+3x+
4
y+2x+
5y+3x+
5y+4x+
5y+5x+
5
y+5x+
3
y+4x+
4y+5x+
4
(a)
(b)
]\^`_ aM��b~n�[�[r�sv�4�Jj{stqN�%�i^7r����[y{q��[y{st�4jl�0�iÅ¿ß�NÀ/`>sů µ� " µ� � ªNµ�j{q�w�h�r�^]h�jl^]yl\_^]h/�i^7r���¡¨q�y{\G^c¯ µ [ " , µN¯ µ� " , µ>¯ µ , " µ[ ^4j/¯ µ , " µ� hkqN�Hj�>w[r�h]A¿ßoMÀ6b*w[rtjls��[r�s��]�Jjls�q��*�i^7!�µ£� �$ªR�R[ªR[ª5� � ���}y2²9µ1� �#R[ªGR�ªGR�ª�� � �¿ß�]À6�n^4��y{q�w��M^]\G^]�Hj��[^]h/jl^]yl\_^mh6¯ µ� " µ+ ^4jn¯ µ� µr[ " µ+ Z\[
�7����h7r��%hkw[stjl^NµM��q�w�hc�i�4��q}jlqN��h7rt^G�&Y[s Åy{^p�i^G�MqNs���hc¾ � �i^Gr��%hkqN\_\G^p�i^]hc�[y{qi�iw[stj{h7���}yljls�^4rvhn����y �#� �;� � � ��-FA^]h7�[y{q��[y{st�4jl�mh�i^]h��Tº�¥»��qi�i^]h6�<^4y{\_^�jljl^4�Hjn�[r�w�hlst^]w[y&h�hks�\_�[rtst¤M�]�Jjls�q��Mh��i^�r � �����is½j{stqN�[�[^]w[y����/�����/�U 5�/7/�6s��Nj{^4y{¬�^]���}�Hj��[�}��h6r�^��4��r��4w[r¼�i^�²��,!��¿ç¤��Nw[y{^c�>À´]b�FA^��&Y[s Åy{^0�[^��MqNs��[h7¾�[7�iw.�[y{q>�[w[s½jn^mh$j��]�N�}r� � � [ �$� � [ � b�F���[y{q��[y{s���jl�G��N��y{���Hjlstj7uHw[^phls � � � � �$ª�ª5� � µ<��rtqNy{h � � � � �$R�R � � � �[^�y{^�j{^4�>w[^p���4���4y{�4^��[����hTrv�¸�4q�r�q��[��^��i^��<q�sv�[h¾ � ^]hkjT�}s���hks£��ylqN���}�N�4^0��� w��[^�hk^]w[r�^c�<qNhlstjls�q���^�jnr�^�o[s½jT�[^0hlq�\_\_^ � � ^]hkjT�i��¤���s¼���}y� � � �%� � � � ' � � � [ ��)&� � � � � � � [bºTq�w�h6\_qN�Nj{ylqN��h6�i^�\�«4\_^�uHw[^ � � � �#� � � � ' � � � [ ' � � � � �*)&� � � � � � � [
xK�}y��4q���hl�]uHw[^]�Njmµ'���L� R[µ �-��+�- R � SGS5S � �4/NrF¼^��4��r��4w[r®�iw �&Y[sÅyl^��i^¸�MqNs��[h�¡¨q�ylj��i^K²��,!� ^mh$j���rtw�h�hkw[o[jls�r®^�jp^�¶i�[r�q�stjl^rv��[y{q��[y{st�4jl��hlw[st¬J���Nj{^�]� � Ë q��>= � � � % z){ �-² �� ��O*�%_���k���:���)��L³¾�:��� �(�;�� �&O�/�6����
ª ² � ª � ² ������ � �� �! ¾ �R¾ � « ª�� ��\³¾�� ��3,'�����¾ �R¾ � « ª��3� ª
���%e��%Í ^�jljl^�[y{q��[y{s���jl�_�[qNw�h��<^4y{\_^�j��i^�\_qN�Nj{yl^]y�uNw � ��w���w[��^�[ylqN���}�H�Jj{stqN���i^y{^�j{^4�>w[^�[^�[y{qJ¬>st^]�Hj��[^r��*�4q�r�q��[�[^�i^�<q�sv�[h
¾�� � Ze����7^��[rtwMh4µ � � � Ze� �$� � � � Z-� � � ¯ �� Z3[ " �� Z\[ ^�j � � � Z-� �|R�Ç"ǹæ�È'É�Ê�Ë�Ì
� ���:��%^�#¿v*^��J���*^���'���2�5�������/%����D�$�:�^�����I*����}���
���<; Ë �>=¨� m � %��c�4�[�]y{�}jls�q��.�i^]h6�[y{qi�iw[stj{h��M�}yljls�^4rvh��<q�w�ynrt^��]�}rv��w[r¼�i^�²��H!��6º pÅ�4� �>Ç ��� �T^]wi¶��Tº�¥»��qi�i^]h2²Î^�j?! �i^�³)�&Y[s<y{^]h@ Ë �°��= Ç � �V¾/³�« ª5�®jl^4y{\_^]h � � + �$� � +, Z3[ S5SGS � � +�ACB � � � � Fί �� " �� P�� � � � Ze� � Fί �� Z\[ " �� Z\[ PR�B � Ë ��� �|R �� � � « ª�� Ì ËË B � Ë ��� �¾R �/Ì ËÊ B � � � � µr[ + F ¯ �� + " �� � µr[�Ze� + )¸¯ �� + µ?[ " �� � Ze� + PÉ B Ç p�Ì � Ë ���B Ç p�Ì�� Ë �� B � Ë ��� � ª ! � � « ª " Ì ËÌ B � Ë ��� �¾R � « ª Ì Ë��B � � � � + Fί �� + " �� � Z-� + )%¯ �� + µr[ " �� � Ze� + Z\[ PA�B Ç p�Ì � Ë �ACACB � � � � � Fί �� � " �� PA R�B Ç p�Ì�� Ë �A Ë B � Ë ��� ��� � µr[� � ³.« ¾ Ì ËA Ê B � Ë ��� �¾R )³.« � « ¾ Ì ËA É B � � � � + Fί �� � µ"� + Z � µr[ " �� Z-� + Z3[ )%¯ �� � µ3� + Z � µ3� " �� Ze� + Ze� PA ��B Ç p�Ì � Ë �A � B � � � � � Z � Z\[ Fί �� Z3[ " �� � Z � µr[ PA Ì B Ç p�Ì�� Ë �A��B � Ë ��� � � Z\[� " )³.« ¾ Ì ËR��B � Ë ��� �¾R )³.« � « ¾ Ì ËR�ACB � � � � µr[ + F ¯ �� � µ3� + Z � µ3� " �� Ze� + Z\[ )¸¯ �� � µ3� + Z � µ"� " �� Ze� + Ze� PRCR�B Ç p�Ì � Ë �R Ë B Ç p�Ì�� Ë �
bU`>s¼¯ �� Z\[ � " �� Z3[ �|R�µi�[q�w�h��4q���hlsv�i�4y{q���hO²��Ã^�j?!�� �4q�\_\_^��i^]h���q�\�o�yl^mh/�[^1�M³�« ª5�/o[s½j&h�^�jnqNoijl^]�[q��Mh
² � ! � �
��� ��! ¾4�V¾ � Z3[�« ª5�
� � � ¾�� � «#¾ � µ3���8�� � ¾ � � Z-� hls3³F^mh$jn�M�}s�y ¾4�V¾ � Z3[�« ª5�3�ª � � � ¾�� � « ¾ � µ?[v�ª� � ¾ � � Ze� hlst��q��
F¼^_�[y{qi�iw[stj�²��H!��U^]hkj���st�Mhks9w[�)�[q�\po[y{^G�i^O�V¾/³´«� �7o[s½j&h0^�j��}wM��w[�[^_y{^�j{^4�>w[^G�[^_�M^]wijk«�j{yl^G���]�[�4y{�4^Gq�w��[ylqN���}�N�4^�[�}�Mh6r��_��qNrtqN�[�[^��i^��<q�sv�[hn¾�� � Z-�}
bU`>sA¯ �� Z\[���G" �� Z\[ � ª�µ[��q�w�h7��q��Mhksv�i�4y{q��Mh�w[�*q��<�4y&�}�M�i^���qN\_\G^�w[�*�[q�\po[y{^��i^k³#o[stj{h7^�jTr � �}wij{yl^��4q�\_\_^�w[�*�[qN\�o[y{^�i^1�M³�« ª��/o[stj{h]�~ns���hlsRµ
² � ! � ��� �! ¾4�R¾ � « ª�� � ¾4�V¾ � Z3[�« ª5� � ¾��
� µr[6«#¾ � µ3�6« ¾ � µr[v�8�� � ¾ � � Ze� hls\³)^]hkj6�M�}s�y¾4�R¾ � « ª��3� ª
� ¾ �R¾ � Z\[�« ª��3� ª � ¾��� µ?[/«#¾ � µr[�« ¾ � �ª� � ¾ � � Ze� hls��[q��
F¼^c�[y{q>�[w[s½j(²��(!�³�4j{���Nj6w[���[q�\po[y{^c�i^E�V¾/³�«�¾ ��o[s½j&h4µ[��w���w��[^7y{^�j{^4�>w[^c�[^0�M^]wij�«�j{yl^c���]�[�4y{�4^7q�w��[y{q��������4^c�[�}�Mh�rv���qNrtqN�[�[^��i^0�MqNs���hT¾ � � Ze�
b|>�st����rt^]\_^4�Hj]µ[hlsů �� Z3[ �I" �� Z\[ ��ª�µ>�[q�wMh/�[�]�iw[svhlq���h6�i^0rv���[y{q���yls���j{�7�GuNw�^´²��H!�� � ¾�� � Z3[°'FA^7�[y{qi�iw[stj�^]hkj6��st��hlsÅw[��[q�\po[y{^c�i^E�V¾/³�«³ª5��o[s½j&h4 Í qN\_\G^�� � � � Ze� � � ¯ �� Z\[ " �� Z\[ ��ª�µi�}wM��w[�[^0�[y{q��M�}�N�}jls�q����i^0y{^�j{^4�>w[^c�[^0�M^]wijn�[y{qJ¬�^4��sty/�i^r��_�4q�r�q��[�[^��i^��<q�sv�[hn¾�� � Z-�}
å d ÇAåji
������������ ������ ���������������� "! �������#�$��%'&(��)�)*��+�#� ,-�.�$���/�0�$�/%1���23! 45�64�7/�/�0�$�/%8��*9�:�����;4 @
s 1s 2s 3
HA
λ0(2)
λ1(2)
y+2x+
0 y+1x+
1
HA
y+0x+
2
λ0(1)
y+1x+
0 y+0x+
1
λ0(0)
y+0x+
0
s 0
y+3x+
0y+2x+
1
λ0(3)
λ1(3)
y+1x+
2y+0x+
3y+4x+
1y+3x+
2
λ0(5)
λ1(5)
y+2x+
3y+1x+
4
c 0(4)
λ0(4)
λ1(4)
λ2(4)
s 4
y+3x+
1y+2x+
2y+1x+
3 y+4x+
0y+0x+
4
FA
λ0(6)
λ1(6)
HA
λ0(7)
λ0(8)
s 8 s 7
c 0(6)
s 5
y+4x+
4+3yx+
4 y+4x+
3 y+2x+
4 y+3x+
3 y+4x+
2
HA
s 6
c 0(5)
y+5x+
1y+4x+
2y+3x+
3y+2x+
4 y+6x+
0y+1x+
5y+0x+
6
(4, 2)
λ0(6)
λ3(6) λ1
(6)λ2(6)
y+4x+
1y+3x+
2y+2x+
3y+1x+
4 y+5x+
0y+0x+
5
λ0(5)
λ2(5)
y+3x+
0y+2x+
1
HA
y+1x+
2y+0x+
3
λ0(3)
λ1(3)
λ0(2)
λ1(2)
y+2x+
0 y+1x+
1
HA
y+0x+
2
λ0(1)
y+1x+
0 y+0x+
1
λ0(7)
λ2(7)
y+5x+
2y+4x+
3y+3x+
4y+2x+
5 y+6x+
1y+1x+
6
λ2(8)
y+5x+
3y+4x+
4y+3x+
5 y+6x+
2y+2x+
6y+6x+
3y+5x+
4
λ0(9)
λ1(9)
y+4x+
5y+3x+
6
λ0(10)
λ1(10)
y+5x+
5 y+6x+
4
HA
y+4x+
6
λ0(11)
y+5x+
6 y+6x+
5
λ0(12)
y+6x+
6
c 0(4)
c 0(5)
c 0(6)
c 0(10)
FA c 0(8)
s 8c 8 s 7c 7s 12 s 11 s 9c 9s 10
λ1(8)
λ0(8)
λ1(7)
c 6 s 6 c 5 s 5 s 3
y+3x+
1y+2x+
2y+1x+
3 y+4x+
0y+0x+
4
FA
λ0(4)
λ2(4)
s 4 s 2 s 1
λ0(0)
y+0x+
0
s 0
λ1(4)λ1
(5)
FA
HA
FA
HA
FA
HA
(a) n = 5
Compresseur
(b) n = 7
Compresseur (4,2)
Compresseur (4,2)
]\^`_�a<� b Í ��r��4w[r¼�i^�²9µH!�µ r�q�y&hluHw[^_¿ß�HÀO³D� ® ^4j�¿¨oMÀO³d���>
F¼qNy{h{uHw[^L³D� ® ¿ç¤��Nw[y{^7�H�NÀ�µHrv���[y{q���yls���j{�c���[q�w�h/�H�}y&�}�Hjlstj/uHw�� �}w��4w[�[^cyl^4jl^]�Hw�^T��� ^]hkj��[ylqN���}�N�4^T�[^7�[r�w�h���� w[�[^c�<qNhls½j{stqN���}ºnqNw�h�q�o[jl^4��q���h6�}s���hls ����� �#� � � � ' � � � [ ��).�%� � � � � � � [ ) � � � � � � � � )&� � � [ � � � � � ���L�$� � � � )&� ��� � � ��� [�
�� �#� ��� � ' � ��� [ ��) � � � � � � � [ ��L�$� � �
^�j � � �¾R[µ �-�(+.- R � S5SGS � ½4/Ni�TZmhnuHw[^�³� �[µ[�i^mh�y{^�jl^]�>w[^]h/�<^4w[¬N^4�Hj����}yn�4q��Hjly{^chl^0�[y{q�������^4y6�i^��[r�w�hlst^]w[y&h��<qNhlstjls�q���h6^4jn�[q�w�h^ Å^]��jlw[qN��h�r�^]h��4��r��4w[rvh6^4�8���/�����/�U 5�/7/���FA��is6°���w[rtjl���4q���hls�hkjl^ _�is�\_^4��hlstqN�[�[^]yn��q�y{y{^]��j{^4\_^4�Hj6rt^mh���qN\G��yl^mhlhl^4w[y&h���� � ¾ ��ºL»¼Ç �1q � Ç x"z�{ ���$��%���³d����&�² µ � �$R�ªGR�ªGR[ª5� � �5��! µ � �$ªR�R[ªR[ª5� � ��Nr��*��;���Hhj���^¿k! �������#�$��%d��®¯ µ� " µ� �5�?��®¯ µ� " µ� ,-�;�/7/�Ga5*^�*�%e�k�;���j�5%^*^��a5*����� ��3,e�;��,'�G��4�����*f 5a5*T! cK�6�_�:���6�/%�%e� ���,^�/�$Q 7¾ ���/��¯ µ� " µ� �¯ µ� " µ� � ¯ µ� " µ� ��ªf�F¼qNy{h{uHw[^L³� ��µ>rt^��4��r��4w[r£�i^ � � ^4jn�i^�� � yl^muNw�st^]ykj/rt^mh/jly{q�svh®j{^4y{\_^]h � � � � µ � � � [ ^4j � � � � µ[�}s���hls¼uNw�^7rv�Gyl^4jl^4�>w[^�� � � � �N�4�[�]yl�]^�[�}�Mh0r��*�4q�r�q��[�[^�i^_�<q�sv�[h�¾ ��¼ºnqNw�h�wij{str�svhkqN��h��}s���hlsKw[� ��q�\_�[y{^]h{hl^4w[yE�M� � ¾ �7�[q��Hj�r � w[�[^��[^]h0^4�Hj{yl�]^]h0^]hkj0jlqNwJ|$q�w[y&h0�4�H�}r�^
¦4�]ylq�¿ç¤��Nw[yl^c��o<À� Í Y����4w[�%�[^]h/\_q}j{h � � � µ�� � � � ³«³ªc��q�\_�<q�ylj{���Nj � � � � µr[� "0o[s½j&h4µi��q�w�h6�4��r��4w[r�q���h�rt^mh6��q�w��[rt^mh�� � ��� � � � �r � ��s��[^��i^p�4q�\_�[y{^]h{hk^]w[y&h7��� ��� ¾0���FA�%hk^]w[r�^��is6°�4w[rtjl�_��q��Mhksvh$j{^ ��i�4jl^]yl\_s��[^4y7rt^p�[q�\po[yl^p�i^�o[stj{hc�i^pyl^4jl^4�>w[^��[ylqN���}�N�4^��i^r��_�4q�r�q��[�[^��i^��<q�sv�[hn¾ � Z3[)]b `is � ���[µi�[qNw�h6yl^m��^]¬�q��Mh�w���w��[s�uHw[^�o�s½jT�i^0y{^�j{^4�>w[^�� � � � bÂ�c�}�Mh�j{q�w�hprt^mh���wijly{^]hG�4�Nh4µ��[q�w�hGh{�°¬�q��Mh�uNw�^¸rv�F��qNrtqN�[�[^¸�[^%�<q�sv�[h_¾ � Z\[�4q�\_�<q�yljl^�w[� ��q�\_�[y{^]h{hl^4w[y$� � ¦�¾�" � ¾0�0uHw[s�N�4�[Z]yl^O� � ¦}¾ "c«` �co[stj{h � � � Z3[ + A~ns���hlsRµ�rtqNy{h{uHw[^ � ^]hkj�����stymµ¼w[�#o[stjp�i^�y{^�j{^4�>w[^_^4�Hj{y{���Nj{^��i^rv�*��q�r�q����[^��i^�<q�sv�[h�¾ �^mh$j��]�N�}r� G¦]�4y{q��`>w[���MqHhkqN��h6���}y�^4¶i^4\_�[r�^ � �|½�'FA�_��qNrtqN�[�[^��i^��<q�sv�[hn¾ � ���]��^mhlhlstjl^0w[�*��qN\G��yl^mhlhl^4w[y��$� � ¾ �����4���4y&�}�Hj
��� ÕKÝRì$ÒVêmØ Ô§×{Ô�ÝRÕ$ÒßÔVÕzì»ÚmÑÓãmÕkè�è�Õ$æ�Ô£ÚmÝR×&Ñ>à}Ù�ê°Ý������ �
Ç"ǹæ�È'É�Ê�Ë�Ì
�m� ���:��%^�#¿v*^��J���*^���'���2�5�������/%����D�$�:�^�����I*����}���
w[��hl^4w�r�o�s½j��i^Gyl^4jl^4�>w[^ � � � � \FA�¸�4q�r�q��[��^p�i^p�MqNs��[h�¾ �4q�\_�MqNykj{^�w[�)��q�\_�[y{^]h{hl^4w[y � ® � ¾0�7���[�[s½j{stqN�[���}�Hj0��s���u�o[s½j&h0�i^�MqNs���h®¾ ^4j��i^]wi¶Go�s½j&hz�i^ny{^�j{^4�>w[^�^4�Hj{y{���Nj{^��F � w����i^4wi¶_^]hkjz�[�m��^]h{h{�}s�yl^]\G^]�Hj9�]�N��r 0¦4�]ylqMµ}�M^]yl\_^4jkj{���Hj��}s���hls[w[�[^n�<^�j{s½j{^hks�\_�[r�s½¤M�]�Jj{stqN���iw*�4q�\_�[y{^]h{hk^]w[y�� ® � ¾0�c¿¨y{^4\_�[rv���4^4\_^4�Hjn��� w[�[^��4^4r�rtw[r�^k>i~�����y/w��[^���^]rtr�w[r�^72T~7À�
`>sO³ ^]hkj�����stymµ � � � ^]hkj�w[� \_q}jG�i^ ³r¦}¾�o�s½j&h4 Í q�\_\_^r��F��q�r�q����[^��i^��<q�sv�[hp¾ � Z\[��q��Hj{st^]�Njpw[�³��qN\G��yl^mhlhl^4w[y1�$³r¦}¾ � ¾0�c^4j�N�4�[Z]yl^#�$³r¦}¾0«1 �6o[stj{hc�i^�y{^�j{^4�>w[^�µM�[q�wMhn�[rv���4q���h��]�N��rt^]\G^]�HjTw[�F�4q�\_�[y{^]h{hk^]w[y7�M³r¦�¾ � ¾ ���[����hnrv����qNrtqN�[�[^��[^��MqNs���hc¾ � -F¼^mh¬N^]��j{^4w[y&h � � � µr[ ^�j � � � µ3� �4q�\_�<q�ylj{�}�Hj��&Y����4w[� �$³r¦}¾�« ª��7o[stj{h]µ£str9¡ß��wij�^4���4q�y{^p��r��N��^4y��i^mh���qN\G��yl^mhlhl^4w[y&h �M³r¦}¾ � ¾ ��¿ß�[q��Hjr � w��[^�i^]hc^4�Hj{yl�]^]h7^]hkj7¤[¶i�]^ %¦4�]ylq>Àn�[����hcrt^mhc�4q�r�q��[��^]hc�i^p�<q�sv�[h�¾ � µr[�^4j�¾ � µ3�}¼��^4\�}y&uHw[q��Mhn\�}s��Hjl^4�M�}�Hj0uHw[^ � � � µ3� � µ?[ ^4j � � � µ3� � µ3� hkqN�Hj��i^]h�\_q�j{h��i^.�M³r¦}¾�« � «Uª���o[s½j&h4 �»rz^]hkj�¡ß�N��s�rt^��i^\_qN�Nj{yl^]y�uHw[^rv�*hkqN\G\_^��i^mh � � � µ3� � µ?[ + h � ^ <^m��j{w[^ r � ��s��[^���� w[�F��qN\_�[yl^mhlhl^4w�y �M³r¦�¾c« �:� ¾0�� Í q���hlsv�i�4y{q���hnr����4q�r�q��[��^��i^��<q�sv�[h �$³_� ¾ � � ª5�� Í ^4r�rt^4¥§�4sAy{^���qNs½j �M³r¦}¾c« � «³¾0�6o�s½j&h�i^0y{^�j{^4�>w[^��7^�j���qN�Hjls�^4�Hj��$³r¦}¾n« � « ª��®o[s½j&h � � � µ3� � µr[ + i�n���[�M^]rtqN��h6uHw�� w��%��qN\_�[yl^mhlhl^4w�y��$³r¦}¾�« ��� ¾0����q�\_�<q�yljl^ �M³�«F¾ � «� �^]�Nj{yl�]^]h] �»r��M^]yl\_^4jc�iqN����r�^�j{y{��s½j{^4\_^4�Hj7�i^mh%�$³r¦}¾0« � «³¾0�?� �M³r¦}¾�« � « ª��(� �$³r¦}¾0«Ã¾ � «1 ��o[stj{hc�i^��<q�sv�[h%�M³_� ¾ � �Qª���xK�}y7��q��Mhk�muNw�^4�Hj]µirv�hkqN\_\G^��[^]h � � � µ3� � µ3� + hl^��4��r��4w[rt^��]�N��rt^]\G^]�Hj��°¬N^]�cw[�´�4q�\_�[y{^]h{hk^]w[y��M³r¦}¾c« ��� ¾ ��[ºTq�w�h��N�4�[�]ylqN��h��}s���hls�i^mh��NylqNw[�M^mh��i^¸�[^4wi¶#��q�\_�[y{^]h{hl^4w[y&hE��� ��� ¾0��µAq�W � � � ³r¦}¾p« � µA^4jp�}y{y:«�jlqN��h0r�^��[y{q>�4^]h{hkwMh0rtqNy{h{uHw[^ � � � �MµA� � ^]hkjk¥ J¥»�is�yl^� ��³r¦�¾T«���'F��_��q�r�q����[^��i^��MqNs���hn¾�� � Z � �4q��Hjls�^4�Hjn�iq��M�7w��.�4q�\_�[y{^]h{hk^]w[y&h��$� � ¾0�����]�[�4y&�}�Hj6w[�.o[s½jT�[^�yl^4jl^4�>w[^�� � � � Z � � uHw[^�[qNw�hn�N�[�istjls�q��[�[qN��h � � � � Z � � ^�j � � � � Z � [ pr � �}sv�i^���� w��[^���^]rtr�w[r�^k>i~
¾ � � � � � ��� � � � Z � � ��� � � � Z � [ ��� � � � Z � �~Tst��hlsVµ9��w���w��[^¸y{^�j{^4�>w[^¸� � ^mh$j_�[y{q��������4^ Fr�����qNrtqN�[�[^¸�i^��<q�sv�[hG¾�� � Z��� � � y{��s�hlq����[^4\_^4�Hjp�����}r�q��Nw[^��<^4y{\_^�j_r � �}r�rtqi�]�Jjls�q���i^mh���qN\G��yl^mhlhl^4w[y&hzr�q�y&h{uNw�^L³´^mh$j6st\_����stym�F � �}r���qNylstjlY[\_^7�py{�]hlw[\_^c�4^]h6��qN��hls��i�]y{�}jls�q���h�hkw�y/r � ���[�istjls�q����i^]h/�[y{qi�iw[stj{h/����ykj{st^]r�h]~TwFj{^4y{\G^�i^Grv�¸�[y{^4\_s�Z4y{^po<q�wM��r�^p�N�4�[�]y{���Nj0�i^]h��4q�\_�[y{^]h{hk^]w[y&h4µM�[q�w�h0hkjlqi�&Ä�qN��hc�[�}��hcrv�¸¬J��ylsv�}o[r�^ . r�^p�[qN\�o[y{^_�i^_o[s½j&h0�i^y{^�j{^4�>w[^ � � � Z\[ � �N�4�[�]yl�mhK�[�}�Mh9r����4q�r�q��[�[^n�[^��MqNs��[h/¾ � Z3[mN�c�}�MhKrt^n�]��hK�N�4�[�]y{��rRµ}�4^��[q�\po[y{^/^mh$j/hks�\_�[rt^]\_^4�Hjz�4�H�}r M³r¦}¾�"�« [X�q�wij{^�¡¨qNs�h]µi��^�jljl^7¡¨q�y{\�w[r�^T^mh$j6s�����qNyly{^]��jl^nr�q�y&hluHw[^�³I���[>ºnqNw�h6�°¬�q��Mhz^4�%^ Å^�j��4��r��4w[rt�7w[�¸o�s½jn�i^7yl^4jl^4�>w[^0hlq�ylj{�}�Hj{^ � � � � �[����hrv����qNrtqN�[�[^n�i^T�MqNs��[h/¾ � Í � ^]hkj®�MqNw[y&uNw�q�s���q�w�hz�[��¤��[svh{hkqN��h . ������*� � � "9«i � ª����~U�&YM��uHw[^�stjl�4y&�Jj{stqN��i^nrv��hk^m��qN���i^�o<q�w��4rt^Nµs�r£^]hkj/���]��^mhlh{�}s�y{^7�i^0�is�\G^]��hlstqN�[�[^4y6r�^c�4q�\_�[y{^]h{hk^]w[y���� � ¾0��iºnq}j{q���h6uHw[^crt^��4q�\_�ijl^]w[y � ^]hkj/s��[stjlsv�}r�s�hl�� �³Kµi�[w�s�h6�i�m��y{�4\_^4�Hjl�N¬Kj&�}�HjT�iqN�[�[��w[�.s����isv��^ � µ[str��}r�q�y&h�^mh$j6¡ß�N��s�rt^��i^0\_q��Hj{yl^]ynuNw�^
������� �� ³ ¾�� ��� � � ª¾�� � � # ³r¦�¾ hls\³�^mh$j�����sty�^4j � ��³ � µ?[� " hlst��q�� ¿R�NÀ
F � ^4¶>��yl^mhlhlstqN�p�4s½¥»�i^]h{hlw�h9^]hkjKjlqNwijl^4¡¨q�svh9^4y{ylqN�[�4^6hks^³D���[�FAq�y&h��[^�r � w[�[svuHw[^�����h{h{�}��^6�[����h9r��0o<q�wM��r�^�µJs�ri¡ß�}wijz^]�_^ Å^�jzs���h$j&�}���4st^]yw[�*�4q�\_�[y{^]h{hk^]w[y��M� � ¾0���¬z�.st�Hjl�]��y&�}�Hjn��^4jkjl^�s��i¡¨qNyl\�Jj{stqN�%�[����h�r � �muHw��Jj{stqN�)¿V��À�µi��q�w�h�qNoijl^]�[q��Mh®¤M���}r�^4\_^4�Hj
� ������ � � ���� �� ³ ¾�� ��� � �ª¾�� � ��� �� ��� �! #" �!��� �%$'&'�(" $*)+ � ���, ��� $*".- �."�� � $/".&10
� �[^�¡¨q�svhpr�^]h_uHw��}jly{^%�[y{qi�iw[stj{h ²9µ,!kµ6µv²8Z�!KZ®µv²9µH! ZQ^�jK²1Z�!)µ �]�}rv��w[r��]h]µ�r�^.yl�mhkw�r½j&�Jjh � q�oij{st^]�Hj_�4q��i¡¨qNyl\_�4\_^]�Nj r � �muHw��Jj{stqN�´¿$�mÀ�NºTq�w�h����[�istjls�q����[q���hz^4�9���/�����/�U 5�/7/��²9µH!�µ)^�jO²1Z�!�Z���st��hls�uHw[^n^4j�²9µ,!�Z´^4j�²1Z�!)µ ¿ç¤��Nw[yl^T���NÀ�^]�jl^]���}�Hj�4q�\_�ijl^��[w%¡ß��s½jTuHw[^
¯ µ� " µ� �³¯ Z� " Z� �¯ µ� " µ� )¸¯ Z� " Z� ¯ µ� Z3[ " µ� Z\[ �³¯ Z� Z3[ " Z� Z\[ � ¯ µ� Z\[ " µ� Z3[ )%¯ Z� Z3[ " Z� Z\[¯ µ� " Z� �³¯ Z� " µ� �¯ µ� " Z� )�¯ Z� " µ� ¯ µ� Z3[ " Z� Z\[ �³¯ Z� Z3[ " µ� Z\[ � ¯ µ� Z\[ " Z� Z3[ )%¯ Z� Z3[ " µ� Z\[
ºTq�w�h�qNoijl^]�[q��Mh��}s���hks¼uHw��}jly{^7¬N^]��jl^4w�y{h��i^1�V¾/³�« ª5��o[s½j&h9D ��2 µQD ��3 µ�J ��2 ^�j9J �43 jl^]r�hTuNw�^¾ D ��3 � D �42 �|² µ ! µ �8² Z ! Z ^4j�¾0J ��3 � J ��2 ��² µ ! Z �8² Z ! µ
q�W [ ��3 � � U ��3 � �N[ �43 � � Z-� �NU ��3 � � Ze� �¾R[iºTq�w�h6^ <^m��jlw�q���h/^4�Mhkw[stjl^0w[��^0hlq�wMh$j{y{�N��jls�q��%�J¤����i^��4�}rv��w�rt^]y�¾�5 ��3 �65 �42 �|²L! ^4����/�����/�U 5��7Q�4�FA^]h6y{^4�[y{�]hl^4�Hj{�}jls�q���h6�4q�\_�[r��4\_^4�Hj _�i^4wi¶.�i^«?J ��2 ^4jT�i^«T¾0J �43 hlq��Hj�yl^mhk�<^]��jls�¬�^4\_^]�Njn�i�4¤��[s�^]h6����y¾ � � Z3[ «TJ ��2 ��ªT� � � Ze��
��� �7 ª�«.U ��2 �98 ¾ � ��ª�� � � Z-����� � VU ��2 �: ë[×&ÝAÐ<;]à}Ù&ÔVÐ>=kÒRÕ@?}ØÛ×�Þ$Ù�ØÓÙ�æ°æ°Õzã°ÕKà}Ù�ÑÓã°Ò'A �CB�DFEHGÅÞkÙ�è�à}Ù�ݨÔVÕzê°æ7Þ$Ù�è�à°ÝRÕ$ÒRÒVÕ$ê°Ý'A �JIFDLKME�BON@P�DQG �
å d ÇAåji
������������ ������ ���������������� "! �������#�$��%'&(��)�)*��+�#� ,-�.�$���/�0�$�/%1���23! 45�64�7/�/�0�$�/%8��*9�:�����;4 �N�
� �W; Ë �>=¨� m � xFÍ ��r��4w[r¼�i^�²��H!��Ãr�q�y&h{uNw�^�³� ��º pÅ�]� �HÇ �N� �7^4w[¶��nºn¥§�4q>�[^]h2²Î^�j?! �i^�³��&Y�s Åyl^mh��°¬�^m�L³� �@ Ë �°� = Ç � xzy{qi�iw[stj�²��H!�� �i^1�R¾�³%« ª��6�&Y�s Åyl^mh6yl^]�[y{�]hl^4�Hjl�c^4�1���/�����/�U 5�/7/�ACB Í �}rv��w�rt^]y�rt^mh/jl^4y{\_^]h � � � Gr � ��s��i^�r � �}r���q�y{stjlY[\_^0�SPR�B Í �}rv��w�rt^]y�^4�.���}y&�}r�r�Z4r�^� � F R �j�e�,+ - R � ª � ¾ � � � � ¾/³_«�� � ¾�³¸«£ � ¾�³¸« ¾ /��� � � � F � � � � � � � [ ) � � � � � � � � )&� � � [ � � � � � � � � � F � � � � � � � [ ) � � � � � � � � )&� � � [ � � � � �Ë B Í �}rv��w�rt^]y�^4�.���}y&�}r�r�Z4r�^ � � F � � � � � ���9F � � � � ' � � � [ ' � � � � ' � � � � �� [ F � � [ � )&� � [ [ � � �9F �#� � � � ' � � � [ ' � � � � � � � � � �� � F �#� � � � )&� � � [ � ' � � � � � � � � Z�� F � � � � Z�� � ' � � � � Z�� [ �� � F �#� � � � ' � � � [ ��) � � � � � � � [ � � � � Z-� F � � � � Ze� � ) � � � � Z�� � � � � � Z�� [ �� � F �#� � � � ' � � � [ ' � � � � ��)&� � � � � � � [ � � � � Z-� F � � � � Ze� � �Ê B � Ë �)� ��� )³�« ª Ì ËÉ B Í �}rv��w[r�^4y Gr � ��s��i^���� w[�´�4q�\_�[y{^]h{hk^]w[y�� � µr[� " � ¾0�
¾�� � � ����� Å��� Z���
+ � � � � � +� � � � � ����� ��� Z3[�+ � � � � � + � ,�
+ � � � � � Z\[ + � q�W . ��������$R � � ¦�¾�"�«�� ���B Ç p�Ì � Ë �� B . F ��� ��� � � "n«£ � ª5� P��7F ³�PÌ B � Ë �)� �|³ � Ì Ë��B � F ��� ���M� � ���� �� � � " � � µr[� ">� � PA�B Í �}rv��w[r�^4y Gr � ��s��i^���� w[�´�4q�\_�[y{^]h{hk^]w[y���� � ¾0�
¾�� ������� Z���
+ � � � � � +� � � ������� ��� Z3[�+ � � � � � + �
, Z\[�+ � � � � � Z\[ +
ACACB . F �2«£SP���F��(� ª(PA R�B Ç p�Ì � Ë �A Ë B � � � Z � F � � � � Z � � ' � � � � Z � [ ' � � � � Z � � P � � � Z � F � � � � Z � � � � � � Z � [ )&� � � � Z � � � � � � Z � � ) � � � � Z � [ � � � � Z � � P
^�j¾ � � Z3[ «#¾ J ��3 ��ª�� [�
��� � ¾ � � �� Ze����� � 7 ª�« U �43 � Z3[ 8 ¾ � � ���
� � Z-����� � VU ��3 � Z\[ ¾ �xK�}yn��qN��hl�]uHw[^4�Hjmµ
¾�5 ��3 � 5 ��2 � ¾ D ��3 � D ��2 «#¾ J ��3 «TJ ��2 � ��� �
� Ze����� � 7 [ �42 � � [ ��3 � Z3[ � VU ��2 � � VU �43 � Z3[ 8 ¾ � �ª�� [�
��� � 7 [ ��2 � � VU ��2 � 8 ¾ � ¿ß�NÀ
FA���[y{^4\_s�Z4y{^Ghlq�\_\_^G�[^Gr � �]uHw��Jj{stqN� ¿R�NÀchk^�]�}rv��w[r�^G�°¬N^]�$�R¾�³´«� �c�4q�\_�[y{^]h{hk^]w[y{h%�M� � ¾0�G¿¨¤���w[y{^�JoMÀ�\¬z�)¤[¶[�}�Hjcrv��yl^4jl^]�Hw�^^4�Hjly&�}�Hj{^��iw �[y{^4\_st^]yp�4q�\_�[y{^]h{hk^]w[y #ª�µ��[q�w�h����[�istjls�q����[q���hp�}wij{q�\�Jj{s�uHw[^]\_^4�Hj�r��F��q��Mh$j&�}�Hjl^K�M � �[^%��^4r�r�w[rt^ 2T~ ^4j�w[�[^��^]rtr�w[r�^I>i~ �]�}rv��w[r�^4�Hjyl^mhk�<^]��jls�¬�^]\G^]�Hj¾��'��3 [ ���'��2 [ � [ ��2 [ � VU �42 [ ^4j�¾��'��3 � ���'�42 � � [ ��2 � � VU���2 � � ª�T>�s����}r�^4\_^4�HjmµK�[q�w�h��qN�H¬N^4yljlsvhlhlq��Mh���^�y{�]hlw[rtj{�Jj7^4�F�4q�\_�[r��4\_^4�Hj ��[^4wi¶ r � ��s��[^���� w[�F�����is½j{stqN�[�[^]w[ynj{y{�N�is½j{stqN�[�[^]rA^�jc^ Å^]��jlw[qN��hnr�^�y{^]�4q>���}��^��i^¢/qHq�jlY.hk^]rtqN�%rv�_\_��j{Y[qi�i^��i�]�4ylstjl^��[����h6rv�Ghl^]��jls�q��.�.¿ç¤��Nw[yl^����]À�
Ç"ǹæ�È'É�Ê�Ë�Ì
�°� ���:��%^�#¿v*^��J���*^���'���2�5�������/%����D�$�:�^�����I*����}���
HA
11s11s
(4, 2)
10s 10sc 9 c 9
(4, 2) HA HA HA HAHA
s 0 s 012s 12s
X Y+ +
X Y− −
0u(s)1u(s)
1u(c)2u(s)
2u(c)3u(s)
3u(c)4u(s)
4u(c)5u(s)
5u(c)6u(s)
6u(c)10u(s)
10u(c)11u(s)
11u(c)12u(s)
s 6 c 5 s 6 c 5 s 5 s 5 s 4 s 4 s 3 s 3 s 1 s 1s 2 s 2. . .
Bits provenant du calcul de
Bits provenant du calcul de
(4, 2)
2u(s)1u(c)
2v(s)1v(c)
1u(s)1v(s)
HA FA
0u(s)0v(s)
(4, 2)
3u(s)2u(c)
3v(s)2v(c)
(4, 2)
12u(s)11u(c)
12v(s)11v(c)
12p(s)12p(c)
3p(s)3p(c)
2p(s)2p(c)
1p(s)0p(s)
1p(c)0p(c)
13p(s)
1 1
Recodage
13p(s)12p(s)
1p(s)0p(s). . . 012p(c)
1p(c)0p(c). . .
Additionneur en base deux traditionnelle
. . .
2n−1 chiffres
XY
2n = 14 bits
(a)
(b)
(c)
]\^`_�a��Eb�xzy{qi�iw[stj0�[^G�i^]wi¶F�nºn¥§�4q>�[^]h0�i^_hk^]�ijc�&Y[sÅyl^mh4K¿ß�HÀ Í ��r��4w[r9�i^)²�µ,!kµ�� ²1Z!KZ®K¿¨oMÀ Í �}rv��w�r��i^�5 � �$²9µH!�µ��²1Z !�Z �9«I�M²9µ,! Z_�1²1Z�!)µ(�/^4�8���/�����/�U 5�/7/�4�¿ß�mÀ Í q��>¬�^]y{hls�q��¸�i^ 5 ^4�*��qN\_�[rt�]\_^4�Hj G�[^4wi¶�^�jny{^]��qi�[����^0�iw�yl�mhkw[rtj{�}j]�� �� � � ��&10 � ���.$'&'� " $*)+ � ��� � � $*"%- �."�� � $*".&10
� �[^��[y{q��[y{st�4jl�0�4¬>sv�i^4�Hj{^��i^]hn�Tº�¥»��qi�i^]hn�M^]yl\_^�jT�[^�yl^]\G��r��N��^4y�r�^]hT�i^4wi¶�\�w[rtjls��[r�st^]w[y&h����}yljlsv��w�rts�^4y&h��4��r��4w[r����Hj2²�µ !)µÃ^4j² Z ! Z ���}y�w[�.\pw[r½j{st��rts�^4w[yT�[q��.hks������7j{y{�N�istjls�q��[�[^]rv]� � Ë q��>= � � � x"z){ �\² *�%i�L�������G���&����6�/��
² � �8² � ² � )_² ºTq�w�h��]�}rv��w[r�q���h���st�Mhks\²L! ^4�*�4q�\_�[r��4\_^4�Hj _�i^4w[¶� Gr � �}sv�i^��i^cjly{q�svh6\�w�r½j{st�[r�sv�4�Jj{stqN��h�^4j��[^��i^4wi¶.hkqNw�hkjly&����j{stqN��h�]
²L! � �M² µ «�² Z � �<! µ « ! Z �T� �$² µ �1² Z � �<! µ � ! Z �9«#¾�² µ ! Z « ¾/² Z ! µ� �M² µ )_² Z � �W! µ )$! Z ��« ¾/² µ ! Z « ¾/² Z ! µ
ºTq}jlqN��h®uHw[^c��^4j��}r���qNylstjlY[\_^�^mh$j6�}����rtqN��w[^ �rv��\_��j{Y[qi�i^T�[y{q��<qNhl�4^��M�}y�±���y{�}j{hlw[o��c�<q�w[y�\�w[rtjls��[r�st^]y��i^7��y&�}���[hz^4�Hjls�^4y&hT¿ß¬�q�s�y����y�^�¶i^4\_�[r�^KBÓ�CçÀ�
�� � ���*& "��J"��! � � �%$1&'� " $*) �'���, ��� $*"%- � " � � $*"%&10FA^]h/�}r���q�y{stjlY[\_^]hz��jlwM�ist�mh��4s½¥»�i^mhlhlw�hz\_s��[s�\Gsvhl^4�Hj/r�^]h®yl^mhlhlq�w�y{�4^]hK�[�m��^mhlh{�}s�yl^mh �rv��y{�]��rtsvhl�}jls�q��_\�Jj{�4y{st^]rtr�^7��� w[��\pw[rtjls��[rts�^4w�y
�<q�w[yG�nº�¥»��qi�i^mh4�XAqNwijl^4¡¨q�svh4µ��[�}��h�rt^¸�]��h���� w[��^¸s�\G��r����Nj&�Jj{stqN�)r�q��Ns��4st^]rtr�^q�w³hlw[y�w��³��s�y{�4w[s½j >�x �0~e¿����$�5�6O�;�:���:�/� ����h��6�� �/�j� �����:� �4À/�isvhk�<qNh{�}�Hj��i^7\pw[rtjls��[rts�^4w�y{h����}o[r��]h]µNs�r¼hl^4\�o�rt^c�[r�w�h/st�Hjl�]yl^mhlh{�}�Hj6�i^��4�}rv��w�rt^]y�² � ²�µ*«D²1Z�^4j�!¹�N!�µ´«L!�Z^]�.�4q�\_�[r��4\_^4�Hj �i^]wi¶� Gr � �}sv�i^��i^��i^]wi¶�hlq�wMh$j{y{�N��jl^]w[y&h4µi��� ^ Å^]��jlw[^]y�w[�[^�\�w�r½j{st�[r�sv�4�Jj{stqN�*hks����[�]^7^4jT�i^�y{^]��qi�i^]y6rt^�y{�]hlw[rtj{�Jjm
� � ,%�����_,��� �/��'% ����36���0������'� !�����4�0����5��� �� ��"$3���365c�0������'8��,��������93
¢/s�^4�³uHw[^�r � �]rt�]¬J�Jjls�q��³��w³�4��yly{��h � ^ Å^]��jlw[^%�}svhk�]\G^]�Hj *r � ��s��i^%�i^]h�\�w[rtjls��[r�st^]w[y{h���j{w��is��]h��4s½¥»�i^mhlhlw�h]µ¼r��F��q��M��^4�[jls�q��³��� w[���rt�Nq�y{s½j{Y[\_^Thk�<�]�4s½¤<uNw�^ ��4^�jkj{^7qN�M�]y{�}jls�q���<^4y{\_^�j�rv��y{�]��rtsvhl�}jls�q������ w[�¸�4sty&��w�s½j/�[�]�4^]h{hkstj{���Nj�\GqNst�Mh��i^7yl^mhlhlq�w�y{�4^]h9\�Jj{�4y{st^]rtr�^]h]ºTq�w�h���q�w�h6�[y{q��<qNhlq��Mh/�[^0�]�}rv��w[r�^4y
² � � �M² µ «£² Z � � � �M² µ � � « ¾/² µ ² Z �G�M² Z � �qNw
² � � �M² µ )_² Z � � «���² µ ² Z^]�%j{sty&�}�Hj�����ykj{s¼�i^�r��G�[y{q���yls���j{�0hlw[s�¬J�}�Hjl^�]
å d ÇAåji
������������ ������ ���������������� "! �������#�$��%'&(��)�)*��+�#� ,-�.�$���/�0�$�/%1���23! 45�64�7/�/�0�$�/%8��*9�:�����;4 �m�
� � Ë q��>= � � ��� z){ �/���r² *�%_���k���:��� ��L³|�:��� �(�;�� 5�� �6�� O,-�;�G�*����$ (²��?²��·���?²9µ?²1Z� 5�/%����;�� 0,'���5�#��7/���K�5%^�(��� �%-�/�Kh��:�� ���%�R¾�³%« ª��)�����R¾�³%« ¾0�)h����# �¨� ¯ �� ¯ �� µr[ �|R'&¼¯ �� ¯ � ��R��5�A¯ �� ¯ + �³¯ �+ ¯ � �¯ �� ¯ + )�¯ �+ ¯ � �F¼^mh®��hkjlwM��^]hzst�Hjl^]yl¬N^4�����Hj®�[�}��h®r � �4r��4¬J�}jls�q��¸��w��4�}y{y{�6�4r��NhlhlsvuNw�^�¿¨¬Nq�s�yz���}y�^�¶i^4\_�[r�^kBÛ�GC¨Àzh � ���[�[r�s�uHw[^]�Nj��4�N��rt^]\_^4�Hj��}w��]�}rv��w[r
�i^ �$²��,���_]�¯ �� ¯ �� � ¯ �� ^�jc¯ �� ¯ �+ �ï �+ ¯ �� ��¾J¯ �� ¯ �+ -FA�_¤M��w[y{^�As�rtr�w�hkjly{^�r�^�\_�]�]�}�[svhk\_^��[^��4��r��4w[r��i^p�[y{q>�[w[s½j&hn����ykj{st^]r�h�MqNw[y2²9µ§²9µ ^4j(²9µ?²1Z�r�q�y&hluHw[^�³D���[iºTq�w�h/��� �°¬�qN��hzj{q�wij{^�¡¨qNs�h6���Nh/^4���4q�y{^n¡¨q�y{\�}r�s�hl�crt^mh/��rt�Nq�y{s½j{Y[\_^]h/�i^0���]�[�4y&�Jj{stqN�¸�i^]h¬�^m��jl^]w[y&h��i^0o[s½j&h � � � �<q�w[y�r � �]rt�]¬J�Jjls�q��*��w��]�}y{yl�N
x+5 0
0
x+3x+
5
x+5x+
3
x+4
x+5x+
2
x+2x+
5
0
0
x+5x+
1
x+2x+
4
x+1x+
5
x+3
x+4x+
2
x+5x+
0
x+4x+
1
x+1x+
4
x+0x+
5
0
0
x+4x+
0
x+3x+
1
x+1x+
3
x+0x+
4
x+2
x+3x+
0
x+0x+
3
0
0
x+2x+
0
x+0x+
2
x+1
0
0
x+0
x+5 x+
5x+3 x+
5x+2
x+4
x+5x+
1
x+4x+
2 x+4x+
1
x+5x+
0 x+4x+
0
x+3x+
1
x+3x+
0
x+2
x+2x+
0 x+1 0 x+
0
x+3
x−3x+
5
x−5x+
3
x+5x+
2
x−2x+
5
x−5x+
1
x−2x+
4
x−1x+
5
x−4x+
2
x−5x+
0
x−4x+
1
x−1x+
4
x−0x+
5
x−4x+
0
x−3x+
1
x−1x+
3
x−0x+
4
x−3x+
0
x−0x+
3
x−2x+
0
x−0x+
2
0
0
0
0
0
x−4x+
3
x−3x+
4
x−3x+
2
x−2x+
3
x−2x+
1
x−1x+
2
x−3x+
0
x−2x+
1
x−1x+
2
x−0x+
3
x−0x+
1
x−1x+
0
x−2x+
0
x−0x+
2
x−1x+
0
x−0x+
1
x−4x+
0
x−1x+
3
x−0x+
4
x−3x+
1x−4x+
1
x−5x+
0
x−3x+
2
x−2x+
3
x−1x+
4
x−0x+
5
x−5x+
1
x−4x+
2
x−2x+
4
x−1x+
5
x+5x+
2
x−4x+
3
x−3x+
4
x−2x+
5
x−5x+
3
x−3x+
5
x−5x+
4
x−4x+
5
0
x−4x+
5
x−5x+
4
0
0
−1x+
4x = 1
−jx = 0x+
i
(b)(a)
]\^`_ arA�b´~n�[��rtsv�4�}jls�q��F�i^Grv�¸�[y{q���yls���j{�p���<q�w�y�³8� ��K¿ß�NÀ Í �}rv��w�r��i^ �M² µ � � K¿ßoMÀ Í ��r��4w[rK�i^ ² µ ² Z ¼`>s9¯ µ� ¯ Z[ �!ªG�}r�q�y&hjlqNw�h�r�^]h/jl^]yl\_^mh��i^�r��p¡¨qNyl\_^0¯ µ� ¯ Z, µ[¯ µ� ¯ Z, µ�¯ µ , ¯ Z� µi¯ µ , ¯ Z� µ[¯ µ , ¯ Z� ^�jn¯ µ [ ¯ Z, hlq��Hj��Hw�r�h]ºL»¼Ç �1q � Ç�� z �TNT%�3! �����.*f ��0�;�����6�� D ����L,e�+� N����/�0�$�/%^ 9��%^�j����7/�5%e�/%��)�}�/�� �"! *�%e�£4��}4�7/�/�0�$�/% �/*¾�������;4�&K���/%' ��$�4��;��%' d�6�£���Q ����³D� ® �-�)�/*f O,'��*�7Q��%' ����/�6�5*��6�5�7�M²��,���bc ��! �/�$���/*9�5���;��*����(����6�LNr��*��;���e���"¿r�´,-�;�:,-���$4��j4��K��%e��$a�*^� a5*T! �/*^�5*�%e�´��45%e45�;�/�0�$�/%��)�;�5�U�5%^*^�k%e�k �*���7G�$�5%^�O�6�/�� 5a5*^��³d� ® �5�O%e�/*f � �*,�,-��������%' k�6�� ��������.*��}�� � �5� � � � N?��*��:���Q������ ��?�� ��(40�H�/�}���K��%��r,'�/ : ��$h5�6���� �����,-�.� N��5��&O��/%' �*�%-���K�/��%e��;�)���� �*��;��&§�}�)���/�}��*��§��L²9µ?²1Z ��Nr��*��;���Qhj���
� � ��'��/��,% �����'
F¼^mh��nºn¥§�4qi�i^]h/hkqN�Hj®jl^]r�h6uHw[^Tjly{q���uHw[^]y®w[�¸jl^4r£�4q>�[^T^]hkj6�]uHw[s�¬°��rt^]�Hj �r � �}y{ylqN���is�y��}w¸�[rtwMh���ylZmh4>ºnqNw�h��°¬Nq���hz\GqN�Hjly{�7uHw�� str��� ^]hkj0�M��h7jly{q��)�is+°��4str�^��� s�\_�[rv�}�Hjl^]y��i^]hcq��<�4y&�Jj{stqN��hc�}y{s½j{Y[\_��jlsvuHw[^]h0�iqN�Hj0r�^]hcqN�M�]y{�����i^]h0hlq��Hj0yl^]�[yl�mhk^]�Hjl�]h7����y0�4^]h0��qi�i^mh4ºnqNw�h_�°¬�q��Mh��[y{q��<qNhl�¸�i^mhG��rt�Nq�y{s½j{Y[\_^]hp�M^]yl\_^4jkj{���HjG�i^.��qN��h$j{ylw�sty{^%�i^mh_���[�istjls�q����[^4w[y&hp^�j�i^]hG\�w�r½j{st�[r�s�^4w[y&h4µ9�[w�s�hGs�rtr�w�hkjly{��i^]hGhlst\_�[r�s½¤<�4�Jj{stqN��hpst�Hjl^]yl¬N^4�����Hj�rtqNy{hp�i^¸r � �]rt�]¬J�Jjls�q�� �}wÃ�]�}y{yl�NAºnqNw�hp�[^4¬�qN��h�j{q�wij{^�¡¨qNs�hp^4���4q�y{^¡¨q�y{\�}r�s�hl^4yp��^�jljl^��i^4y{�[s�Z4y{^q��<�4y&�Jj{stqN�¼AºnqNw�h�^]�>¬Hsvh{�}��^]q���hc�]�N�}r�^4\_^]�NjG�i^¸�i�]¬�^4r�q����M^]y�w[�³�}w[jly{^��}r���q�y{stjlY[\_^��i^�\pw[rtjls��[rtsv�4�}jls�q�� o��Nhk��hlw[ypr��.�[y{q���yls���j{�hkw�st¬J�}�Hj{^k]��}w��rtw�h®w[���i^mh9jl^]yl\_^]h®¯ µ� " µ+ µ�¯ Z� " Z+ µ�¯ µ� " Z+ ^�j®¯ Z� " µ+ ^mh$j��4�N��r ª�Nºnq�wMhKst\_�[rv�}�Hj{^4y{q���hK^]��hlw[s½j{^n�[qHhz�}r���qNylstjlY[\_^mh^4���927��F �}¤��*�i^0rt^mhn��qN\_���}y{^4ym
�����J�6�9��'�����
B���C�~�Å~�¬Hs�¦4s�^4��s�h]7`>s����[^m�>¥»�ist�Ns½j7�Hw�\�o<^4yTy{^4��yl^mhk^]�Nj&�Jj{stqN��h6¡¨q�yT¡ß��hkjT����y{��rtr�^4r��}y{s½j{Y[\_^�jlsv�}� ;��� ���;��%' 5���5�0�$�/%' ��/%��O�6���5�#�;��%^�$�' �/�L,e*��j���� �µ¼�m�'] ����@5bH�H����µ[�G@�A[��BÓ�QC�²M ¥ Í H¢6�}|k�}y&�£µH²M>�Tw[��y{�}j]µ[`Åi±0r���µH�����¸²M ¥§b >b*w[r�r�^4ym�`>q�\_^cq��<^4y&�Jj{q�y&h9¡¨q�y6q��[¥Vr�st��^Ty&���ist¶>¥»¾��4q�\_�[wij&�Jj{stqN��h4��f��*���%e���-��¥
(���;�/���6�5�v�/%e��k�~ ��0���$h�*��j�� ' �/�L,e*��0��%H��µM���'] �N��A>b>�}�>�iµM�G@�@���Ç"ǹæ�È'É�Ê�Ë�Ì
�]� ���:��%^�#¿v*^��J���*^���'���2�5�������/%����D�$�:�^�����I*����}���
0x+1x+
4x+1x+
3x+4 x+
4x+2 x+
2x+0 x+
0x+3x+
1x+4x+
0 x+3x+
0x+2
s 1s 2s 6s 7 s 3 s 0s 5 s 4s 8
HA
x−3x+
1 x−4x+
0 x−3x+
0x−2x+
1x−1x+
2x−0x+
3x−1x+
3x−0x+
4
HA
s 4
x−4x+
1x−3x+
2x−2x+
3x−1x+
4
s 5
x−3x+
3 x−4x+
2
HA
s 6 s 2
x−4x+
3−3xx+
4
s 7s 8
0
s 1
x−1x+
0 x−0x+
1
s 0
0x−2x+
0 x−0x+
2
s 3
(a)
(b)
]�^`_�a£�7b Í �}rv��w[r¼�i^�¿R�NÀ��M²�µ(���0^�jT�[^�¿¨oMÀO²9µ§²8Z)r�q�y&hluHw[^´³d� ®
B ��C�~�¼���¢�q>q�jlY¼G~ hls����[^m�Fo[s�����ylg*\pw[r½j{st��rtsv�4�}jls�q���jl^]�&Y��[s�uHw[^N��2*^�/���j�5��� �O�f��*���%e���v��¥ �9�:���'��%^�$�� i�/%e ��,�,-�.�$��O�9�����^����K�/�0�$�� &µ��M¿R�NÀ�]Ó�}��A5bi�}�N��µ��G@H�i�N
B ��C�~� Í �}wM�&YHgN9`iw[y�rt^mh6\_qJg�^4�Mh/��� �]¬Hstjl^]y6rt^mh6^4y{yl^]w[y{h6�[����h6r�^]h��4��r��4w[rvh��>w[\_�]ylsvuHw[^]h] ' ����,-�U�� ��L�5%e�*f ������! �������45� �$� ��� { ���$�5%e���� �& (�/���~ �µ¼�N��]Û�J��@>b[��@��[µi�m�}�H�[��n^4�[w[o�rtsvhkY�^]�%st�.~nw[�Nw�hkjls�� Í �}wM�&YHgNµ>qH^]w[¬>yl^mh6��q�\_�[r�Z�j{^]h]µÅ�4Z4y{^0hk�]yls�^�µiX�q�\_^�pµi�[��H�[�4¥V�N�H�iµH�°¬J�}s�r���o[rt^��}j������������� ���������������������������������� �!��� ����"�#�� �%$��&���'�(�%#���)�#&$����+*',�*�-�.'/�0�1�2�143!5�6M
B �/C��p'¬nM±0�Hw[jlY¼ �-�'� �����2��¥ ' �/�L,e*��j��� 2�;�;���;���)� ��%H��µ�¬Nq�r�w[\_^��[K~��[�isvhkqN�%Á)^]hlrt^]g�µM�}���%^]�istjls�q��¼µA�G@�����B A�C0x9�±7q�y{�[^]ylw[�%�}���.²M ¥»b [b*w[r�r�^4ym9�Tº�¥»��qi�is��[�_q}¡��>w[\poM^]y{hG]K�i^�¤M�[s½j{stqN�*�}����hkqN\_^c��ylqN�M^]ykj{st^mh49XA^m�&Y[�[sv�4��r¼��^4�<q�yljT�}�����}¥�H�[µ-FA��oMqNy{�}jlq�s�y{^0�[^�r � �»�i¡¨qNyl\�}jlsvuNw�^p�iw´xK��y{��rtr��4r�s�hl\_^�µe¬®��qNrt^GºnqNyl\�}r�^�`>w[�<�4y{s�^4w[y{^p�i^ F¼gNq��¼µ<��A�~Trtr��4^G��� �§j{�}r�s�^�µ\A�@N��A}�FAg�qN� Í ^m�i^�¶%�>�>µ�6c��jlq�o<^4yn�����}�M
BÓ�QC�¢c[xK��ylY���\_sR ' �/�L,e*��j��� �������M�f�K�5�#�$�4�6T¶>¡¨q�y&� � ��st¬N^4y&hkstj$g�xzy{^]h{h4µ������N�[B ��C�²M?¬nK��q�o<^4ylj{hlq��¼)~2��^4© ��rv��h{hpq}¡T�is���stj{��r/�is�¬>s�hls�q��³\_^�j{Y[qi�[h] ;��� �'�;��%' 5���5�#�$��%' b��% �O�6�����0�;��%^�$� ' �/�L,e*��j���� �µT¬ Í ¥��] �[�]�>bi���N�iµ[�@N���[A��^]�[y{st�Hjl^m��s��_¬T�¬n�`>©6�}yljl¦4rv�}�M�i^4ymµ ' ����,-*��U�5� �������������5�#�$��µ �Kq�rV£��µ �j¬T¬�¬ Í qN\_�[wijl^]yn`>qi��s�^�j$gxzy{^]h{hX�wij{q�y{s���rRµ'FAqNh�~nrv�}\_stjlqHh4µ Í ~�µ¼�G@�@��[
B @�C�±_K�pKXAqi�&Y[^]y]#XA^m�&Y[�[svuHw[^]hpq}¡T\pw[rtjls��[rtsv�4�}jls�q�� �}��� �ist¬>svhks�q��³¡¨q�y_�}wij{q�\�Jj{s��¸o[st�M�}y{g#�[st¬>sv�i^4ym7�O*^�����U�5��� � �5�/*���%-�/����¥�K���:�^�/%��$�� )��%ei��,�,-�.�$��i�K�/�M�'�5�K�/�0�$�� �µA�N�N¿R�NÀ�] ��A��>bi���}�Mµ[�G@H�}��
B��]��C���8<s�\_\G^]yl\���[�¼�J���%e����� �����5� ���;���f���j�����0*��;�� �¥���� ' �����.�#J��Q 5��:9-¿ { d��%e1�M�'�5��� { �/%^�M�'�� ��~ � xzY���jlY[^mhksvh4µT`i©�s�h{h>�^]�i^]y{��r��»��hkjlstjlwij{^�q}¡KXA^m�&Y[�[q�r�q��Ng;8<w[y{s��&YAµ 27�}yljlw[���>b �cqNyly{^��K^]ylrv�}�Mµ£�G@�@H�>
å d ÇAåji
������������ ������ ���������������� "! �������#�$��%'&(��)�)*��+�#� ,-�.�$���/�0�$�/%1���23! 45�64�7/�/�0�$�/%8��*9�:�����;4 �°�
� � �9��,¸5��� �� �_,%�����_,��� ����������������� �� 2�/� -%�9��-�����3/��3�
O�;�5*�7Q� �����! ��� ��������������� ���n�n^4\�}y&uHw[q���h�j{q�wijT��� �}o<q�y&��uNw�^
² � ! � �� Z\[���� � ¾ �
��+ � � ¯ �+ " �� Z + � �
� Ze����� � ¾ �
� Z\[�+ �3� Z � µr[ ¯ �+ " �� Z + ¿¨�HÀ
FA�G�[y{^4\_s�Z4y{^0�[q�w[o[r�^�hkqN\_\G^��[^cr � �]uHw��Jj{stqN�)¿ß�HÀ/�<^4wijThl^�y{�4�]�4yls�y{^� Z\[���� � ¾ �
��+ � � ¯ �+ " �� Z + � ¯ �� " �� � ��� Z3[ ��
��� [ ¾ � �� ��+ � � ¯ �+ " �� � Z + � � � Z\[ ��
��� � ¾ � � µr[ �� µr[�+ � � ¯ �+ " �� � µ?[�Z +
� ¯ �� " �� � � � Z3[ ����� [ ¾ � �
��¯ �� � " �� �
� � Z3[�+ � � ¯ �+ " �� � Z + � � � � Z\[ ��
��� � ¾ � � µr[ �� µr[�+ � � ¯ �+ " �� � µr[�Z + ¿R��À
F¼^mhA�i^]wi¶0hlq�\_\_^]h¼st���[s��4�4^]hA���}y � �4q�\_�MqNykj{^4�HjAw[����q�\�o�yl^K����sty��i^Kj{^4y{\G^mh¼uHw[^z�[qNw�h¼�<q�w�¬�q��Mh¼�}�[�M�}y{st^]y£��y ����^ �rv���[y{q���yls���j{�®�� � Z3[�+ � � ¯ �+ " �� � Z + �
� Z\[�+ � � �¨¯ �� + " �� � Ze� + � ¯ �� + µr[ " �� � Ze� + Z\[ �T�
� Z\[�+ � � �߯ �� + " �� � Z-� + )¸¯ �� + µr[ " �� � Ze� + Z\[ �
^�j� � µr[�+ � � ¯ �+ " �� � µr[�Z + �
��+ � � �߯ �� + " �� � µ?[�Z-� + �³¯ �� + µr[ " �� � Ze� + �T�
��+ � � �¨¯ �� + " �� � µ?[�Z-� + )%¯ �� + µr[ " �� � Z-� + �
F � �]uHw��}jls�q��#¿R��À6h � �]��y{stj���st�Mhks� Z\[���� � ¾ �
��+ � � ¯ �+ " �� Z + � �'� � � � � � Z3[ ��
��� [ ¾ � ��� � � � � � �
� Z\[�+ � � �'� � � + � � � � Z3[ ��
��� � ¾ � � µr[��+ � � �'� � � µr[ +
q�W � � � � � ¯ �� " ��� � � � µ?[ + � ¯ �� + " �� � µr[�Ze� + )¸¯ �� + µ?[ " �� � Ze� +� � � � + � # ¯ �� + " �� � Z-� + )%¯ �� + µr[ " �� � Z-� + Z3[ hls � �� �¯ �� � " �� hlst�[qN�
Í ^]hK�muHw��Jj{stqN��hÅ|$w�hkjlst¤�^4�Hj®r�^]hnªm¾c�[y{^4\_s�Z4y{^]h9r�st�N�[^]hz�[^�r � �}r���qNylstjlY[\_^��[�¬z���}�[��rtsvuHw��}�Hjzr�^n\�«4\_^��[y{st���4st�<^ 0rv��hk^m��qN���i^��iq�w�o[rt^hkqN\_\G^��i^�r � �]uHw��Jj{stqN�)¿ß�HÀ�µ[�[q�wMh6q�oij{^4�[qN��h
� � Ze����� � ¾ �
� Z3[�+ �"� Z � µr[ ¯ �+ " �� Z + � ¾ � � Ze� ¯ �� Z3[ " �� Z\[ �
� Ze����� � � � �
¾ � �� Z3[�
+ � � � Z � µr[ ¯ �+ " �� � Z + �� Ze����� � � Æ�� �
¾ � � µr[� Z\[�
+ � � � Z � µ3� ¯ �+ " �� � Z + µr[� ¾ � � Ze� ¯ �� Z3[ " �� Z\[ �
� Ze����� � � � �
¾ � ���¯ �� Z3[ " �� � Z � µr[ �
� Z-��+ � � � Z � µr[ ¯ �+ " �� � Z + � �� Ze��
��� � � Æ�� �¾ � � µr[
� Z\[�+ � � � Z � µ3� ¯ �+ " �� � Z + µr[
Ç"ǹæ�È'É�Ê�Ë�Ì
�A ���:��%^�#¿v*^��J���*^���'���2�5�������/%����D�$�:�^�����I*����}���
FA^]hc�i^4wi¶*hlq�\_\_^]hns����isv���]^]h7���}y � �4q�\_�<q�yljl^4�Hj ¸�[q�w�¬�^]��w.w[�´�[qN\�o[y{^�����styc�i^�j{^4y{\G^mh4 �»r�^mh$jTjlq�w[jl^�¡¨qNs�h7wij{str�^p�i^p\_q>�[s½¤�^]yr � s����is4�4����^��°¬J�}�Hjn��� �}�[�[r�svuNw�^4y�rv�p��ylqN�[yls���j{���
� Z-��+ � � � Z � µ?[ ¯ �+ " �� � Z + � � � Ze� � Z-��
+ � � ¯ �� � µ + Z � µr[ " �� Z + Z3[�
� Z � Ze��+ � � �߯ �� � µ3� + Z � µr[ " �� Ze� + Z\[ )¸¯ �� � µ3� + Z � µ"� " �� Ze� + Ze� �
^4j� Z\[�
+ � � � Z � µ3� ¯ �+ " �� � Z + µr[ � � � Ze� � Ze��+ � � ¯ �� � µ + Z � µ3� " �� Z + Z\[
�� Z � Z-��+ � � �¨¯ �� � µ3� + Z � µ3� " �� Z-� + Z3[ )%¯ �� � µ"� + Z � µ3� " �� Z-� + Z-� �
�»r¼^mh$j�¡ß�N��s�rt^��i^�\_qN�Nj{yl^]y�uNw�^���^]h6�muHw��Jj{stqN��h6��q�y{y{^]hl�MqN���i^4�Hj��}wi¶%rts����[^mh0ªG ¾�_�i^�r � ��rt�Nq�y{s½j{Y[\_^c��O�:��*�7/�)��)�6�´,-�;�:,-���$4��j4 ���T`is3³�^mh$j�����stymµi�[qNw�h��i�]�[w[s�hlq��Mh��i^�r��_�i�4¤��[stjls�q��*�iw*�nºn¥§�4q>�[^0uHw[^
² ������
� � ��� � Z\[���� � ¾ � � µ?[ � ªR[ªGR�S5S5S&ªRÀ Á� Ã����� ������
¿$AHÀ
`>s2³·� ¾iµ9s�r/^mh$jp¡ß���4str�^��[^%\_q��Hj{yl^]ypuHw[^�r����[y{q��[y{st�4jl�¸��^]hkjG�4q�y{yl^m��j{^9]§²�������
� �kªGR � � ��¾£� ¾4�V¾���« ª��:¦/[K`>w[���MqHhkqN��h\�}s��Hjl^]���}�HjpuHw[^�rv�*�[y{q���yls���j{��hkqNs½j�¬>y&�}s�^�MqNw[y�³ ^4j�\_q��Hjly{q���h�uNw � ^]rtr�^�r � ^mh$jG�4�N��rt^]\_^4�Hj��MqNw[y�³d� ¾[�ºTq�w�hp�i�]�iw�s�hlq���h��i^r � �muHw��Jj{stqN�)¿#ANÀ6uHw[^
² ������
�U¾ � µr[ � � ��� � Z\[���� � ¾ � � µr[
�U¾ � µr[ � ¾4�R¾� « ª�� � ¾
� µ3���þ � µr[v� ¾ � µ?[�«#¾
� ¾ �R¾� µ3�6« ª5�
F��p��yl^]w[¬�^0^]hkjT�}�M�}r�q���w�^c�[����h6r�^0�]��h6q�W_³)^]hkj�st\_����stym[xK�}y��i�4¤��[stjls�q��¼µ
² ������
� � � Z\[ � ����� � ¾ � � ��ªGR[ªR(S5SGS&ªGR[ªÀ Á� Ã����� ������
FAq�y&hluHw[^K³¾��ª�µA��q�w�h�¬��4y{st¤�q���h���s�hl�4\_^]�NjGuNw�^L�R¾ �R¾�[7«Qª��T� ª��;¦�9� ªN"¬z� hkw[���MqHhl���Nj�rv�*�[y{q���yls���j{�¬Hy&�}s�^�<q�w[y)³KµA�[qNw�h\_q��Hj{ylqN��h G�[q�w[¬N^]��w%uHw�� ^4r�rt^�r � ^mh$j��<q�w[yL³K� ¾
² ������
�Q¾ � µr[ � � � Z3[ � ����� � ¾ � �
�Q¾ � µr[ � ¾4�V¾� « ª5�3�Uª � ¾
� µ3���þ � µr[v� ¾ � µ?[�«#¾�� ª
� ¾4�R¾� µ3�6« ª��3� ª
å d ÇAåji
������������ ������ ���������������� "! �������#�$��%'&(��)�)*��+�#� ,-�.�$���/�0�$�/%1���23! 45�64�7/�/�0�$�/%8��*9�:�����;4 �J�
O�;�5*�7Q� ����6��,e�;��,e���$45�U4����nºnqNw�h��i�m�iw[svhkqN��hn�i^0rv�p��ylqN�[yls���j{���GuHw[^
² � � ² � ���� ��! ¾4�V¾ � « ª5�
� � � ¾�� � µ3�6« ¾ � µ"���1�� � ¾ � � Z\[ hks"³F^]hkjn����sty ¾4�V¾ � « ª5�3�ª � � � ¾ � � µ3� «#¾ � µ3� � ª� � ¾ � � Z3[ hks��[qN�
`>w[�[�<qNhlq��Mh6\_��st�Hj{^4�����NjnuHw[^�²��£�|²�����[xK�}yn�i�4¤��[stjls�q��¼µ'²� � ¾
� « ª�«�²�����^�j
² � ² � ¾� ²
�����«�²
�����«�² �
�����
��� �! ¾�� � µr[�« ¾ � µ3�O�þ � ¾ � � Ze� hks\³)^]hkj����}s�y¾�� � µr[�« ¾ � µ3�n«#¾ � � ¾� � ¾ � � Z-� hks��[qN�
FA�_�i^]wi¶istZ]\G^��M�}yljls�^0�[^�r��G�[y{q��[y{s���jl�0�i�m��q�w�rt^0s�\G\_�m�is��}jl^]\G^]�HjT�i^�r��_�i�4¤��[stjls�q��*�iw*�nºn¥§�4q>�[^c�4jlw��[st���[����hn�4^�jn�}yljlsv��r�^�
Ç"ǹæ�È'É�Ê�Ë�Ì
Unité de recherche INRIA Rhône-Alpes655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)
Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)
Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)
Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)
Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)
ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)��������� ���� ���������� ��� ���
ISSN 0249-6399
Recommended