Modular Multiplication for FPGA Implementation of the IDEA Block Cipher

Texte intégral

(1)Modular Multiplication for FPGA Implementation of the IDEA Block Cipher Jean-Luc Beuchat. To cite this version: Jean-Luc Beuchat. Modular Multiplication for FPGA Implementation of the IDEA Block Cipher. RR-4558, INRIA. 2002. �inria-00072030�. HAL Id: inria-00072030 https://hal.inria.fr/inria-00072030 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Modular Multiplication for FPGA Implementation of the IDEA Block Cipher Jean-Luc Beuchat. No 4558 September 2002. ISSN 0249-6399. ISRN INRIA/RR--4558--FR+ENG. ` THEME 2. apport de recherche.

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(20) d1>*"%/406h#&7 H :<; >13* (1). (1). K1. α. β. K2. (1) β. α−β or 0. 0 or β−α. . 0 or β−α. 16. γ α. K4. α. γ (1) K5. acgbRn¢(jevxojfmPg9o·hWVhvb_lor¢*h_lvg_`hÌ¿W¸_cUdemnmTva¡hxW_`}nXka`g _cWv¬ o·hRnjkh<hxjkfW je~´jk^LhjkdeWbkhRT_lo Í Ì«À]| Ê { oWvx_`Wªo*°vxbeU Djhh_lg W i À h n R W i v ¸ u Z { º k j U c _ ` a c _ V W o beQ<v§RTWVA_`hvvj¢*h_l_`~A¥¯hR °vxbe_coU&oU{(jea`ha`cWWvvj ÍhRPjejkac^ ob WU+jks?^PWV~¦~½_¡hoU_loijkhxac(RTWU+vxW mT°a`bfhv_c}TWac_`^TWVWVvxg oV ¬ hWVRnoxoxWVjkovxW q;sThxa`bb;gW¼?oWªÌºg9ohxW_`W´fW°acbfq¦v*sT_`^nmTor_chxa ~Jje^nhg RTW WKB @vxWVD Í tL¬ mT_cvWª~U+mna¡hx_`}Tac_ Wv]°vbfU. (1). K3. FIFO (depth = α ). α. α+β (1). K6. α. 2α+2β+γ β. 2α+2β+γ. CLB. β. Slice 3 First carry chain. Round 1. β. Tiockiq. γ γ. γ. LUT. LUT. (9). (9). K1. α. β. K3. (9). (9). K2. β. α−β or 0. α. K4. Output round. LUT. 0 or β−α. . ×. (216 + 1). . FF. Slice 2. FF. FF. FF. (b) Embedded multiplier. FF. LUT FF. net. Critical path. FF. LUT. Tmultck FF. Tioock FF. MULT18x18S FF. FF net. (c) Embedded multiplier with an internal pipeline stage. ¸_`dfmTvxW XT²N¹i_cvrhxW ¥A rjev_`hRnUKWh_lg]°WVjkhmTvxWVo·béWVvÁ_cW¢ ¬ (a) Virtex−II CLB. T}¸_`_c}?deWmnacv_ W^TW+eor²ÎhxjkzdfiWVoV¬ {±gbeU}Tmhjh_cbe^ }njhxR jk^n~ _¡hoJbf}h_cbe^njea mTgjkvvxWvx{ Xeqj a`¹iÍ bf¬ de_cv_lhgW ¥A~ WVr~ _lg~jkWVh´LWª_l~§g W;hxbjkaljeor~Tb¦~WV_`hU+_cbes?^¦WV~Tjeo+^n~:UorjemTÂsTq-hvªje g9hx_`Vbf;^ hzÌ¿¢·¸b_`deo bksTg be1_¡hHUhmTRT}T^nWVacU oW_cUdeor^TWVmTWª^f}n~ h<}P¢*U+bfvr_cmn~Thxa¡_`Wehx^T_`¬d}TQ<achz_ ¢·RnWv_cboo]¢*_cÌ°^Th_c}TRT~mW hRh·F }P_loD\ bf_lvr~hQWªo*jkVVafkk°¥1«besnªvp_¡hhxRTsToW·_àcdfb_`^TUgWV}T~o acÍ Wbf¨LUv(WªjeWg^TR hxUKjbfhx_`vbfWf^:¨PWªbkjeDghxR¦RTW UUmTa¡bhx_`~}nmTa`_cacWb v]Rnjeo]jk^Jbe}U+hxmT_`bfa`^nh_cjk}Taac_ _`gV^LjhhxWV_`vbf^P^pjk¬waD¸T}nmT_`}?vhWRnac_`W^nvW _òzjk^hsTjkachxdeWKRTWfW(h¬§b¹iAjk_cmTvmThvxhW }Tbe¥AvxU _corjk_`N^nh~Tde_lgjkacjkqehxac¨%j qhxoRTRT~_cWo WVWjeh°WVaNjkj¢*^nhxmT_¡~hxvxR oWqA_`_c^fhVo hx¬RT}?Q<WVbAoRTbe_loHvxW a`hq bAbe~\ albowg oQmTWUVWVkU6W¥%^LVmThxfWV^T~ g beU}?be^TWV^LhV¨nj´jk_caljksTacW_c^hRTW ac_ sTvxjevqbeNqL^nF }Ta`_ °q§uvbP¨njka` 0 or β−α. FF. FF net. Slice 0. Round r (r = 2, ..., 8). Tioock. MULT18x18. FF. LUT. Second carry chain. γ. Tmultck. FF. Slice 1 LUT. Critical path. FF. LUT. .

(21) hh_c_c}TbeWV^µoac_`_`WVdfjkv:^T^P_`_c~J^no;d:~hWVRnje}P^W WVje^nfvW~TWªq½oj«be}Ph^J_cbfU_`h^LWRTh:WKW?bkIh jkgvxhx_`deRTWVW ^LW h hRP¸UKjkuvb~Z~¢·mn{ jea`b vvWJWªor_`bfUmT}Tvg acWWVUoV¬WU+^LQ<hmTRTja`W GnvozhhRTvxWWjkacdebevx_`hRTUo(_cÂébfa`´fWorUjkacDUmTa`h_c}Ta`_cWvo]jk^n~jevW hdeRAvmnjk}noRn}no8jk@Tvh¬c]_lg mThb aljk@Tvxac¬ q@ ~¬NWV» ~_lWgjkjkhalWªor~Jbg hxbfb:^n¹io_c_c~vrWVhxW v(¥AjK«U+]mT~Ta`WhÁ_c}T_cgac_`WVWVov(Ìºsn}njejkovWVj~ be| ^¬ U?_cbU~UmTacWb vxUje^T^ Í °bev jf~T¹ ~WVvxAo(iÌº_c}nUjk}TvjkacWdfUvxjeW}T^LR hxjk@nh¬_cbeÍ^no}TvxBcbe}? D«bf¬oWVA~KmnsAgRq WVjkor^¦h_cbf^T}PdWVjkvxjka`hhWVbfvv*^PvjWªhtf_cémnW]_`vx°WVbfo*v<bfh^TRTacW q ¹]_`\vhQW¥Lo jk-^nº~¦jkUgbe_`mTacqeal~:¬ s?Wjk^:_c^LhWv . (2n + 1). (2n + 1). . . î fDî.

(22) .

(23) "!$#&%(') *'+*,- ./0213*%54"6$#&78 9:<;= 31>*? . |*WV}Tacjfg _c^Td¦Ì@ Í jk^n~ °Ì Í _`^³Ìz Í qL_cWal~To Ì« Í Ub~ UKb~ _`_` _lÀÌ°g_c^LjkmThaNvWV}P¹dejWLihvRµ }nsAjkqvdfjkW_cU^n^ToWWVWhvxvWjkhvh_cbf^Tvpd:jk}Tac`¨w_`b}?¢(jkWðo1ach_ Whx^TRTv+WWhozRTmPhWorjkWVdfvporWVmnhosb_ch^ orvRTjehxbfg RTvrhWhxWWvU+^]Ì°hmns?Rna¡bAhxWbe_`}Tg acWVvxac_jk_`Whr^ v }nvjkjeUvxjeW UKhxWWv hWVv ¬Í Q<¨pRTjkW^n~ }PjkjvðjkhxÍ UWvW hhxRTWvZW Gn^njeaNje~T~WVvKÌºsPbAbfa`vxWªWVjktL^JmT_c}PvWªjo °jmT´vjkh_cRTa jeWVsTv<acW W+¥_c}T^JalÍjk¹i^n_cjkvhh_cW be¥A^n oV ¬H~WV_`^nÁg_cgW WVF oV¨1\ hRTW QVk¥%V sTacUbgmTa¡ohx_`je}nva`W_cWv^Tbk_lho h_coUKRnmnWVg}nRoa`_cWVoj(UKhg bAWV_cv^Lbeg alhmTo·WV_`~;jkhDvx_csAWi^§qdeacUWVbe^TdebW_lÁgKv_cjk^TÌ acÊd qKjeE sTo·a`vWijkWVde^nhb_l~ozhx}TºWje_`vD}?ororh(Whxacgje_ ^TjkdeW]vxWªvxo%jeq_`m^Lachbehxbfb*dfU_chg RTjÍ hW¬w_lgU+AjeqAmTa`^ca`q hjk_csn}TaàcW _`WV_cvV^¦¬wbeQ<mTRnv*Wvx_cUW °}TbevxacWWeU¨LmTW};^LhxhjkbKh_chbeRn^pv¬<WVWA_c}T^n_cg }?W Wachx_`RT^TWWoorUhxjejedea`WªDoWjeU+vs?WWVj~n´~jkWV_ca¡~ U+}T_cmn}Pa¡WVhxa`_`_c}T^TacW_ WorvhxopjebkdeTWe¹¨ _`vhW¥A_lo<«WV1tL~TmnWjkÁa1_cghxWVb oDRnjbeév Wwje°^bev<be}hhxRn_`_cbfo<^nºjkjeaeU_`^L_àchqeWV¬v^Pjka.

(24) . Q<~WVRToW _`dfp^TbW¢<vi bkL(H_chdeRTR W+jezaìdfbevx{_¡hxRTsTU acbA¢<gjeo%g bf_c}Tv_cRTdeWV_cv^njeB a`cD qi¬·~Q<WªRTog_loiv_cjesPa`Wªdf~bevxsA_¡qihxRThxU:RTW ¨ ~j+W hxGnbA^nbeWVa%~hxbKsAq+}?WvtL°bemnvxjkU h_cbeU^bÌr~ mTÍ ac¨fb }Tvxb´L_l~WªoHhRTU+W*mT}na`hv_cbf}Tdeacv_ gVjkjUh_cUbe^:WvH¢*¢*RT_¡WVhxR^ ¬ _ÙU bb~~ ~T_`´ ~~_c_c´´ Ub~ Ìz Í _` ~T_`´ Ub~ RnL(jkb^n¢·~TWa`éWªWV~vV¨(oWh}PRTjkWµvjghjfWVora`qfWª¬o§i¢*bkRThxWVWvWhRnjkh bev U+mnorh¦sPW Ub~ Ub~ Ì¿X Í ¢*or}?RTWVWgvx_cWjeapW^ngb~_`^ndKbe¬w {oo¨TmTUtLWmn^Tjhxb_`¢¾bf^ hRPÌ«Xjh s?WVgbeU²NWV~o mTW]hbKhxRT_co Í_` _`bk hxRTWvx¢*_lorW Ê be^noWVtLmTWV^fhxa`qf¨ UUbb~~ _`_` − − + > ¸ c _ e d n m v W M @ ~W}T_lg9ho¯hxRTW RPjkv~¢·jevW¾jevxgRT_`hWªg9hxmTvW bkj − UKpbb¢<~ mnL(a`b _cdeRJjea`dfbevx_¡hxRTUU:mT¬ia`h _c}Tha`}T_lgvjk_cUh_cbejk^µvx_àcbeq¦}?g Wbfvj^nhxorbe_lorv hxo(snjf_c^ orWªje~ ^ be^µhRnW mT¢*^nRTobf_`dfoW^TWV_`~µ^n}TUmhxmTo a¡hx_`}na`_cjeW^nvK~ jk^n~µj¦jevUWbo~WmTacWVacg9b hxWV~ sAq jU+omnmTa¡shxh_`}TvjeacW g ¥hWVWVvv ¸ _`dfmTvxW @T² F b~mTacb U+mTa`h_c}Tac_`WVvsnjforWª~Kbe^KhRTW pb¢< L(c_ deR¦jea`dfbevx_¡hxRTU:¬ ~W}?W^n~T_`^Tdbf^ jk^n~ ² UUbbA~~ _`_` @ Í Ì ! Ä#"%$'& ( *Z*) 22*)( 5 -* d1>*,+* 13 .- ` d1 /) ?2 5?2 #)?'+* Ub~ _` je^n~ *10 ' H*2)23 jk^n~ _`_` ] 465 * 1> .-* 7 #8) ° Ì Í 4 7 ~_c´ _` jk^n~ f]f hï gij(j(k . x

(25) y =. (2n + 1). .  xy   . 2n − xy 2 n + 2n + 1 2n > xy 2n , x

(26) y = n n xy 2 − xy 2    xy 2n ≤ xy 2n . x = 0. . . m1. y=0. ( (2n + 1 − x) x

(27) y = (2n + 1 − y). m1. x = 0, x = 1,. m1. .. n. 2 2n. y = 0, x = 0.. n×n. (2n + 1). cH.   0 = 0   xy. 2n. y = 0, x = 0, x 6= 0. y = 0, x = 0, x 6= 0. n. Modulo 2 subtracter 15:0. cL. 31:16. cH. n. Modulo 2 adder. 1. x == 0 y != 0. y. (xy) mod m. m3. 1. m2. y == 0. x. m1 = 0, 1, 2 or 3 pipeline stages m2 and m3 = 0 or 1 pipeline stage. (2n + 1). cH. 2n 2n. 1. Latency: α = m1 + m2 + m3. x != 0 y != 0. y. (2 + 1). y  n  (2 + 1 − x) cL = (2n + 1 − y)   xy 2n. 0. x. n. cL x. 16 × 16. . (2n + 1) = (−j) (2n + 1) = 2n + 1 − j,. 2n  1 x

(28) y = 0  n 2 +1−x. m ∈ {0, 1, 2, 3} 1. m3. y = 0. 1 ≤ j ≤ 2n. cH > c L , cH ≤ c L .. m2. n. (2n · j). (cL − cH + 1) 2n (cL − cH ) 2n. m1. xy. Modulo 2 subtracters. x, y ∈. Z∗2n +1. (. . ". y 6= 0,. •. y 6= 0.. x=y=0 cL = 1 cH ≤ c L x

(29) y = c L − c H = 1. cH = 0.

(30) ¤. <9H " "!$#&%' H*?'*?3 .`

(31) d1>*"%/406h#&7 H :<; >13* L. n. n. n. H. n. L. H. H. L. L. H. L. H. . L. n.

(32) . H. * .

(33) . _ì{or^LmTohsTWVh~vvWªjejeog9a`g_lhxj]v_`_cbfs?h^nRTWVoVW]~©¬ g jebf{3sPUKb}P´fga`jkWeWVvx¨é_cWVohbeRTv^KW vWVbkp¢*1bhzv¢<¢·_`h b_cL(^TVd¯_cde¤kR«jesTa`cjk_`bhwac¢(de_c^fbfohxvW_`mnhdfoRTWU vhxowb¯jk_cÂde^néW~Kbfha`hz´fv¢·_lWV~b o bkjkw^n~4hxRThWKb©omTvxWsUhxvxbjf´fg9W¦hx_`bfh^nRToiWµjkg hibfhURn}nWjk}Tvjvx_lhxg beW+vª¬6be{ijoxalojkmTvxUdeWVWJv(hxU+Rnmnja¡h hx_`}TacW je¥^nWV~ v ¬H» W RnjéW Ub~ UbA~ ~_c´ ~_c´ UbA~ ~_`´ . x. n. y ∈ {1, . . . , 2 − 1} xy. 2n ≥ xy 2n 2n − xy ⇔ xy. ⇔ xy. jk^n~. ⇔ xy. UbA~. n. n. 2n ≥ 0. 2 + 2 − xy 2 n ≥ 2n | {z } "!$#&% ' !$' () n. 2 + xy. ~_c´. 2n. Ìº¤ Í. n. +1≥2 ,. tLmnjkh_cbe^no Ì@ Í jk^P~ Ìº Í jeo. _¡_¡ jeje^n^n~~ ¡ _ e j n ^ ~ Ub~ bkhRnWvx¢*_coW jk^P~ _`_` jkjk^n^n~~ ` _ k j n ^ ~ ~_c´ behRTWVv¢*_loW ¸jhx_`bedfvmTvxÌðW;hRn W§_`}nac`jemPvxozjehxvxUKjkWhhWVWVovxohxRTWg¨ bevxvWªor}?jkbe^n^P~ ~_`^ndJjeRnvjeWvx~_l~¢<WVjk^fvxhxW_cgVbfjk}PaWVhvrb hRnbfoW~TWVoxg vx_`s?WV~_c^:AWªg9hx_`bf^ @n¬` Í ¬  0    0 cL = 1    xy cH. x 6= 0 x=0 x=0. 2n.  x    y =  0    xy. ,. x 6= 0 x=0 x=0. 2n. y = 0, y 6= 0, y = 0,. .. m1 m2. m3. Latency: α = m1 + m2 + m3. x x != 0 y != 0. y. y = 0, y 6= 0, y = 0,. 1 x == 0 y == 0. 1. cH y & "0...0". x == 0 y != 0. n. Modulo 2 adder. cL. 15:0. m2. +. n bits. 31:16. + m3. x != 0 y == 0. x & "0...0". (xy) mod m. x = 0 y = 1 45 * 1> -P* c = c = 0 2*d1>*-P 3* /) 5*- H R* + d1 /) x

(34) y = 07 52* )($/) ( * ? 2 '+ (22 + 1) = 2 7 0 4 x = 10 y = 9 4 5 * +?- 2 c = 90 • # ) c ≤ c 4 x

(35) y = c − c = 90 7 c = 07 x = 16384 y = 8 4 5 *13. -* c = 0 • ;h )?* ?3*, 4 x

(36) y = (c − c + c = 27 ' 1) 2 = 65535 7 •. most significant bit m1 = 0, 1, 2 or 3 pipeline stages m2 and m3 = 0 or 1 pipeline stage. m1. ~U_c´ b~ Ub~ ~T_`´ Ub~ _H¸ Gn_`WVdf~ mTvxpW·bP¢<² FL(_cbde~Rmnjka`b acdebfv_`hRTU:¬ U+mTa`h_c}Tac_`WVvwsnjforWª~+bf^hxRTW*Ub~A Ub~ ~_`´ Ub~ Ì« Í ( *" * )+! 22Ä#*)("% 5$ -* d1>* +* 13 .- ` )?* 2 2 |*W}Taljeg_`^nd:Ì¿¤ Í je^n~ Ì Í _c^³Ì¿ Í qA_cWal~To d1>*Z* 0 ') * ) 5* R )(3* +.-* 3 ] * 1> -P* U_` b~ UKb~ ~_c´ ~_`´ UbA~ 4 4 #7 4) 5 4 '+9 7 U_` b~ UKb~ ~_c´ ~_`´ UbA~ ] 4 5 *h)?1>* .3-* *?3 4 4 h ; 7 4 '+ 2* bkjk^n~hRThWJRTW*hz¢g bfb¯Uo}nmTjksvhxjvxhxjfbeg9vphx_lWovo^Tb_co§acbeoz^nhxdevxjeWv_`df^TRfWVhg°beWVvxox¢·ojejevvxqf~p¬D²¦Q<¢RTW·WvvxWVWU¢*bvx´_`hjeW a 213* - 3* /) 5* R* + d1 /) * )( /)7 ( 5*? (xy. Ub~. n. 2 − xy. = (xy. = (xy. n. 2 ). 2. n. 2 + 2n − xy 2n + xy. (2n + 1). . n. 2n ). 2n + 1). 2n. 2n .. ". +*. ,.  (xy   . 2n + xy 2n + 1) 2n n n xy 2 + xy 2 + 1 ≥ 2n , x

(37) y = n n 2 + xy 2 + 2) 2n  (xy  n n xy 2 + xy 2 + 1 < 2n ,. •. x = y = 0 c L = 1 cH = 0 cH = 2 n − 1 c L + cH + 1 = 2 n + 1 x

(38) y = (2n + 1) 2n = 1. •. x=0 y=1 c L = 0 cH = y = 1 cH = 2 n − 2 cL + cH + 1 = 2n −1 x

(39) y = (cL +cH +2) 2n = 0 n 2 04 7. î fDî.

(40) .

(41) "!$#&%(') *'+*,- ./0213*%54"6$#&78 9:<;= 31>*? )* +? 2 '+9 x = 16384 y = 8 45 *

(42) 1> -P* c = 0 4 c = • /) ;h )?* 3*?3 24 *?3 h c 2 =−2 2 −5 31 7 1C/ ) )( ( )(4 '+c 2+H*? c d1>+ 12 x

(43) y = (c +c +2) '+9 2 = 2 −1 = 65535 7 •. . . x = 10 y = 9 4%5 90 4 cH = 0 4 ) c + c c+L = cH = 2 n − 1 7 1 = 90 + 2n 4 L H x

(44) y = (cL + cH + 1) 2n = 90 7 H. L. n. H. L. n. L. '. #. n. H. H. n. n.

(45) $ × % . . _c^fhxb¦Ìº Í qA_cWal~To Ub~. (2n + 1). x ˜y˜. =. M + dn −. n−1 X. di 2. i. !. UKb~. (2n + 1). U b~ ÌÍ sA|*q WVUKWVU+s?Wv^nb¢ hxRnjh+j:Ub~mTacb vxWV~mng h_cbe^µ_lo~WGP^TWV~ UKb~ ÌzV Í hh» _cb }TWiac_ jkgV}Tjhx}n_à`bfq^ ² ÌzVjkacdeÍ bfhvb§_`hÌRnU

(46) Í jk°^Pbf~v bfshxje_`^j«sTU_`hb_`~^LmThWVacbdeWVvxoisPWVa`bf^TU+de_cmT^Ta`d UbA~ =. M + dn +. i=0 n−1 X. d¯i 2i + 2. i=0. !. (2n + 1).. . m. hQ<_c}TRTa`WW¥bfW}P¬:v:WVvx» hjkbhW;bfRnvD~je~WV^noxWg ~}nvxac_c_`WJg s?hWhxWVRT~+RTW³beWVv^+orW§}?¸WVRT_cgdeb_cmT¢jevxaWhxgb¦]jeoozhWVhxvxo;_`WVacjkA¢*h+g beRTg UWVbfvv}TW vxvxWV_lg orhWªacoDqj(hURnWVmTbfoa`Wv hzg¢jeob WVo¢*_¡hxR;je ^sT_`h*_`^LhWVdeWVvxo jk^P~ U+mTomna`hg_cR¦}Tacje_`WVo vV¬ pWh·mPo~W?Gn^TW _¡_¡ jk^n~ _¡_¡ _` U+mTa`h_c}Tac_ gVjhx_`bf^;_lo<hRnW^:W ¥}TvxWVoxorWª~§jfo F b~mTacb Ub~ Ub~ _` Ub~ Ìº Í UKb~ ¢*RTWvxW _` Ub~ je^n~ ~_`´ Ìzf Í AmTsnorh_`hmhx_`^nd Ub~ Ub~ _` jk^n~ Ub~ Ub~ { Wª~orhxUbje}Pa`WVvrU°bfbv~U _ G?hxgRTjW hx_Ùbf^ b~be_ GPWV~tLUmnjkbAh~T_cbemT^ a`b Ìze Í _lo¦orhU+_ca` mTa`vxhW _` K U b ~ U b ~ t L T m c _ v Ub~ }Tca _ gVjhx_`bf^ bez{¬|*WUWUsPWVvhRnjkh °bfv x = 0. y =0. (n + 1) × (n + 1) (n + 1) x ˜ y˜ ( x 1 ≤ x ≤ 2n − 1 x ˜= x=0 2n y˜ =. (. (2n + 1) = (M + 2n D). =. x ˜y˜. =. (2n + 1). 2n. n−1 X. di 2i. i=0. !. D=x ˜y˜. n X. di 2i .. =. i=0. 2n. (2n + 1) = (−1). 22n (2n + 1) = ((2n (2n + 1)) · (2n = ((−1) · (−1)). ï. f]f hgij(j(k. n. (2n + 1). (2n + 1). (2n + 1)  n−1 X    M + d¯i 2i + 2 − 2n − 1     i=0   n−1  X    d¯i 2i + 2 ≥ 2n + 1 M +   i=0. n−1 X    d¯i 2i + 2  M+    i=0    n−1  X    d¯i 2i + 2 < 2n + 1 M +  i=0. (2n + 1),. 2n =. m = x − km).. Z∗2n +1. 1 ≤ y ≤ 2n − 1 y=0. y 2n. M + dn 22n + 2n ·. M =x ˜y˜. . (n + 1). . (2n + 1). x˜y˜. (km ≤ x < (k + 1)m) ⇔ (x. !  n−1 X   i ¯  di 2 + 1 M+ 2n     i=0   n−1  X    M + d¯i 2i + 1 ≥ 2n   i=0. n−1 X     d¯i 2i + 2 M +    i=0    n−1  X    d¯i 2i + 1 < 2n M +  i=0. (2n + 1))). (2 + 1) = 1,. (2n + 1). (2n + 1) x

(47) y = 0. x=.

(48) . <9H " "!$#&%' H*?'*?3 .`

(49) d1>*"%/406h#&7 H :<; >13*. » Ê beW^nRnoWVjtLémTW WVjk^f^Phx~ a`qf¨k¢*¨ RTWV^ ¨je^n~Ì. UKb~. UmTa¡hx_`}na`_lgjkh_cbe^p¬ Í ¬¬ È ° " $ & Ä F b~mna`b ë¢W<besThxjk_c^ Äi É Ä Ub~ j Ä É UbA~ Ì «sn_¡h(_c^LhWdfWv _¡_¡ ~_c´ Ì sT_`h(_`^LhWVÍ deWVv Í k je^n~ bevÌ bkhRnWvx¢*_coW Í Ä ¸_c^njea`cqe¨°bf¬wv Q<RnWvxW °bevxW ¨¢W RPjéW ¨ ¨Tjk^P~ UbA~ Ä º Ä Ub~ Ub~ UbA~ UKb~ ÄÉ » ~WVWoxg bevxsT_`s?hxWVjk~_c^sLhxqRTW{(acUdebfbAv~T_`mThRna`b U @n¬ìje^n~U_ca`cmTmna¡orhxh_`v}nja`_lhxgWVjk~h_cbebe^^¸be_c}?deWmTvvxjWhbfTv ¬ ] 4 5 * 13 .-* * +?- 2 4 x = 1, y = 0 (2n · 1) (2n + 1) = 2n Pn−1 ¯ i D=0 M =1 M+ d 2 +1 = 2n −1 Pn−1 ¯ ii=0 i n M + i=0 di 2 +1 < 2 ! n−1 X i ¯ di 2 + 2 x

(50) y = M + 2n. 0, y = 1. i=0.   0    0 = n−1 X    d¯i 2i + 2 M +  . (x = 0, y = 1), (x = 1, y = 0),. " . M+. n−1 X. d¯i 2i + 2. i=0. !. n−1 X. 2i + 2. i=0 n. = (2 + 1) = x

(51) y.. !. . 2n. 2n = 1. y. Latency: α = m1 + m2 + m3 1. 15:0. =0. n. M. + 31:16 m1. n bits most sig− nificant bit. 32. m2. + m3. ¸_cdemnvW² F b U ~mTmna¡a`hxb _`}na`_cWvª¬ U+mTa`h_c}Tac_`WVv(snjforWª~be^:jk^ ! Ä " $" (]*? *) 22 *) 5 -*)213*8+ *1> -9 0/8d1 /) d1 2 5?2 52 0. 1) × (n + 1). ". *10 ' H*2)23. . x

(52) y ←. M+. (n +. n−1 X. d¯i 2i + 2. !. 2n. d¯i 2i + 1. !. 2n. i=0. x

(53) y ←. M+. n−1 X i=0.

(54) . x = y = 0 M dn = 1 4 di =P0 ∀i 6= n 7 n−1 ¯ i x

(55) y = M + i=0 di 2 + 2 (2. n. ]. dn =. i=0. . ". . n. '. '+9 + 1). 2 = 17. =. 0. 2n. =. n. y = 1 4 5 * 1> -P* M = 0 4 d = • x = 0 . # ) + P d¯ 2 + 14 d = 2013* ∀i 56= 07 /) *M 9 3 $* +& x

(56) y = 1 = 2 P− 1 4 '+9 '+9 2 = 0 7 M+ d¯ 2 + 2 2 =2 ] x = 10 y = 9 4 5 *

(57) +?- 2 M = 90 4 D = 0 4 • M + P d¯ 2 + 1 = 2 + 90 7 ; )?* 3*?3 4 ' P x

(58) y = M + d¯ 2 + 1 2 = 90 7 ] x = 16384 y = 8 4%5 * 13. -* M = 0 4 D = 2 4 • M + P d¯ 2 + 1 = 2 − 2 7 x

(59) y / ) d1>*? ' P "* ?3 M+ d¯ 2 + 2 2 = 2 −1 = n. i. n−1 i=0. . n−1 i=0. i. n−1 i=0. m1 = 0, 1, 2 or 3 pipeline stages m2 and m3 = 0 or 1 pipeline stage. (2n + 1). . •. (xy) mod m. x. 2n n 2 (n + 1) ! n−1 X i n dn = 0 d¯i 2 + 1 < 2 M+. M ← xy D ← xy. . Modulo 2 adder 15:0. 15:0.

(60) . . (2n + 1). =0.

(61) . y=0 y ← 2n. . . . dn = 1 M = 0. 2n =. (n+1)*(n+1) multiplier. . . 1. x=y=0. di = 0 ∀i 6= n. x=0 x ← 2n.

(62) . . i=0. (2n + 1).

(63) . . . .. . . . n−1 i=0. i. n. i i n−1 i=0. i i n−1 i=0. n. i. 0 i. n. n. i. i. n. n. i. i. n. n. 7 { e ´ V W v q g ` a sAq Ê mTv_cdeWVvZ*?bL)orW< jk7 vB gRRTD«¬]_`hWV¶^ng hormThWVvxjfW·~¦RnjebkoHNs?hxRTWWVW ^K }nmT sT0 ac_ o *RTdLWV~jhx_cWe^¨?hRTnWVq 65535. . î fDî.

(64)

(65) "!$#&%(') *'+*,- ./0213*%54"6$#&78 9:<;= 31>*?. mnje~TorW~WVjkvV^¬(Q< HRndf_cjkohbeW·}?hWbvjgbehxbeUvi}TomWhxWVW*UhoiRnW(RTb_`^n¢·}TWméhWVv*gVjkUvxv_corq hxjebee?Whx^DRT¬(Wi¸ToWVbegvbe_`^n^T~ ozh¨njkje^n^ng~ We¨k_` ¢·W<RnjéW UbA~ ¨ ~_c´ UKb~ ~_`´ °Q<beRTvxW]WWªbetLmmnhx}Tjka1mh(hxbgjebev^TvxW qjkbe^PD~§hxRT¢·)W WGPbevxsorh·hjkU_c^ b~mTacb jf~T~WVv_lo·hRTWVvW Ub~ UbA~ ¢*bkhRTRn_cWgv*R½_c^T_lo}Tmg hiacWV´jejevaca`q³mTWª¢*o¬ vxbe^TdP¬4Q<RT_coWvxvbfvKbgVg mTvo°bev§oW´fWvjka. . (g vxbeWVhjfW orWªh~RnsAjkh;qbfhxRT^T_lWeo;¬wbfA}P_c^nWVg vxWijkhhbfRTv§W]g g bf_cvxUKgmT}n_`mhhbeWª^;o§¸hxRT_cdeWµmnorvmTWiU ¤+_`Â´fbeacéWVo _c^ hU_cbeb^³~{imTÌroacªb X hÍRTW_lo*´jkjema`hxmTbeW Ujfj~Thx~_cgV_`_cho§jk_cacbev`q;^nWVoV}Tor¨?vmTWªhUorRTWVWU^LhhWVWVWV~p~-v¬U sAq °bf¨<mTj ^n~_c^g bevxtLvxmnWVjkg9 h?_c_cbeU^µUKmTWV^nv_¡Uh+je_lo ^T^;vxWVRPtLjemTo*_cv~Wª~?W Gnhx^Tb¦WV~ Rnjk^P~a`WhxRTWVoWo}PWªg _ljkagVjeoWVoV¬| ¬ _`_` jkjk^n^n~~ _` {-U+je~TmT~Ta`hW_c}Tv acjeW ¥gVg WbevHvor~WV_ca`^TWªg9dhophxb hxRTW_cjk^T^P}T~ mhHbk¬ (hxRTbeWhWKGnh^nRnjkjkaLhU|b~¬ mT1ac_`b UUWv Uje~njk~^TW^¦v§RPorhjevxoimn~Tg hWVmTo_`vxdfW^ThWVb³~:}?jk^¦WvWV°be^nvx~AU ¶jkvxhbeRnmT_co^P~Aalje gVorh§jkvxjevq~n~}P_¡jkhx_`vbfjk^pac`¬ WVa¡ u}Tvvx?WW GTGT¥¥ 6. % . |U+ ¬mTa`h?_c}T_cUKac_ gVUjhWV_cvbeU^©jkbe^n}?^ WvjRnhjebfoËvxo}Tmnvxorbe_c^T}?dbfoUWV~ b~mTUKacbkb ~vxmnWVa`~b mngWV~µ}njevrhx_cjea °je~be~nWVac´L~b_lWg ¢*vWªo<_ò ^Tjkdjkvx^nWivx~ WRT¢*bbevx¢·mn_¡hx0vW_`´f^nGnWd^nv·jkbevxaNjkhURTtLbWVmnv~jkmT_ch^Tac_cb WbeI^µgÌz_`VWV^fh<² °bfv<_logmTsnvxjfvorWVWª^f~Jh*¸bf^JuZhxRT{ W Í }GnTv^nbjk~amPjfg9~ThxoV~¨:Wv UBcbA~TmTD ¬ a`b gjebv~vxqLmn a`boxjéW¾je~n~Wvo¨:U+jkmT^Pa`h~ _c}Tacj _ gVjU h_cbbe~^6mTac_cb o Ub~ ~WGn^nWV~sAq?² F UbA~ _`_¡ UKb~ uwu UbA~ ÌrªX Í ¢*RTWvxW uwu Ìr9@ Í + ¸_cdeUmnvmTWKa`h¤_c}T~a`_cWW}nv_cg jehx^no~¯hRT_`hxWKojk°vbfgmTRTv_`hbeWVg }hhxmT_`bfvx^nWjkbea·N}Th_cRT}P_lWVoia`_cU^TW¦b~ozmThjkacb dfWVoV¬¯{ GnvxorhorhxjedeW*_cUK}na`WVUKWV^LhxowtfmPjh_cbe^Ìr9 @ Í hb+gbeU}TmhxW*hRTW+V¤ + UhK_c}Tba`~_lgmnjka`hbe_c«bevx^¦WV~_lmPo*g hxWVRT~+W^J}njkgVvjkhvx_ljkv_caAWV}T~;vxbbe~mTmnhig sAhxqoV¬ oF mTUb~Umn_à`^Tb dhRTWªorW hxWUvxUmTa¡o UKjkjk^nvxWVbe~U+mTh^ns?RT~WW v*gVg hxjkbfRnvx^nvjozqh*hjkhxjeRT^L~nh_l~o(Wbev*¢*}?¢*W_¡hx_`vhRjR hx_`bfÛbgVjk~^mTacsPb W+² vWªjkacH_ KVWV~;jesL~Tq;~TWjkv^:o¬HW|*^n~W ¸_`dfmTvxW]¤n² F b~mTacb UWvxUjk^T^ o*jkacdebfv_`hRnU¬ U+mTa`h_c}Tac_`WVv(snjforWª~be^:| ¬ ?_cUK Ub~ UKb~ Ìz Í x=y=1. 2n = 1 xy. xy. . a+b. n. 2n =. (2n + 1). n. 0. 2n + xy. xy. 2n. 2n + 1 = 1 00 . . 00} 1. | .{z. 2n. 0. (n−1)×. 2n. (2n + 1) = (1 + 1). xy. . . . . 2n = 2. $ . .  ¯  (Y , 1) ? ? ¯ 1) (C , S ) = (X,   (0, 0). . . 1). X = 2n Y = 6 2n , n X 6= 2 Y = 2n , X = Y = 2n .. x. y. (2n +. .

(66) . (2n ± 1). (2n + 1). (2n + 1). (a + b + 1). =. n+2+. n−1 X. i. i=0. !. (2n + 1),. 0 or 1 pipeline stage. (2n +. y. .... x ¯i · 0 · · · 01 · · · 1.. m2. Modulo (2 n +1) carry−save adder. =xi · yn−i−1 · · · y0 y¯n−1 · · · y¯n−i +. Modulo partial− product generator. x. Latency: α = m1 + m2 + m3 + m4. 2. m1. i. a + b ≥ 2n , a + b < 2n .. m3. 1. S. ★. S. 0 1. m4. 0 1. 2 correction. 1). n. ★. C. n. (2 +1). C. 0 1. Modulo (2n +1) adder. 2. n. (2n + 1). . c ( = c¯ ). (a + b + 1). ï. f]f hgij(j(k. (2n + 1) = (a + b + c¯ ) ). 2n .. (2n + 1). (xy) mod m. (2 + 1). 2n. (a + b + 1) a+b. n. xy. (2n + 1) =. (.

(67) V . <9H " "!$#&%' H*?'*?3 .`

(68) d1>*"%/406h#&7 H :<; >13* . . .

(69) . . acQorjWjeqhxsTo·Í a`bkW+bkDw}PhxoRTjkmTvWKUjk°UUbemTW jehxv vW_ vjkKoacWªde¬oHbfQ<hvRn_`RThWiWRnUU¹"o]jkLi_c~^Wªµoro}?gg vWVb_cgsP~_ G?WªWi~Jg ¢<_`hje_cjeWVsPo·o]bjk´fÌºmTjkWhvx°beWVbeUj v jejko^nhW_l~´fgWjk~vacWjkq a de}TWa`WV^nUWvWj^LhhWªWª~%~¨beorqA^K^L¹ihRn_cvrWVhxoW _HKV¥AWV«~¯mnjeo^n_`^n~Kd¹iA_cqAvrhx^TW }T¥Aac _¡r°q ~TuWvÁb]_cgWV¬ oNn¬ WV@nU¨ªje}T^na`b~ qA_c_cUK^Td i_ca`_çT^be¥v{i¹ia`_ccv_ljkhW ^n¥Ag W+rDA~TWvxW_`ÁWª_co<gWVnoV¬cëh¬ RRT@eW·_¿hx¬ RT_`v~Ub~mTacb UmTa¡ hor_cac}T_ ga`WV_lgojkg hbe_cbeU^}njejea`vdfWªbe~;vx_¡hxhRTbU hxRTjeWa`chzb¢·¢(b opj*pbo_`¢<df^TL(_ G?_cdegR¦jk^LjkhDacdLdejkbf_cv^_`h_`RnÛhxoVWvx¬*UQ<oDRTbk_lo df¢*jkRT_cW^ vxW_lohRnbkWU+gbemTmTa`vho_c}TWJac_àcWVWVv+oxo;_lo _cU_cUK}P}nbfa`vrWVhUKjk^LWVh^Lh°WVbe~µvbf¹i^ _cvrÊhxW ¥A« oje~^nW~µÁ_lg vWªWo thfRTmn_l_ò·vxWVºojkUhxRT_ca`WqfÜ¢jkW]_c^½jkalor}nbKjkvbehsnobkWvxéhxWRTWjRno_`jedfvx^T~H_ ¢<GPgjkjevx^fWh·v_`Wª^Porg bfvxmTEWVvxjegoWVWioVbe¬ D¸ThxRTbfWv vh~_cWVW}Tacaljka`j_chq+WWVv;~s?jkW hhzmb¢·hxhbeWRTUWVW^jjkhxjev_ca`gVgdfjkRTbeac_`vx`hq¯_¡WVhxg RTdfhUWmT^Tvxo WiWVvxbejkjkph^PWªh~~©RnW sA¬q¯Q<hxRTRT_lWowdLojkqL}K^Lhx_cRToWVg oWV_coKvrhjkhxU+bA_c^TmTbeaàla`q o mnbkÌ°be(g9^nh}noW+jeÍ ¬Uvxje¸beUKvx_c^nW+WhjeorWVa`hxcvxqejko ¨%df_lWbfo mT_c^¦vWh¥RT}PW+WVhvvx_cWÜW+Wo^LmThxUo]orURn_`b^n¨N¢Îdje^nhhRn~RnjkW+h}nhxjeRTvrWhx_cjesPaDWªoz}T_`h (vxbor¢·~AWWh ¢·je^fQ<h<RThxWJbbegje}vhxvx_ÙqL _ oxKjW]éW:hRTjeW~nhx~RTWvxv;bemTsndfjeRTo}TWV~½mh(bfbe}PDWVhvxRnjkWhbevacbeWV}?jfW~Tvojhxhxbeb¯vª¬ hxRTW ac}Tjevvbfdf}nWVjkordLh:jjehx^n_`bf~ ô_c^Ja`b¢·gVjkWVvxorvh¦qL¶g o_cjvxg´fmTW]_¡hjfo~T¬£~WAvoV_c^n¨Pg oWqL^LhhxRnRTWWVvxoW _co(_lho:bA^Tbealb¾oi~Tgjebev^ vxq h hmnob¦gorvW;_csT}~mThWV_c_calbe~~³^_lgj§jkjkhac°`Wªmnb~³¢(a`·ogjejkmn~Tvxo~vxqJWVhvbacbeg hxdeWVje_laè«gW¬§acje_`{ ^T~Wª´éo+jkWV^LjevhqJ^njk~³dea`bWijk¢<acbk` pba`WVgVhxéRTjWVh_co<aW"¹gVhzjk¢Livxbv q4ac\ibe~deQW_lgo shPjkRT^nWªW§g ~;beghUvRn_`WªWho _lvxgW jeU°beawbevx}nvxWjkWhhbgRpbe¬vUWªQ<~}TmnRTac_gWVgVWojWhxhxWVRTv~µWWªorjkjkmn^nvxa¡WVh~ ojn=¬_ch^nWVLi~^nbmn~ ¢gWVhWVb¦oéWVjeWªvVôz¨Lhhjk_cRn^nsTW gac_vovWªRbfjemohhWKRn_c^Tjk_c^d h hjkbacde¸bevxu_`hZRT{iUoVo*¬ _c^LhW^P~WV~:°bfv]{]A Ê oijevW+^Tbkhjkac¢·jqo*je~njk}hxWV~ ! H. '* ; ' + #?0®24! "Y ·' . h(jeRT~TbW~¢½_`~hhWª_cbeRPor^J_cjdeh<^;UK¢bkbWH~RnmnjKja`bgé_`W(}nRT?W IWhxv<jkg vx_cfWbeWV^LmToh^nje~;U~_lbó·jk~^LormThhacvbjkjkde_cW deRLbehH°beºjevx¢·orhrje¶Uvxg~%jkmTvx²a¡hxvx_`_`q§^L}nhxaàc_cWbeWdedevWV_logv ¨ jkorRT^nbf~ vrhx]W\^§QhRTo]W _cUKg vx}n_`ha`WV_lgUKjkWVa%^L}PhjhsnRD_¡hz¨¢*WV_ljeorgWKR;W be¥T}?gWa`mnvjo_chxébeWKv·beRnvªje¬ o<¶j^ }nbfjkvxv~jkWVUviW hxhb . . . . . . . . (2n +1). . . 2. 3. (n + 1) × (n + 1). m1 = 2 m 2 = 1. m3 = 1

(70). . . . . . . (2n + 1). 2n. . . vx¹_cg§L^LmnU+dfs?WW^Tv+WVbevxjk(h_`be^LvhxWjkvxmT^nhjkbea·U}Tjk_c}Ph_lWVga`jk_c^Tac W:qµozjfh~Tjk~TdeoKWªovxWÌºQde_ljkorsnhWa`W:voX _`Í ^¬³WVÀjfgmTR v vxg bebfmTvx~T^n_`~4^Td:_`^ hxb:bevh~RTWWªv§orW§hb½}njkg vbfjkvUvxWVW g hxhWacvqooÌº¸qA^n_cdegmTRTvxvxW¦be^T _ K¬WJ¸nhRTmTWµvrhxRT~TWjvxhUj-bejfvxg9We ¨ bes?mnW hzv¢·hWbAWbeâhz_c¢·o;bKjksTomnacW¦gVg hxWVb¯oxo_`_`´f^PWorWVvxvrbehmTvx^PW~Tdfo_c¬orhWVvxobf^Í hRTW sPbfmT^n~Tjev_cWVo deQvxjkbesTmnac};WiXbf² }P(WVvxmTjkUh_cbesP^nWVoVvw¬ bk1_`^LhxWvx^njkaT}n_`}?Wac_`^nWiozhjkdfWVobk?hRTW(hRTvxWW â 9ã áïªì â 9ã i . . . . . α ∈ {0, . . . , 3}.

(71). . α ∈ {0, . . . , 4}. . ⊕. β ∈ {0, 1}. γ ∈ {0, 1}. Pipeline stage Optional pipeline stage Round 1. Round 1. Round 1. Round 1. Round 2. Round 4. Output round. Output round. Output round. Round 8. Output round. ¸g Wª_òdfomTbevxvªW¬ ²w¸TbfmTvijkvgRT_`hWªg9hmnvWªo·omT_`hxjesTa`W°bfv(jk^:z{}Tvxbk Uj jev~WV¸Wde_`_ldfozmTmThxWvx} WÎvª¨bejkj ^Pbe~ ~^TWVW j }T_cvUg bfhxmTmTo½^na`h~%j6_c}Tä`snWhxje¥RToWW _cvªg ¬)bfRnmje{&h}nvx~m}T¢<h©aljkjkhx_cvxvx^LW jeh^nWje¥Lozvx°h gbfRTvsTU_`hacbAWªjkg9ghhx_cmTbev^p_lWo ¨ omT}T}Tac_`Wª~hxb4hRnWg_`vg mT_`hJhxRTvxbemTdfRÎhRTWU+mTa`h_c}TacW ¥Wvª¨hxRTW Gnvozh³vxbemT^P~ bkzi{)_co W´jea`mnjkhWª~%¨jk^n~ hRTW½vxWVomTa`hµ_lo horjebµho+bevxj¦hxWVRT~µg W¦be_cURn^¯jesTvxhx_c~RT^n¢<W§jhxjkvx_`vxbfWW§df^n_cjkvxorbeawhWVmngvV^n_`v¨H~%g ¢*¬¾mTRT_`h§ bL]orÌ°W;_«h¬ RTWeg ¬W¦be^LvxhxbeWmT^L^Ph~©_lo_lohxRT_ÙW^¯}Tac°WWVU~¯WV¨psP^fhjehxRnWVg_c~ o sngRnjfjeor_l_`g^T_c^T_`d hWVvxUKjkbh_c~éWªW oKjea`_cvxegW RT_`Ê hWªg9ÊhxmT²vWbe^na`_cq³o6bfhx^TjeW§_àcbe}TvxalWVjk~Ë_c^LhhxWb3¥Lh°WVsTWVacÍ b~gsn jfg_l o WsT^Pacbg gvxqL}TjhWVðhx~-Wv¯jh^T_cj^TW4hx_Ùga`W:bgjk ^nE ~©gqA¢ga`WWªo4gjkÌ°^-WV_`df}TRLvxbh¯´L_lv~bfWmT^nj~To³^TWVje¢ ^n~ _`^n}ThxRTmWh bemTh}Tmhihvjk^nor°bevxUjh_cbe^ Í ¬ pWh ~TW^TbehWhRnW+g acbgvxjkhW be (a) 1+1 rounds. (b) 2+1 rounds. (c) 4+1 rounds. (d) 8+1 rounds. . α =β =γ =0. f. î fDî.

(72) e.

(73) "!$#&%(') *'+*,- ./0213*%54"6$#&78 9:<;= 31>*?. Q jesTacWKf² Ê beU}njkvx_lorbf^§bkhRnW°bemTv*Ub~mTacb È ° ;Ä º Ä ÊÊ XkXk¹i¹iLL¶¶¤¤ p´fbWV¢<~ L(Db_cde¢<R -Li_`dfR ¶ K U n } v b ÊÊ XkXk¹i¹]LXfk¶¤ ¶¤ Ê jkvxvqL¶oj´fWije~n~Wv*snjforWªU~;mTjea¡vxhxg_`Rn}n_¡a`hx_cWWVv g9hxmTvxW ÊÊ XkXk¹i¹iLL¶¶¤¤ p´fbWV¢<~ L(Db_cde¢<R - Li_`dfR ¶ K U n } v b ÊÊ XkXk¹i¹]LXfk¶¤ ¶¤ Ê jkvxvqL¶oj´fWije~n~Wv*snjforWªU~;mTjea¡vxhxg_`Rn}n_¡a`hx_cWWVv g9hxmTvxW ÊÊ ¹¹VVffkkNN¶¶¤¤ p´fbWV¢<~ L(Db_cde¢<R - Li_`dfR ¶ K U n } v b ÊÊ ¹¹VVffkkNN¶¶¤¤ Ê jkvxvqL¶oj´fWije~n~Wv*snjforWªU~;mTjea¡vxhxg_`Rn}n_¡a`hx_cWWVv g9hxmTvxW ÊÊ ¹¹VVffkkNN¶¶¤¤ p´fbWV¢<~ L(Db_cde¢<R -Li_`dfR ¶ K U n } v b ÊÊ ¹¹VVffkkNN¶¶¤¤ Ê jkvxvqL¶oj´fWije~n~Wv*snjforWªU~;mTjea¡vxhxg_`Rn}n_¡a`hx_cWWVv g9hxmTvxW ÊÊ ¹¹VVffkkNN¶¶¤¤ p´fbWV¢<~ L(Db_cde¢<R - Li_`dfR ¶ K U n } v b ÊÊ ¹¹VVffkkNN¶¶¤¤ UmTa¡hx_`}na`_cWv p b < ¢ L(_cdeR ÊÊ ¹¹VVffkkNN¶¶¤¤ ¶UK}nvb´fWV~ Db¢<U- LimT_à¡dfhx_`R }na`_cWv . " . . . . (n + 1) × (n + 1). . . . . (n + 1) × (n + 1). . . . . (n + 1) × (n + 1). . . . . (n + 1) × (n + 1). . . . . (n + 1) × (n + 1) . . ï. . f]f hgij(j(k. (n + 1) × (n + 1). (2n + 1). U+mTa`h_c}Tac_`WVvxo·°bfv(¹i_cvhW ¥A je^n~¹i_cvrhxW ¥A rº ~T$ WÁ_cg;ÄWVoV¬ ¿ Å ¿ Ä È p XX X@ @ @ . m1. m2. m3. m4. . ". 66 48 25 393 75 58 25 474 241 204 178 392 241 226 177 466 325 318 305 371 365 314. ! " " . " . 15.7 15.9 15.8 15.7 6.9 7.1 6.3 6.9 24.7 24.3 32.1 31.1 15.4 15.2 22.5 11.9 10.9 10.8 15.2 11.0 10.8 14.6.

(74) ª X <9H " "!$#&%' H*?'*?3 .`

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(76) sTacW(hRT W Í ¬NvxW Q<hxRT_Ù_co_c^TvWªdormnjka¡a`h ' D deg bbf~v_`WhRndeU®WV^TWbkv<jhxhxRTbeW§vHhobKqL^L~hxWVRToxWVg ovx_c_co sPW(hbAWbe¥al}Toa`¬_lg »_`hacW§qRTRnbj¢´fWKhURTW]bU+~_HmnGna¡WVhx~¯_`}Tbeac_ mTWvv % be{(âcZ nj]W ¥A}?mnbeW^wvxUK_`F UWV_lWVUg ^fvxbebfhovxoqq~ozWªhx¬oWg¶Uv^ _co=s?je\(WV~n~Ka`~h_`_¡v^hxj_`bfhxV^RT(_lowhx¢·b}nbejehvxRn}PWKWVozhvwzj¢hx_`WVbfv^§{W<Ì°}?fg WbffvU°beF}TvxUmL hxWVjkKf~ ¨ _lmn}To]vxg behxhob}njesPdfWjkh_cWUKjf}n~Ta`WV~UK¨ WVvxWVo^LhÌºWV¸~p_cde¬mT» vxWW ormT¬KU"Q<RTmnW}J¢*~hx_¡hxH_RTIR³W gjmTa`hxhz}nvqjeWVWvra`hx_cWV_cbejeo]·a_cg}T^JjevxvbhxvxRT~AqLW h_cbe^ E }PjhRD¨(WVjfgR4}TÍ vxbAgWVoxorbfv§g bf^Lhxjk_c^no;j©omTsTeWVq½UKWVUbevxq ~WhWvxU_`^Pjh_cbe^-bk(hxRTW:¢*_c~hRÍ bk(hxRTW:_`^LhWVvUWª~_cjkhW:ormnUoV¬ . . . ' '

(77) " =. =. . . ·f. . . 64 f. 9. . !. (n + 1) × (n + 1) ". . . . ( ( ' '

(78) " = 8f.. #. %$ '& ) (. . . . . . . (2n + 1). . "*. ( ( ' '

(79) " =. = 64f.. ·f. . . . ". . . 55. 107. !. 107. . 55. . . . . . .

(80) . n. Pi = xi y i ∈ {0, . . . , n − 1}. î fDî.

(81) 9@.

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(83) dfz_ì´fvxWWVo{RTmTWVa`hx}TvoW+vxbAj§g}TWVsTmToxvxsnor_`bfa`W_lvxHoroVRnb¬ WV´f~ WvxÁsA_`WVq ¢ obebeNUhRTW/WbkUhRnjeW_`^ v °bebe}?vWhvRTjW(hbfoxvVjkÜhRnW·WacjkzhiWV^ng { q be}Tnvx°bbeg mnWªvHoogbea`bv g¢*_`ghqAR³ga`°Wªmnoa`¬·Q<a`bARnbfjk}³^nAmTo^Tvxhxbebachx`_cRT^T_ld o f¬ Ê vqA}hxb bAbforhWv Ì bLojeÂqfj *? B` @PD jk^n~ E h Bc(D _lo j F RT_cdeRTacq"UbA~TmTacjev 7 je^n~"vxWVgbe^ GThxo_`^Lhxb§be^nW Ê ¹+ªeee4¸uZ{¬%Q<RT_loiW¥}PWVv_cUW^Lhi_ca`cmnor V W n m g n R k j E hvjhWªohRnW_`^nW Ig _cW^ngq behxRTW§jfg9hmPjkaoqA^fhxRTWVo_lohbAbealo°bfv GndemTvjksna`W gbeÍ }Tvxbg Wªoobev hxjkA_c^Td °mna`³je~T´je^LhxjkdfW be . m1 = m 2 = m 3 = 1 β = 1 " . . . . . . " " " !. (n + 1) × (n + 1) . . (n + 1) × (n + 1) . . ". (n + 1) × (n + 1). γ=1 " .

(84). " ! " . " . "*. 4845 * 4199 * 3077 "* 2905 * 2436 "* 1983 * 10024 * 9586 * 8745. . . . . . . ∼ 8.0 ∼ 7.9 ∼ 8.5 ∼ 4.0 ∼ 4.0 ∼ 4.1 ∼ 4.3 ∼ 4.3 ∼ 2.8. 8.0 8.1 7.5 7.9 8.0 7.7 14.9 14.9 22.9. . . max(2Pi+1 + Pi ) =. n−1 X j=0. 2j + 2 ·. n−1 X. 2j. j=0. = 2n − 1 + 2 · (2n − 1) = 2n+1 + 2n − 3.. $ & . . (. . . . . . n. (n + 2). max = (22 · (2Pi+3 + Pi+2 ) + (2Pi+1 + Pi )) {z ) } | {z ) } | 2. = 2 · (2. n+2 n+1 n. + 2 − 3) + 2. n+2 n+1. . + 2n − 3. = 2n+3 + 2n+2 + 2n+1 + 2n − 15, (n + 4) 16 × 16 (2n + 1). . . 29. . . ï. f]f hgij(j(k. 10.8. . . . . . . . . .

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