Destek ayırma algoritması ve McEliece şifreleme sisteminde zayıf anahtarlar

(1)

)(1%ø/ø0/(5ø(167ø7h6h

**'(67(.$<,50$$/*25ø70$6,9(**

0F(/,(&(ùø)5(/(0(6ø67(0ø1'(

ZAYIF ANAHTARLAR

<h.6(./ø6$167(=ø

Ekrem EMRE

(QVWLW$QDELOLP'DOÕ : 0$7(0$7ø.

(QVWLW%LOLP'DOÕ : &(%ø59(6$<,/$57(25ø6ø 7H]'DQÕúPDQÕ : 3URI'U0HKPHWg=(1

Haziran 2014

(2)

(3)

ii

7(ù(..h5

(÷LWLPKD\DWÕPVUHVLQFHEHQL|]YHULLOH \HWLúWLUHQWP|÷UHWPHQYH|÷UHWLP\HVL

KRFDODUÕPDWHúHNNUELUERUoELOLULP%LOKDVVDWH]oDOÕúPDPÕQKHUDúDPDVÕQGDELOJL

YH WHFUEHOHUL\OH EHQL \|QOHQGLUHQ 6D\ÕQ 3URI 'U 0HKPHW g=(1¶ H HQ LoWHQ

WHúHNNUOHULPLVXQDUÕP

$\UÕFD H÷LWLP KD\DWÕP ER\XQFD PDGGL YH PDQHYL GHVWHNOHULQL ]HULPGHQ

esirgemeyen aLOHPHGHWHúHNNUHGL\RUXP

(4)

iii

ødø1'(.ø/(5

7(ù(..h5 ... LL ødø1'(.ø/(5 ... LLL 6ø0*(/(59(.,6$/70$/$5/ø67(6ø ... Y 7$%/2/$5/ø67(6ø ... YL g=(7 ... YLL 6800$5< ... YLLL

%g/h0

*ø5øù ...

7DQÕPYHgQHUPHOHU ...

0F(OLHFHùLIUHOHPH6LVWHPL ...

*RSSD.RGODUÕQ'HNRGODQPDVÕ

%HUOHNDPS-0DVVH\DOJRULWPDVÕ...

3DWWHUVRQDOJRULWPDVÕ ...

%g/h0

'(67(.$<,50$$/*25ø70$6, ...

%g/h0

0F(/,(&(ùø)5(/(0(6ø67(0ø1'(=$<,)$1$+7$5/$5 ...

%g/h0

$/7(51$7ø)0(727 ...

%g/h0

6218d/$59(g1(5ø/(5 ...

(5)

iv

.$<1$./$5 ...

g=*(d0øù ... 

(6)

v

**6ø0*(/(59(.,6$/70$/$5/ø67(6ø**

ॲ_௣ : ݌ HOHPDQOÕVRQOXFLVLP ݓݐ +DPPLQJD÷ÕUOÕ÷Õ ܥ^ୄ : ܥ lineer kodunun duali

ܹ஼ : ܥ NRGXQXQ+DPPLQJD÷ÕUOÕNVD\DFÕ

ܵ_௡ ^«Q`NPHVLQLQVLPHWULJUXEX

ܣݑݐ_஼ : ܥ kodunun otomorfizm grubu

ܥ_௜ : ܥ kodunun ݅¶LQFLELOHúHQHJRUHGHOLNOLNRGX ܦܣܣ 'HVWHN$\ÕUPD$OJRULWPDVÕ

࣢ : Hull kod

: )UREHQLXVG|QúP

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vi

ù(.ø//(5/ø67(6ø

Tablo 1.1. ܨ_ଶ^ర FLVPLQLQHOHPDQODUÕ ... 9

Tablo 1.2. ݃^ିଵሺߙ_௜ሻ HOHPDQODUÕQÕQGH÷HUOHUL ... 10

Tablo 1.3. ݀௜ GH÷HUOHULQHNDUúÕOÕNJHOHQ ߪ^ሺ௜ሻሺݔሻ SROLQRPODUÕ ... 13

Tablo 2.1. ܥ ve ܥ^ᇱNRGODUÕLoLQܥ௜ǡ ܥ_௜^ᇱǡ ܹ஼_೔ሺݔሻ ve ܹ_஼_೔^ᇲሺݔሻ GH÷HUOHUL ... 20

Tablo 1.3 ݀௜ GH÷HUOHULQHNDUúÕOÕNJHOHQߪ^ሺ௜ሻሺݔሻ SROLQRPODUÕ ... 24

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vii

g=(7

$QDKWDUNHOLPHOHU*RSSD.RGODU0F(OLHFHùLIUHOHPH6LVWHPL'HQN.RGODU'HVWHN

$\ÕUPD$OJRULWPDVÕYH=D\ÕI$QDKWDUODU

%XWH]G|UWE|OPGHQROXúPDNWDGÕU%LULQFLE|OPGHED]ÕWHPHOWDQÕPYH |QHUPHOHU

YHULOPLúWLU

øNLQFLYHoQFE|OPOHUGHGHQNNRGODUDUDVÕQGDNLSHUPWDV\RQXHOGHHWPHNLoLQ

NXOODQÕODQELU\|QWHPRODQ'HVWHN$\ÕUPD$OJRULWPDVÕ6XSSRUW6SOLWWLQJ$OJRULWKP YH0F(OLHFHùLIUHOHPH6LVWHPLQGH=D\ÕI$QDKWDUODUNRQXVXQDGH÷LQLOPLúWLU

6RQ RODUDN G|UGQF E|OPGH 'HVWHN $\ÕUPD $OJRULWPDVÕQD DOWHUQDWLI ELU PHWRW

YHULOPLúYHEHúLQFLE|OPGHGHED]Õ|QHULOHUGHEXOXQXOPXúWXU

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viii

SUPPORT SPLITTNG ALGORITHM AND THE WEAK KEYS IN THE MCELIECE CRYPTOSYSTEM

SUMMARY

Key Words: Goppa Codes, McEliece Cryptosystems, Equivalance Codes, Support Splitting Algorithm and Weak Keys.

This thesis consists of five chapters. In the first chapter some essential definitions and theorems are given.

In the chapters two and three The Support Splitting Algorithm which is used to find permutation between equivalent codes and The Weak Keys in The McEliece Cryptosystem are mentioned.

At last, in the chapter four an alternative method to Support Splitting Algorithm is given, and in the chapter five some suggestions are given.

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**%g/h0*ø5øù**

7DQÕPYHgQHUPHOHU

gQHUPH [1] Pozitif bir ݊ WDPVD\ÕVÕYHULOGL÷LQGHKHU݅ WDPVD\ÕVÕ

݅ ൌ ݍ݊ ൅ ݎ

ELoLPLQGHLIDGHHGLOHELOLU%XUDGDݎǡ Ͳ ൏ ݎ ൏ ݊ െ ͳ úDUWÕQÕVD÷OD\DQELUWDPVD\ÕYHݍ LVH KHUKDQJL ELU WDP VD\ÕGÕU $\UÕFD ݎ VD\ÕVÕ ݉݋݀ െ ݊ kalan, ݍ VD\ÕVÕ LVH E|OP

RODUDNDGODQGÕUÕOÕUYH݅ ൌ ݎ݉݋݀݊ ELoLPLQGHLIDGHHGLOLU

7DQÕP [1] Pozitif bir ݊ WDPVD\ÕVÕ LoLQ RODVÕ WP ݉݋݀ െ ݊ NDODQODUÕQÕQ

NPHVLQLܴ_௡ ൌ ሼͲǡͳǡ ǥ ǡ ݊ െ ͳሽ LOHJ|VWHULUVHNYHKHUݎǡ ݏ א ܴ_௡ LoLQܴ_௡]HULQGH

ݎ ൅ ݏ ൌ ሺݎ ൅ ݏሻ݉݋݀݊ ve ݎ כ ݏ ൌ ݎݏ݉݋݀݊

úHNOLQGH݉݋݀ െ ݊ toplama ve ݉݋݀ െ ݊ oDUSPDLúOHPOHULQLWDQÕPODUVDNܴ௡ ]HULQGH

EXLúOHPOHUOHLúOHP\DSPD\D݉݋݀ െ ݊ aritmetik denir.

7DQÕP [1] ܩ ൌ ሼܽǡ ܾǡ ܿǡ ǥ ሽ NPHVLYHULOGL÷LQGHEXNPH]HULQGHDúD÷ÕGDNL

|]HOOLNOHULVD÷OD\DQELU۩ LúOHPLYDUVDܩ NPHVLQHJUXSGHQLU

i. ׊ܽǡ ܾ א ܩ LoLQܽ۩ܾ א ܩGLUNDSDOÕOÕN|]HOOL÷L

ii. ׊ܽǡ ܾǡ ܿ א ܩ LoLQሺܽ۩ܾሻ۩ܿ ൌ ܽ۩ሺܾ۩ܿሻ ELUOHúPH|]HOOL÷L

iii. ׊ܽ א ܩ LoLQܩ NPHVLQGHͲ۩ܽ ൌ ܽ۩Ͳ ൌ ܽ RODFDNúHNLOGHELUͲ HOHPDQÕYDUGÕU

ELULPHOHPDQ|]HOOL÷L

iv. ׊ܽ א ܩ LoLQ ܩ NPHVLQGH ሺെܽሻ۩ܽ ൌ ܽ۩ሺെܽሻ ൌ Ͳ RODFDN úHNLOGH ELU Ȃ ܽ HOHPDQÕYDUGÕUWHUVHOHPDQ|]HOOL÷L

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(÷HU׊ܽǡ ܾ א ܩ LoLQܽ۩ܾ ൌ ܾ۩ܽ |]HOOL÷LVD÷ODQÕ\RUVDEXJUXEDGH÷LúPHOLJUXS\D

GD$EHOJUXEXGHQLU$\UÕFDH÷HUܩ NPHVLVRQOXLVHEXJUXEDVRQOXJUXSYHJUXSWDNL

HOHPDQVD\ÕVÕQDGDJUXEXQPHUWHEHVLGHQLU

gQHUPH [1] ܴ௡ ൌ ሼͲǡͳǡ ǥ ǡ ݊ െ ͳሽ NPHVL݉݋݀ െ ݊ WRSODPDLúOHPLDOWÕQGD

bir gruptur. Bu grup Ժ_௡ úHNOLQGHGHJ|VWHULOLU

7DQÕP [1] ܩ sonlu bir grup ve ݃ א ܩ ROPDN ]HUH EX JUXEXQ KHU HOHPDQÕ

݃۩݃۩ ǥ ۩݃ úHNOLQGH݃ HOHPDQÕQÕQVRQOXWRSODPODUÕúHNOLQGHLIDGHHGLOHELOL\RUVDܩ JUXEXQDVRQOXGDLUHVHOJUXSGHQLU%XQDJ|UHܩ ൌ ሼ݃ǡ ݃۩݃ǡ ǥ ǡ ሽ \D]ÕODELOLU

gQHUPH [1] Mertebesi n olan vHJLOHUHWLOHQGDLUHVHOJUXS ሼͲ݃ǡ ͳ݃ǡ ǥ ǡ ሺ݊ െ ͳሻ݃ሽ úHNOLQGHGLU%XUDGD

݅݃ ൌ ݃۩݃۩ ǥ ۩݃ᇣᇧᇧᇧᇤᇧᇧᇧᇥ

௜௦௔௬పௗ௔

ǡ ͳ ൑ ݅ ൑ ݊ െ ͳǡ Ͳ݃ ൌ Ͳ

úHNOLQGH WDQÕPODQÕU %XQD J|UH D\UÕFD ݅݃ ՜ ݅ G|QúP DOWÕQGD ܩǡ Ժ_௡ ye izomorf olur.

7DQÕP [1] ܩ bir grup ve ܵ ك ܩ ROVXQ(÷HU6NPHVLGHD\QÕLúOHPDOWÕQGDELU

grup ise bu gruba G grubunun alt grubu denir.

gQHUPH [1] Sonlu bir grubun her alt grubunun mertebesi grubun mertebesini E|OHU.

gQHUPH [1] ܩ sonlu bir grup ve ݃ א ܩ ROPDN]HUHܵሺ݃ሻ ൌ ሼ݃ǡ ݃۩݃ǡ ǥ ሽǡ ܩ QLQVRQOXGDLUHVHOELUDOWJUXEXGXU$\UÕFD݉ א Ժ௡ ROPDN]HUH

ȁܵሺ݉ሻȁ ൌ ݊

݃ܿ݀ሺ݉ǡ ݊ሻ

\D]ÕODELOLU%XUDGD݃ܿ݀ሺ݉Ǥ ݊ሻǡ ݉ ve ݊ WDPVD\ÕODUÕQÕQHQE\NRUWDNE|OHQLGLU

%XQDJ|UHܵሺ݉ሻ ൌ Ժ_௡ ROPDVÕLoLQJHUHNYH\HWHUúDUW݃ܿ݀ሺ݉ǡ ݊ሻ ൌ ͳ ROPDVÕGÕU

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7DQÕP [1] ॲ bir NPH ۩ ve כ LúOHPOHUL DúD÷ÕGDNL |]HOOLNOHUL VD÷OD\DQ ॲ

]HULQGHWDQÕPOÕLNLLúOHPROVXQ%XGXUXPGDॲ NPHVLQHFLVLPGHQLU

i. ॲ NPHVL ۩ LúOHPL DOWÕQGD GH÷LúPHOL bir gruptur (birim eleman 0 ile J|VWHULOLU

Toplamsal grup da denir.

ii. ॲ^כ ൌ ॲ െ ሼͲሽ NPHVLכ LúOHPLDOWÕQGDGH÷LúPHOLELUJUXSWXU ELULPHOHPDQÕLle J|VWHULOLUdDUSÕPVDOJUXSGa denir.

iii. ׊ܽǡ ܾǡ ܿ א ॲ LoLQሺܽ۩ܾሻ כ ܿ ൌ ሺܽ כ ܿሻ۩ሺܾ כ ܿሻ \D]ÕODELOLU

gQHUPH [1] Her ݌ DVDO VD\ÕVÕ LoLQ ܴ௣ ൌ ሼͲǡͳǡ ǥ ǡ ݌ െ ͳሽ NPHVL PRG-p toplama ve mod-SoDUSPDLúOHPOHULDOWÕQGDELUFLVLPGLUYHॲ௣(asal kalan cismi) ile J|VWHULOLU

gQHUPH [1] ॲ VRQOX ELU FLVLP ROVXQ %XQD J|UH ܵሺͳሻ ൌ ሼͳǡͳ۩ͳǡ«ሽǡ ॲ FLVPLQLQ VRQOX ELU DOW FLVPLGLU $\UÕFD ܵሺͳሻǡ ݌ VD\ÕGD HOHPDQ LoHUL\RUVD ݌ bir asal VD\ÕGÕU YH EX FLVLP ॲ௣ cismine izomorftur. Burada ݌ VD\ÕVÕQD ॲ cisminin NDUDNWHULVWL÷LGHGHQLU

7DQÕP [1] .DWVD\ÕODUÕ ELU ॲ FLVPLQLQ HOHPDQODUÕ RODQ WP SROLQRPODUÕQ

NPHVLॲሾݔሿ LOHJ|VWHULOLU

7DQÕP . ݃ሺݔሻǡ derecesi ݉ olan ELU SROLQRP ROPDN ]HUH H÷HU ݔ^௠ teriminin NDWVD\ÕVÕ¶HHúLW ise, ݃ሺݔሻ¶HPRQLNSROLQRPGHQLU

gQHUPH [1] ݃ሺݔሻǡ derecesi ݉ olan monik ELUSROLQRPROPDN]HUHKHU݂ሺݔሻ polinomu

݂ሺݔሻ ൌ ݍሺݔሻ݃ሺݔሻ ൅ ݎሺݔሻ

ELoLPLQGHLIDGHHGLOHELOLU%XUDGDݎሺݔሻ, derecesi ݉ GHQNoNRODQELUSROLQRPGXU

Bu durumda ݎሺݔሻ polinomuna ݉݋݀ െ ݃ሺݔሻ kalan denir ve ݂ሺݔሻ ൌ ݎሺݔሻ݉݋݀݃ሺݔሻ

\D]ÕOÕU

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7DQÕP [1] ݂ሺݔሻ ve ݃ሺݔሻ SROLQRPODUÕLoLQ݂ሺݔሻ ൌ ݍሺݔሻ݃ሺݔሻ RODFDNúHNLOGH

bir ݍሺݔሻ polinomu bulunabiliyorsa ݃ሺݔሻ polinomuna ݂ሺݔሻ SROLQRPXQXQELUE|OHQL

denir.

7DQÕP [1] Bir ݂ሺݔሻ SROLQRPXQXQNHQGLVLQGHQYHGHQIDUNOÕPRQLNELU݃ሺݔሻ E|OHQLYDUVD݃ሺݔሻ polinomuna ݂ሺݔሻ SROLQRPXQXQELUoDUSDQÕGHQLU$\UÕFDGHUHFHVL

¶HHúLWYH\DGDKDE\NROXSoDUSDQÕROPD\DQELUSROLQRPDLQGLUJHQHPH]SROLQRP

YHLQGLUJHQHPH]ROXSD\QÕ]DPDQGa monik olan bir polinoma asal polinom denir.

gQHUPH . [1] ॲ KHUKDQJL ELU FLVLP ROPDN ]HUH ׊݂ሺݔሻ א ॲሾݔሿ monik polinomu ॲሾݔሿ GH DVDO SROLQRPODUÕQ oDUSÕPÕ úHNOLQGH WHN WUO RODUDNoDUSDQODUÕQ

VÕUDVÕGúQOPHNVL]LQ\D]ÕODELOLU

gQHUPH. [1] ॲ KHUKDQJLELUFLVLPROPDN]HUHGHUHFHVL݉ olan monik bir polinom ॲ ]HULQGHHQID]OD݉ WDQHN|NHVDKLSRODELOLU

gQHUPH. [1]ॲ KHUKDQJLELUFLVLPROPDN]HUHॲ^כ oDUSÕPVDOJUXEXYHULOHQELU

pozitif ݊WDPVD\ÕVÕLoLQPHUWHEHVL݊ RODQHQID]ODELUWDQHGDLUHVHODOWJUXSLoHUHELOLU

YH H÷HU E|\OH ELU GDLUHVHO DOW JUXS YDUVD HOHPDQODUÕ ߚ א ॲ^כ ROPDN ]HUH

ͳǡ ߚǡ ߚ^ଶǡ ǥ ǡ ߚ^௡ିଵ úHNOLQGHGLU YH ݔ^௡െ ͳ ൌ Ͳ GHQNOHPLQL VD÷ODUODU $\UÕFD EX DOW

cisme ߚ WDUDIÕQGDQ UHWLOHQ oDUSÕPVDO GDLUHVHO DOW JUXS GHQLU (÷HU EX JUXS ॲ^כ JUXEXQDHúLWVHߚ ya primitif eleman denir.

gQHUPH . [1] ॲ_௤, ݍ HOHPDQOÕ ELU FLVLP ROPDN ]HUH ॲ_௤^כ oDUSÕPVDO JUXEXQXQ

ݍ െ ͳ VD\ÕVÕQÕWDPE|OHQSR]LWLIKHU݀WDPVD\ÕVÕLoLQPHUWHEHVL݀ olan oDUSÕPVDOELU

alt grubu YDUGÕU

7DQÕP [1] ߚ א ॲ௤ ROPDN]HUHॲ௣ሾݔሿ de ݃ሺߚሻ ൌ Ͳ úDUWÕQÕVD÷OD\DQGHUHFHVL

HQNoNRODQPRQLN݃ሺݔሻ polinomuna ߚ QÕQ minimal polinomu denir.

gQHUPH . [1] ߚ א ॲ_௤ HOHPDQÕQÕQ PLQLPDO SROLQRPX ݃ሺݔሻ ROPDN ]HUH ॲ_௤ cisminin ߚ HOHPDQÕQÕLoHUHQHQNoNDOWFLVPLॲሾݔሿȀ݃ሺݔሻ cismine izomorftur.

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gQHUPH. [1] .DUDNWHULVWL÷L ݌ olan ݍ HOHPDQOÕKHUॲ௤ cismi bir ॲሾݔሿȀ݃ሺݔሻ cismine izomorftur. Burada ݃ሺݔሻ א ॲ_௣ሾݔሿ ELU DVDO SROLQRPGXU %XQD J|UH ݍ ൌ

݌^{ௗ௘௥௚ሺ௫ሻ} \D]ÕODELOLU$\UÕFD ݃ሺݔሻ א ॲ௣ሾݔሿ, derecesi m olan asal bir polinom olmak

]HUH݌^௠HOHPDQOÕWPFLVLPOHUॲሾݔሿȀ݃ሺݔሻ cismine izomorftur.

gQHUPH. [1] ݃ሺݔሻǡ ݌^௠ HOHPDQOÕELUॲ cisminin minimal bir polinomu olsun.

%XQDJ|UH݃ሺݔሻ SROLQRPXQXQN|NOHULߚǡ ߚ^௣ǡ ߚ^௣^మǡ ǥ ǡ ߚ^௣^೙షభ úHNOLQGHGLU%XUDGD݊ǡ ݉ VD\ÕVÕQÕ E|OHQ SR]LWLI ELU WDPVD\ÕGÕU 'DKDVÕ ݃ሺݔሻ polinomu ݔ^௣^೙െ ݔ polinomunu E|OHU

gQHUPH. [1] ݔ^௣^೘െ ݔ polinomu ॲ௣ ]HULQGHWHNUDUVÕ]RODUDNॲ௣ሾݔሿ deki asal SROLQRPODUÕQoDUSÕPÕúHNOLQGHGLU$\UÕFDॲ௣ሾݔሿ GHNLDVDOSROLQRPODUÕQGHUHFHOHUL݉

\LE|OHU

gQHUPH. [1]+HUPSR]LWLIWDPVD\ÕVÕ ve asal ݌ WDPVD\ÕVÕLoLQ LoLQGHUHFHVLP

olan asal bir ݃ሺݔሻ א ॲ_௣ሾݔሿ SROLQRPXYDUGÕU

gQHUPH. [1] ݌^௠ HOHPDQOÕKHUFLVLP݉ VD\ÕVÕQÕE|OHQSR]LWLIKHU݊ WDPVD\ÕVÕ

LoLQ݌^௡ HOHPDQOÕELUDOWFLVLPLoHULU

݌^௠ HOHPDQOÕELUFLVLPॲ ve bu cismin ݌^௡ HOHPDQOÕELUDOWFLVPL ࣠ ROVXQ%XQDJ|UH݊

SR]LWLI WDP VD\ÕVÕQÕQ ݉ SR]LWLI WDP VD\ÕVÕQÕ E|OPHVL JHUHNWL÷L úX úHNLOGH

J|VWHULOHELOLU: ݉ ൌ ݍǤ ݊ ൅ ݎǡ Ͳ ൑ ݎ ൏ ݊ ROPDN]HUH࣠^כ oDUSÕPsal dairesel grubu ॲ^כ oDUSÕPVDO GDLUHVHO JUXEXQXQ DOW JUXEX ROGX÷XQGDQ ݌^௡െ ͳ VD\ÕVÕ ݌^௠െ ͳ VD\ÕVÕQÕ

E|OHU'ROD\ÕVÕ\OD

݌^௡ ؠ ͳ݉݋݀ሺ݌^௡െ ͳሻ

\D]ÕODELOHFH÷LQGHQ

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݌^௠െ ͳ ؠ ݌^{௤Ǥ௡ା௥}െ ͳ ؠ ሺ݌^௡ሻ^௤݌^௥െ ͳ ؠ ݌^௥െ ͳ ؠ Ͳ݉݋݀ሺ݌^௡െ ͳሻ

ฺ ݌^௥െ ͳ ൌ Ͳ ฺ ݎ ൌ Ͳ

elde edilir. O halde ݊ SR]LWLIWDPVD\ÕVÕ݉ SR]LWLIWDPVD\ÕVÕQÕE|OHU

gUQHN1.1. ॲ_ଶ^ర FLVPLQLQWPHOHPDQODUÕ ݔ^ଶ^ర െ ݔ SROLQRPXQXQN|NOHULGLU$\UÕFD

ॲ_ଶ ]HULQGH GHUHFHVL ¶ Q E|OHQL RODQ DVDO SROLQRPODU ݔ^ଶ^ర െ ݔ polinomunun asal oDUSDQODUÕGÕUODUYHEXSROLQRPODUÕQGHUHFHOHUL Ͷ¶ E|OHU 'ROD\ÕVÕ\ODEXSROLQRPODUÕQ

dereceleri ͳǡʹ veya Ͷ RODELOLU $\UÕFD ߚ א ॲ_ଶ^ర HOHPDQÕQÕQ PLQLPDO SROLQRPX

݉ఉሺݔሻ ൌ ݔ^ସ ൅ ݔ ൅ ͳ ROVXQ%XQDJ|UHߚ \ÕLoHUHQHQNoNDOWFLVLPॲሾݔሿȀ݉ఉሺݔሻ polinomal kalan cismine i]RPRUIRODFD÷ÕQGDQ ߚ SULPLWLIHOHPDQGÕU'L÷HUPLQLPDO

SROLQRPODUDúD÷ÕGDNLJLELGLU

݉_଴ሺݔሻ ൌ ݔ

݉_ଵሺݔሻ ൌ ሺݔ ൅ ͳሻǡ

݉_ఉ^ఱሺݔሻ ൌ ሺݔ െ ߚ^ହሻሺݔ െ ߚ^ଵ଴ሻ ൌ ݔ^ଶ൅ ݔ ൅ ͳ

݉_ఉ^ళሺݔሻ ൌ ሺݔ െ ߚ^଻ሻሺݔ െ ߚ^ଵସሻሺݔ െ ߚ^ଵଷሻሺݔ െ ߚ^ଵଵሻ ൌ ݔ^ସ൅ ݔ^ଷ൅ ͳ

݉_ఉ^యሺݔሻ ൌ ሺݔ െ ߚ^ଷሻሺݔ െ ߚ^଺ሻሺݔ െ ߚ^ଵଶሻሺݔ െ ߚ^ଽሻ ൌ ݔ^ସ൅ ݔ^ଷ൅ ݔ^ଶ൅ ݔ ൅ ͳ

$\UÕFD ݔ^ଶ൅ ݔ ൌ ݔሺݔ ൅ ͳሻ ve ݔ^ସ൅ ݔ ൌ ݔሺݔ ൅ ͳሻሺݔ^ଶ൅ ݔ ൅ ͳ \D]ÕODELOHFH÷LQGHQ

ॲ_ଶ^ర cisminin alt cisimleri {0 , 1} ve {0 ,1 , ߚ^ହǡ ߚ^ଵ଴ሽ olarak bulunur.

7DQÕP 1.1.11. [2] ܪ girdileri Ͳ ve ͳ OHUGHQROXúDQKHUKDQJLELUPDWULVROVXQ%XQD

J|UH

ܪݔ^௧ ൌ Ͳ

GHQNOHPLQLVD÷OD\DQWPݔ YHNW|UOHULQLQROXúWXUGX÷XNPH\HSDULWHNRQWUROPDWULVL

ܪ RODQELUOLQHHUNRGGHQLU%XUDGDLúOHPOHU݉݋݀ʹ \HJ|UH\DSÕOÕU

7DQÕP 1.1.12. [2]ݔ ൌ ݔଵݔଶǥ ݔ௡ NRG V|] LoLQ +DPPLQJ D÷ÕUOÕ÷Õ ݔ௜ ് Ͳ úDUWÕQÕ

VD÷OD\DQVHPEROOHULQVD\ÕVÕRODUDNWDQÕPODQÕUYH ݓݐሺݔሻ úHNOLQGHJ|VWHULOLU

(16)

7DQÕP1.1.13. [2] /LQHHUELUNRGXQLNLNRGV|] ݔݒ݁ݕROPDN]HUHEXLNLNRG V|]

DUDVÕQGDNL+DPPLQJX]DNOÕ÷Õ ݔ െ ݕ NRGV|]QQ+DPPLQJD÷ÕUOÕ÷Õ\DQL ݓݐሺݔ െ ݕሻ RODUDNWDQÕPODQÕU

7DQÕP1.1.14. [2] Parite kontrol matrisi ܪ YHUHWHoPDWULVL ܩ olan lineer kodu ܥ ile J|VWHULUVHN parite kontrol matrisi ܩ YHUHWHoPDWULVL ܪ olan koda ܥ kodunun duali denir ve ܥ^ୄLOHJ|VWHULOLU

/LQHHUNRGODULoLQELUGL÷HU|QHPOLSDUDPHWUHD÷ÕUOÕNVD\DFÕGÕU

7DQÕP 1.1.15. [2] ܣ௜ ile ሾ݊ǡ ݇ሿ െlineer ܥ NRGXQGD D÷ÕUOÕ÷Õ ݅ RODQ NRGV|]OHULQ VD\ÕVÕQÕJ|VWHUHOLP%XQDJ|UH

ܹ_஼ሺݔǡ ݕሻ ൌ ෍ ܣ_௜ݔ^௡ି௜ݕ^௜

௡

௜ୀ଴

ൌ ෍ ݔ^{௡ି௪௧ሺ௨ሻ}ݕ^{௪௧ሺ௨ሻ}

௨א஼

RODUDN WDQÕPODQDQ ܹ஼ሺݔǡ ݕሻ polinomuna ܥ NRGXQXQ D÷ÕUOÕN VD\DFÕ GHQLU $÷ÕUOÕN

VD\DFÕQGD ݔ ൌ ͳ DOÕQDELOLUYH

ܹ_஼ሺͳǡ ݕሻ ൌ ෍ ܣ_௜ݕ^௜

௡

௜ୀ଴

ൌ ෍ ݕ^{௪௧ሺ௨ሻ}

௨א஼

\D]ÕODELOLU

7DQÕP1.1.16. [3] ܺǡ ݊ HOHPDQOÕ KHUKDQJLELUNPHROVXQ%XGXUXPGD ܺ ]HULQGH

WDQÕPOÕWP SHUPWDV\RQODUÕQNPHVLELOHúNHLúOHPL\OHEHUDEHUELUJUXSROXúWXUXUYH

bu gruba ܺ in simetri grubu denir YHNÕVDFD ܵ_௡ LOHJ|VWHULOLU

7DQÕP 1.1.17. [3] ܩ ELU JUXS ROPDN ]HUH ܩ ൈ ܺ ՜ ܺ úHNOLQGH ELU G|QúP LoLQ

݃ א ܩǡ ݔ א ܺ olmak ]HUH ሺ݃ǡ ݔሻ א ܩ ൈ ܺ HOHPDQÕQÕQ EX G|QúP DOWÕQGDNL

J|UQWVQ ݃Ǥ ݔ úHNOLQGHJ|VWHULUVHNDúD÷ÕGDNLLNL úDUWÕQVD÷ODQPDVÕGXUXPXQGD ܺ NPHVLQHELU ܩ -set denir.

(17)

i. ݁Ǥ ݔ ൌ ݔǡ ׊ݔ א ܺ (݁ǡ ܩ JUXEXQXQEULPHOHPDQÕGÕU ii.ሺ݃ଵ݃ଶሻǤ ݔ ൌ ݃ଵǤ ሺ݃ଶǤ ݔሻǡ ׊݃ଵǡ ݃ଶ א ܩve ׊ݔ א ܺ.

7DQÕP 1.1.18. [3] ݔ א ܺǡ ܩǤ ݔ ൌ ሼ݃Ǥ ݔȁ݃ א ܩሽ NPHVLQH ݔ HOHPDQÕQÕQ ܩ DOWÕQGDNL

orbiti denir.

%XQD J|UH ܺ ]HULQGH ׽ ED÷ÕQWÕVÕQÕ ݔǡ ݕ א ܺǡ ݔ ׽ ݕ ֞ ݕ ൌ ݃Ǥ ݔǡ ׌݃ א ܩ olarak WDQÕPODUVDN ܺ ]HULQGHELUGHQNOLNED÷ÕQWÕVÕWDQÕPODPÕúROXUX]YH ݔ א ܺ HOHPDQÕQÕQ

GHQNOLN VÕQÕIÕ ܩǤ ݔ ROXU %XQD J|UH ܩǤ ݔ orbitleri ܺ NPHVLQLQ ELU SDUoDODQÕúÕQÕ

WDQÕPODU

1.2. 0F(OLHFHùLIUHOHPH6LVWHPOHUL

7DQÕP9HULOHQELUDOJRULWPDLoLQJLUGLQLQX]XQOX÷X݊ ve ܲሺݔሻ herhangi bir SROLQRP ROPDN ]HUH DOJRULWPDQÕQ WDPDPODQPDVÕ LoLQ JHUHNHQ VUH HQ ID]OD ܲሺ݊ሻ NDGDULVHEXDOJRULWPD\DSROLQRPDO]DPDQOÕGÕUGHQLU

7DQÕP.2. [4] ߞ, SROLQRPLDO]DPDQOÕELUW-KDWDG]HOWHQDOJRULWPDVÕELOLQHQ ܨ_ଶ

]HULQGH WDQÕPOÕሺ݊ǡ ݇ሻ െNRGODUÕQ(݊ X]XQOX÷XQGD݇ boyutlu lineer kodlar) bir ailesi ROVXQ%XQDJ|UH0F(OLHFHWLSLQGHELUúLIUHOHPHVLVWHPLDúD÷ÕGDNLDGÕPODUGDQROXúXU

I. ߁߳ߞ kodu(gizli kod) ve ߨ߳ ܵ௡ SHUPWDV\RQXELUOLNWHJL]OLDQDKWDUÕROXúWXUXU

II. ܥ ൌ ߨሺ߁ሻ NRGXQXQELUUHWHoPDWULVL ܩ DoÕNDQDKWDUÕROXúWXUXU

III. ݉ܩ ൅ ݁ úHNOLQGHúLIUHOHQHQ ݉ PHVDMÕ ߨ SHUPWDV\RQXYHSROLQRPDO]DPDQOÕ ݐ- hata G]HOWHQ DOJRULWPD NXOODQÕODUDN NROD\OÕNOD HOGH HGLOHELOLU %XUDGD ݁ Hamming D÷ÕUOÕ÷Õ ݐ olan hata vekW|UGU

>@GHJ|VWHULOGL÷L]HUH݉ܩ ൅ ݁ YHNW|UQGHQKDUHNHWOH ݉ܩ YHNW|UQEXOPDN zor ELU SUREOHP ROGX÷XQGDQ, McEliece parametreleri [6] LoLQ VLVWHP EX QRNWDGD JYHQOLGLU 6LVWHPLQJYHQOL÷LD\UÕFD ܥ kodundan hareketle ߨ SHUPWDV\RQXQXYH߁ kodunu elde HWPHQLQ ]RUOX÷XQD ED÷OÕGÕU %X QRNWDGD VLVWHPLQ JYHQOL÷L VHoLOHQ ߞ kod DLOHVLQHED÷OÕGÕU [7].

(18)

7DQÕP .3. [4] ܮ ൌ ሺߙଵǡ ߙଶǡ ǥ ǡ ߙ௡ሻ, ܨଶ^೘ HOHPDQODUÕQÕQ VÕUDOÕ ELU NPHVL YH ݃, t dereceli ܨଶ^೘ ]HULQGHWDQÕPOÕ ܨଶ^೘ GHKLoELUN|NROPD\DQELUSROLQRPROVXQUHWHo

SROLQRP%XQDJ|UH ߁ሺܮǡ ݃ሻ LOHJ|VWHULOHQ*RSSDNRGX

׊݆ǡ Ͳ ൑ ݆ ൏ ݐǡ σ_ఈא௅ݖ_ఈ_௚ሺఈሻ^ఈ^ೕ ൌ Ͳ (1.1)

(1.1) denklemini VD÷OD\DQ ܿ ൌ ൫ܿ_ఈ_భǡ ܿ_ఈ_మǡ ǥ ǡ ܿ_ఈ_೙൯ א ܨ_ଶ^௡ YHNW|UOHULQGHQROXúXU Bu denklem matris formunda

ൣܿఈ_భܿఈ_మǤǤǤ ܿఈ_೙൧ ᇣᇧᇧᇧᇧᇤᇧᇧᇧᇧᇥ

௖

ۏێ ێێ

ێۍ ݃^ିଵሺߙଵሻ݃^ିଵሺߙଶሻǤǤǤ݃^ିଵሺߙ௡ሻ

݃^ିଵሺߙଵሻߙଵ݃^ିଵሺߙଶሻߙଶǤǤǤ݃^ିଵሺߙ௡ሻߙ௡

ǤǤ

݃^ିଵሺߙ_ଵሻߙ_ଵ^௧ିଵ݃^ିଵሺߙ_ଶሻߙǤ_ଶ^௧ିଵǤǤǤ݃^ିଵሺߙ_௡ሻߙ_௡^௧ିଵےۑۑۑۑې ᇣᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇥ

ு

௧

ൌ Ͳ

úHNOLQGH de ifade edilebilir 'ROD\ÕVÕ\OD ܪ matrisi ߁ሺܮǡ ݃ሻ Goppa kodunun parite kontrol matrisidir. %XQDJ|UH߁ሺܮǡ ݃ሻ Goppa kodunun boyutu ݇ YH+DPPLQJD÷ÕUOÕ÷Õ

݀ ROPDN]HUH݇ ൒ ݊ െ ݉ݐ ve ݀ ൒ ݐ ൅ ͳ \D]ÕODELOLU [8].

gUQHN .1. ݂ሺݔሻ polinomunu, ݂ሺݔሻ ൌ ݔ^ସ൅ ݔ^ଷ൅ ͳ úeklinde WDQÕPODUVDN ݂ polinomu ܨ_ଶ ]HULQGHDVDOGÕU%XQDJ|UHߙ א ܨ_ଶ^రǡ ݂ሺߙሻ ൌ Ͳ ROPDN]HUHܨ_ଶ^ర cismi Tablo 1.1. deki gibi olur.

Tablo 1.1. ܨ_ଶ^రFLVPLQLQHOHPDQODUÕ

ߙ^଴ൌ ͳ ߙ^ସൌ ߙ^ଷ൅ ͳ ߙ^଼ൌ ߙ^ଷ൅ ߙ^ଶ൅ ߙ ߙ^ଵଶൌ ߙ ൅ ͳ ߙ^ଵൌ ߙ ߙ^ହൌ ߙ^ଷ൅ ߙ ൅ ͳ ߙ^ଽൌ ߙ^ଶ൅ ͳ ߙ^ଵଷൌ ߙ^ଶ൅ ߙ ߙ^ଶൌ ߙ^ଶ ߙ^଺ൌ ߙ^ଷ൅ ߙ^ଶ൅ ߙ ൅ ͳ ߙ^ଵ଴ൌ ߙ^ଷ൅ ߙ ߙ^ଵସൌ ߙ^ଷ൅ ߙ^ଶ ߙ^ଷൌ ߙ^ଷ ߙ^଻ൌ ߙ^ଶ൅ ߙ ൅ ͳ ߙ^ଵଵൌ ߙ^ଷ൅ ߙ^ଶ൅ ͳ

߁ሺܮǡ ݃ሻ *RSSD NRGX LoLQ ݃ሺݔሻ ൌ ݔ^ଷ൅ ݔ ൅ ͳ ve ܮ ൌ ܨ_ଶ^ర ROPDN ]HUH 7DEOR 1.2.

GHNLGH÷HUOHUNXOODQÕOÕUVD ܪ VHQGURPPDWULVLDúD÷ÕGDNLJLELHOGHHGLOLU

(19)

Tablo 1.2. ݃^ିଵሺߙ௜ሻ HOHPDQODUÕQÕQ GH÷HUOHUL

݃ሺͳሻ^ିଵ ൌ ͳ ݃ሺߙ^ସሻ^ିଵ ൌ ߙ^ଵ଴ ݃ሺߙ^଼ሻ^ିଵ ൌ ߙ^ହ ݃ሺߙ^ଵଶሻ^ିଵൌ ߙ^ସ

݃ሺߙሻ^ିଵ ൌ ߙ^ଵ଴ ݃ሺߙ^ହሻ^ିଵ ൌ ߙ^ଵ଴ ݃ሺߙ^ଽሻ^ିଵ ൌ ߙ^଼ ݃ሺߙ^ଵଷሻ^ିଵൌ ߙ^ଵସ

݃ሺߙ^ଶሻ^ିଵ ൌ ߙ^ହ ݃ሺߙ^଺ሻ^ିଵ ൌ ߙ^ଶ ݃ሺߙ^ଵ଴ሻ^ିଵൌ ߙ^ହ ݃ሺߙ^ଵସሻ^ିଵൌ ߙ^଻

݃ሺߙ^ଷሻ^ିଵ ൌ ߙ ݃ሺߙ^଻ሻ^ିଵ ൌ ߙ^ଵଵ ݃ሺߙ^ଵଵሻ^ିଵൌ ߙ^ଵଷ

ܪ ൌ

ۉ ۈۈ ۈۇ

ͳߙ^ଵ଴ߙ^ହߙߙ^ଵ଴ߙ^ଵ଴ߙ^ଶߙ^ଵଵߙ^ହߙ^଼ߙ^ହߙ^ଵଷߙ^ସߙ^ଵସߙ^଻ Ǥ

ͳߙ^ଵଵߙ^଻ߙ^ସߙ^ଵସͳߙ^଼ߙ^ଷߙ^ଵଷߙ^ଶͳߙ^ଽߙߙ^ଵଶߙ^଺ Ǥ

ͳߙ^ଵଶߙ^ଽߙ^଻ߙ^ଷߙ^ହߙ^ଵସߙ^ଵ଴ߙ^଺ߙ^ଵଵߙ^ଵ଴ߙ^ହߙ^ଵଷߙ^ଵ଴ߙ^ହی ۋۋ ۋۊ

hUHWHo PDWULV ܩܪ^௧ ൌ Ͳ HúLWOL÷LQL VD÷OD\DFD÷ÕQGDQ ܩ UHWHo matrisi DúD÷ÕGDNL JLEL

elde edilir.

ܩ ൌ ۉ ۈۇ

ͳͳͳͳͳͲͳͲͳͳͲͲͳͲͲ ͳͳͲͲͳͳͳͳͲͳͲͲͲͳͲ ͲͳͲͲͳͲͳͲͲͳͳͳͲͲͳی

ۋۊ

%XQDJ|UHȞሺܮǡ ݃ሻ *RSSDNRGXDúD÷ÕGDNLJLELROXU

ͲͳͳͳͳͳͳͳͳͳͳͳͳͳͳͲͲͳͳͲͳͲͳͳͲͲͲͳͳͲ ͳͲͳͳͲͲͲͲͳͲͳͳͳͲͳ ͳͳͳͳͳͲͳͲͳͳͲͲͳͲͲ ͳͲͲͲͲͳͲͳͲͲͳͳͲͳͳ ͳͳͲͲͳͳͳͳͲͳͲͲͲͳͲ ͲͳͲͲͳͲͳͲͲͳͳͳͲͲͳ ͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲ

(20)

1.3. *RSSD.RGODUÕQ'HNRGODQPDVÕ

ሺͳǤͳሻ denkleminin VD÷ODQPDVÕLoLQJHUHNYH\HWHUúDUW σఈא௅_௭ିఈ^௖^ഀ ؠ Ͳ݉݋݀݃ሺݖሻ denkleminin VD÷ODQPDVÕGÕU 'ROD\ÕVÕ\OD aOÕQDQ PHsaj ݎ LOH NRGV|] ܿ ile ve hata YHNW|U de ݁ LOHJ|VWHULOLUVH ݎ ൌ ܿ ൅ ݁ ROPDN]HUH

෍ ݎఈ

ݖ െ ߙ ؠ

ఈא௅

෍ܿఈ൅ ݁ఈ

ݖ െ ߙ ؠ

ఈא௅

෍ ܿఈ

ݖ െ ߙ ൅ ෍

݁ఈ

ݖ െ ߙ ؠ

ఈא௅

෍ ݁ఈ

ݖ െ ߙ ݉݋݀݃ሺݖሻ

ఈא௅

elde edilir. %XQDJ|UH

ܵሺݖሻ ؠ σ_ఈא௅_௭ିఈ^௥^ഀ ݉݋݀݃ሺݖሻ (1.2)

ߪሺݖሻ ൌ ςఈא௅ሺݖ െ ߙሻ

௘_ഀஷ଴ (1.3)

ߟሺݖሻ ؠ ܵሺݖሻߪሺݖሻ݉݋݀݃ሺݖሻ (1.4)

ߟሺݖሻ ൌ σ_ఈא௅_௭ିఈ^௘^ഀ ςఈא௅ሺݖ െ ߙሻ ൌ

௘ഀஷ଴ σఈא௅ ݁_ఈ

௘ഀஷ଴ ςఈא௅ሺݖ െ ߚሻ

௘ഀஷ଴

ఉஷఈ

(1.5)

6ÕUDVÕ\OD ܵሺݖሻ , ߪሺݖሻ ve ߟሺݖሻ SROLQRPODUÕ ሺͳǤʹሻ , ሺͳǤ͵ሻ ve ሺͳǤͷሻ ifadeleriyle WDQÕPODQÕUVDܿ NRGV|]QHOGHHGHELOPHNLoLQሺͳǤͶሻ GHQNOHPLQLVD÷OD\DQHQNoN

dereceli ߪ ve ߟ SROLQRPODUÕQÕQEXOXQPDVÕ gerekir [9].

1.3.1. Berlekamp-Massey aOJRULWPDVÕ

Algoritma I. [10] ሺͳǤͶሻ denkleminde ݃ሺݖሻ ൌ ݖ^ଶ௧ |]HO GXUXPX LoLQ ߪሺݖሻ ൌ ߉_଴൅ ߉_ଵݖ ൅ ڮ ൅ ߉_௩ݖ^௩ǡ ߉_଴ ൌ ͳ ROPDN]HUH ߟ polinomunun derecesi ሺͳǤͷሻ ifadesine J|UHHQID]ODݒ െ ͳ RODELOLU'ROD\ÕVÕ\OD

(21)

σ^௩_௞ୀ଴߉_௞ܵ_௝ି௞ ൌ Ͳǡݒ ൅ ͳ ൑ ݆ ൑ ʹݐ (1.6)

HúLWOL÷L\D]ÕODELOLU%XHúLWOLNWHQKDUHNHWOH ߪ polinomunu úXúHNLOGHEXOXQDELOLU:

ߪ^ሺ଴ሻሺݔሻ ൌ ͳǡ ߪ^ሺ௜ሻሺݖሻ ൌ ෍ ߉௞ݖ^௞

௅_೔

௞ୀ଴

ǡ ͳ ൑ ݅ ൑ ʹݐ

ROPDN]HUH

݀௜ ൌ ෍ ߉௞ܵ௝ି௞

௅_೔

௞ୀ଴

GH÷HULQHJ|UHDúD÷ÕGDNLHúLWOLNOHUWDQÕPODQVÕQ.

ߪ^{ሺ௜ାଵሻ}ሺݖሻ ൌ ߪ^ሺ௜ሻሺݖሻǡ݀௜ ൌ Ͳ

ߪ^{ሺ௜ାଵሻ}ሺݖሻ ൌ ߪ^ሺ௜ሻሺݖሻ െ ݀௣ିଵ݀௜ݖ^௜ି௣ߪ^ሺ௣ሻሺݖሻǡ݀௜ ് Ͳ

Burada ߪ^ሺ௣ሻሺݔሻ polinomu, ݀_௣ ് Ͳ NRúXOXQXVD÷OD\DQ ߪ^ሺ௜ሻሺݖሻ polinomundan |QFHNL

herhangi bir polinomdur. $\UÕFD ሼܵଵǡ ܵଶǡ ǥ ǡ ܵேିଵሽ NPHVLQL UHWLS

ሼܵ_ଵǡ ܵ_ଶǡ ǥ ǡ ܵ_ேିଵǡ ܵ_ேሽ NPHVLQL UHWHPH\HQ PLQLPDO SROLQRPXQ X]XQOX÷X ܮ ile ve ሼܵ_ଵǡ ܵ_ଶǡ ǥ ǡ ܵ_ேିଵǡ ܵ_ேሽ kPHVLQLUHWHQPLQLPDOSROLQRPXQX]XQOX÷X ܮԢ LOHJ|VWHULOLUVH

ܮ^ᇱ൒ ሼܮǡ ܰ െ ܮሽ

\D]ÕODELOLU 'ROD\ÕVÕ\OD yukaUÕGDNL LWHUDV\RQODUOD HOGH HGLOHFHN PLQLPDO SROLQRPXQ

derecesi en fazla ݐ kadar olur.

gUQHN . Algoritma I¶ i kullanarak ሺߙ^ସǡ ߙǡ ߙ^଼ǡ ͳǡ ߙ^ଵଶǡ ߙ^ଶǡ ߙ^ହǡ ߙ^ଵଵሻ NPHVLQLQ

minimal polinomu Tablo 1.3.¶ e J|UH ߪሺݖሻ ൌ ͳ ൅ ߙ^ଵଵݖ ൅ ߙ^ଶݖ^ଶ൅ ߙ^ଵଷݖ^ଷ൅ ߙ^ଵସݖ^ସ olarak bulunur.

(22)

Tablo 1.3. ݀_௜GH÷HUOHULQHNDUúÕOÕNJHOHQߪ^ሺ௜ሻሺݔሻ SROLQRPODUÕ

i ࡿ_࢏ ࣌^ሺ࢏ሻሺ࢞ሻ ࢊ_࢏

1 ߙ^ସ ͳ ߙ^ସ

2 ߙ ͳ ߙ

2 ߙ ͳ ൅ ߙ^ଵଶݖ Ͳ

3 ߙ^଼ ͳ ൅ ߙ^ଵଶݖ ߙ^ଷ

3 ߙ^଼ ͳ ൅ ߙ^଻ݖ Ͳ

4 ͳ ͳ ൅ ߙ^଻ݖ Ͳ

5 ߙ^ଵଶ ͳ ൅ ߙ^଻ݖ ߙ^ଶ

5 ߙ^ଵଶ ͳ ൅ ߙ^଻ݖ ൅ ߙ^ଵସݖ^ଶ൅ ߙ^ଵଵݖ^ଷ Ͳ

6 ߙ^ଶ ͳ ൅ ߙ^଻ݖ ൅ ߙ^ଵସݖ^ଶ൅ ߙ^ଵଵݖ^ଷ ߙ^ଷ

6 ߙ^ଶ ͳ ൅ ߙ^ଽݖ ൅ ߙݖ^ଶ൅ ߙ^ଵଵݖ^ଷ Ͳ

7 ߙ^ହ ͳ ൅ ߙ^ଽݖ ൅ ߙݖ^ଶ൅ ߙ^ଵଵݖ^ଷ ͳ

7 ߙ^ହ ͳ ൅ ߙ^ହݖ ൅ ߙ^ଵଶݖ^ଶ൅ ߙ^ଵ଴ݖ^ଷ൅ ߙ^ସݖ^ସ Ͳ

8 ߙ^ଵଵ ͳ ൅ ߙ^ହݖ ൅ ߙ^ଵଶݖ^ଶ൅ ߙ^ଵ଴ݖ^ଷ൅ ߙ^ସݖ^ସ ߙ^ଵସ 8 ߙ^ଵଵ ͳ ൅ ߙ^ଵଵݖ ൅ ߙ^ଶݖ^ଶ൅ ߙ^ଵଷݖ^ଷ൅ ߙ^ଵସݖ^ସ Ͳ

Algoritma II [9] ሺͳǤͶሻ denkleminin, derecesi ݊ olan herhangi bir ݃ SROLQRPXLoLQ

o|]PDúD÷ÕGDNLJLELHOGHHGLOHELOLU

DGÕP Ͳ ൑ ݅ ൑ ݊ െ ͳǡ ݖ^௜ܵሺݖሻ݉݋݀݃ሺݖሻ SROLQRPODUÕQGD ݖ^௡ିଵ terimlerinin NDWVD\ÕODUÕܽ_௜ ROPDN]HUH݄ሺݖሻ ൌ ܽ_଴൅ ܽ_ଵݖ ൅ ڮ ൅ ܽ_௧ିଵݖ^௡ିଵ SROLQRPXWDQÕPODQÕU

DGÕP

݄ሺݖሻߪ^כሺݖሻ ؠ ߟ^כሺݖሻ݉݋݀ݖ^௡ , ݊ oLIWLVH

݄ሺݖሻߪ^כሺݖሻ ؠ ߟ^כሺݖሻ݉݋݀ݖ^௡ିଵǡ ݊ tek ise

GHQNOHPLQL VD÷OD\DQ GHUHFHVL HQ NoN RODQ ߪ^כ ve ߟ^כ polinRPODUÕ Algoritma I NXOODQÕODUDNEXOXQXU

(23)

DGÕP ߪ^כሺݖሻ ൌ ܿ଴൅ ܿ_ଵݖ ൅ ڮ ൅ ܿ_௥ݖ^௥ǡܰ ൌ ݉ܽݔሼ݀݁ݎߪ^כǡ ݀݁ݎߟ^כ൅ ͳሽǡ ݎ ൑ ܰ ROPDN]HUH

ߪሺݖሻ ൌ ܿ_଴ݖ^ே൅ ܿ_ଵݖ^ேିଵ൅ ڮ ൅ ܿ_௥ݖ^ேି௥

polinomu elde edilir. %|\OHFH HQ ID]OD ݐ KDWD G]HOWHQ SROLQRPDO ]DPDQOÕ ELU

DOJRULWPDHOGHHGLOPLúROXU

1.3.2. Patterson aOJRULWPDVÕ

߁ሺܮǡ ݃ሻGoppa koduܨ_ଶ]HULQGHWDQÕPODQGÕ÷Õ]DPDQ

ܵሺݖሻߪሺݖሻ ؠ ߪ^ᇱሺݖሻ݉݋݀݃ሺݖሻ (1.7)

\D]ÕODELOLU%XQDJ|UHܽ א ߁ሺܮǡ ݃ሻ ROPDVÕLoLQJHUHNYH\HWHUúDUW

ܵ௔ሺݖሻ ൌ Ͳ݉݋݀݃ሺݖሻ ฻ ݃ሺݖሻȁߪ௔ᇱሺݖሻ

ROGX÷XQGDQ YH ܨ_ଶ de herhangi bir polinom ߪ ൌ ߙ^ଶ൅ ݖߚ^ଶ úHNOLQGH LIDGH

HGLOHELOHFH÷LQGHQ ߪ^ᇱൌ ߚ^ଶ polinomu her zaman bir tam karedir'ROD\ÕVÕ\OD

݃ሺݖሻȁߪ_௔^ᇱሺݖሻ ฻ ݃^ଶሺݖሻȁߪ_௔^ᇱሺݖሻ

\D]ÕODELOHFH÷LQGHQ߁ሺܮǡ ݃ሻ ൌ ߁ሺܮǡ ݃^ଶሻ elde edilir. Bu VRQXFDJ|UH ݀݁ݎ݃ ൌ ݐ olmak

]HUH ݀ ൒ ʹݐ ൅ ͳ \D]ÕODELOLU [10]

ሺͳǤ͹ሻ GHQNOHPLQHJHULG|QOUVH ߪ ൌ ߙ^ଶ൅ ݖߚ^ଶROPDN]HUH

ሺߙ^ଶ൅ ݖߚ^ଶሻܵ ؠ ߚ^ଶ݉݋݀݃

\D]ÕODELOLU $\UÕFD ܵ SROLQRPXQXQWPN|NOHULܨଶ^೘ FLVPLQLQHOHPDQÕROGX÷XQGDQYH

݃ polinomunun ܨଶ^೘ ]HULQGH N|N ROPDGÕ÷ÕQGDQ ܵ ve ݃ SROLQRPODUÕ DUDODUÕQGD

DVODGÕU 'ROD\ÕVÕ\OD݄ܵ ؠ ͳ݉݋݀݃ RODFDNúHNLOGHELU݄ polinomu bulunabilir. Buna J|UH

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݄ܵሺߙ^ଶ൅ ݖߚ^ଶሻ ؠ ݄ߚ^ଶ݉݋݀݃

veya

ሺ݄ ൅ ݖሻߚ^ଶ ؠ ߙ^ଶ݉݋݀݃ (1.8)

\D]ÕODELOHFH÷LQGHQ ve ሺͳǤͺሻ denkleminin VD÷ WDUDIÕ WDP NDUH ELU LIDGH ROGX÷XQGDQ

GHQNOHPLQ VRO WDUDIÕ GD WDP NDUH ELU LIDGH ROPDOÕGÕU %XQD J|UH ݄ ൅ ݖ ؠ ݀^ଶ݉݋݀݃

RODFDNúHNLOGHELU݀ poOLQRPXEXOXQDELOLU'ROD\ÕVÕ\OD݀^ଶߚ^ଶ ؠ ߙ^ଶ݉݋݀݃ veya

݀ߚ ؠ ߙ݉݋݀݃ (1.9)

\D]ÕODELOLU Berlekamp-0DVVH\ DOJRULWPDVÕ NXOODQÕODUDN ሺͳǤͻሻ denkleminde ߙ ve ߚ SROLQRPODUÕ EXOXQDELOLU YH E|\OHFH ߪ ൌ ߙ^ଶ൅ ݖߚ^ଶ poOLQRPX EXOXQPXú ROXU [9].

(25)

**%g/h0 DESTEK AYIRMA $/*25ø70$SI**

7DQÕP 2.1. [11] ܫ௡ ൌ ሼͳǡʹǡ ǥ ǡ ݊ሽ LQGLV NPHVL YH ݔ ൌ ሺݔ_௜ሻ_ூ_೙ א ܥ ROPDN ]HUH

݀݁ݏሺݔሻ ൌ ሼ݅ א ܫ_௡ȁݔ௜ ് Ͳሽ NPHVLQHݔ koGV|]QQGHVWH÷L denir.

7DQÕP 2.2. [11] ܥ NRGV|] YHULOGL÷LQGH ݀݁ݏሺܥሻ ൌ ڂ_௫א஼݀݁ݏሺݔሻ úHNOLQGH WDQÕPOÕ

NPH\Hܥ NRGXQXQGHVWH÷L denir.

7DQÕP. [11] ܨ_ଶ ]HULQGHWDQÕPOÕ ݊ X]XQOX÷XQGD ܥ ve ܥԢ NRGODUÕYHULOGL÷LQGH

H÷HU ܥԢ NRGXQXQ NRGV|]OHUL ܥ NRGXQXQ NRGV|]OHULQLQ NRRUGLQDWODUÕQD ELU ߨ א ܵ_௡ SHUPWDV\RQX X\JXODQPDVÕ\OD HOGH HGLOHELOL\RUVDEX GXUXPX NÕVDFD ܥ^ᇱ ൌ ߨሺܥሻ úHNOLQGHGHLIDGHHGLOHELOLU.) bu iki koda denktirler denir ve ܥ̱ܥԢ \D]ÕOÕU.

<XNDUÕGDNL WDQÕPD J|UH ܥ NRGXQXQ LQGLV NPHVL ܫ_௡ ise ߨሺܥሻ kodunun indis NPHVLnin ߨሺܫ௡ሻ RODFD÷ÕDoÕNWÕU Bu tezde DNVLEHOLUWLOPHGLNoH ܥ ile ݊ X]XQOX÷XQGD ve ܨ_ଶFLVPL]HULQGH lineer bir NRGXLIDGHHGLOPLúWLU.

7DQÕP . [11] ܥNRGX LoLQ ߪሺܥሻ ൌ ܥ NRúXOXQX VD÷OD\DQ ߪ א ܵ_௡ SHUPWDV\RQODUÕQÕQROXúWXUGX÷XJUXED ܥ nin otomorfizm grubu denir. Bu tezde, ܥ

kodu YHULOGL÷LQGHܥ nin otomorfizm grubu ܣݑݐ_஼ile J|VWHULOPLúWLU.

7DQÕP . [11] ܬ ك ܫ_௡ROPDN ]HUH ܥ NRGXQXQ NRG V|]OHULQLQ ܬ ile indislenen NRRUGLQDWODUÕQÕQ \HUOHULQHVÕIÕU\D]ÕOPDVÕ\ODHOGHHGLOHQNRGD ܥ kodunun delikli kodu denir ve ܥ௃ile J|VWHULOLU ܬ ൌ ሼ݅ሽROPDVÕGXUXPXQGDNÕVDFD ܥ௜ \D]ÕOÕU

(26)

gUQHN2.1. gUQHNGHNL*RSSDNRGXQGD ܬ ൌ ሼͳǡͳͷሽ ROPDN]HUH, ͳ¶LQFLYH

ͳͷ¶LQFLELOHúHQOHUVÕIÕURODUDNDOÕQÕUVD ߁_௃ሺܮǡ ݃ሻ kodu

ͲͳͳͳͳͳͳͳͳͳͳͳͳͳͲ ͲͲͳͳͲͳͲͳͳͲͲͲͳͳͲ ͲͲͳͳͲͲͲͲͳͲͳͳͳͲͲ ͲͳͳͳͳͲͳͲͳͳͲͲͳͲͲ ͲͲͲͲͲͳͲͳͲͲͳͳͲͳͲ ͲͳͲͲͳͳͳͳͲͳͲͲͲͳͲ ͲͳͲͲͳͲͳͲͲͳͳͳͲͲͲ ͲͲͲͲͲͲͲͲͲͲͲͲͲͲͲ

olarak elde edilir.

gQHUPH 2.1. [11] ܥ NRGXYHULOGL÷LQGHKHUߨ א ܵ௡SHUPWDV\RQXܬ ك ܫ_௡ LQGLVLLoLQ

ߨ൫ܥ௃൯ ൌ ߨሺܥሻగሺ௃ሻ

ÇǤ

øVSDW ܥ NRGXQXQ NRG V|]OHUL DOW DOWD \D]ÕOÕUVD ݉ݔ݊ boyutlu bir ܯ matrisi ve ܥ௃

NRGXQXQNRGV|]OHULDOWDOWD\D]ÕOÕUVD yine ݉ݔ݊ boyutlu bir ܯ௃ matrisi elde edilir.

Burada ܯ matrisi ile ܯ_௃ matrislerinin ܬ ile indislenen VWXQYHNW|UOHULKDULo GL÷HU WP

VWXQ YHNW|UOHUL D\QÕ ROXS ܯ௃ matrisinin ܬ LOH LQGLVOHQHQ VWXQ YHNW|UOHUL VÕIÕUGÕU.

yHQLGHQVÕUDODQPDVÕQGDQúHNOLQGHGLU'ROD\ÕVÕ\ODܯԢԢ PDWULVLQLQVWXQYHNW|UOHUL ߚ_௝ LOH J|VWHULOLUVH ׊݆ ב ܬǡ ͳ ൑ ݆ ൑ ݊ LoLQ ߚ௝ ൌ ߙఙሺ௝ሻ ve ׊݆ א ܬ LoLQ ߚ௝ ൌ Ͳ

\D]ÕODELOHFH÷LQGHQ ve ܯԢԢ ile ܯ_ఙሺ௃ሻ^ᇱ PDWULVOHULQLQER\XWODUÕD\QÕROGX÷XQGDQ

ܯԢԢ ൌ ܯ_ఙሺ௃ሻ^ᇱ ฺ ɐ൫ܥ௃൯ ൌ ߪሺܥሻఙሺ௃ሻ

elde edilir.

(27)

6RQXo ܥ ve ܥԢ NRGODUÕGHQNLVH\DQL׌ߨ א ܵ௡ LoLQܥ^ᇱ ൌ Ɏሺܥሻ ise ܬԢ ൌ ߨሺܬሻ ROPDN]HUHܥ_௃ ve ܥ_௝ᇱ^ᇱ NRGODUÕGDGHQNWLUOHU

7DQÕP. [11] ׊ߪ א ܵ௡SHUPWDV\RQXLoLQ ܶሺܥሻ ൌ ܶ൫ߪሺܥሻ൯ NRúXOXQXVD÷OD\DQ

bir ܶ G|QúPQHLQYDU\DQWGHQLU

TanÕP. [11] ܨ KHUKDQJLELUNPHYHܵǡ ܨ]HULQGHGH÷HUOHUDODQELUG|QúP

ROVXQ(÷HU ׊݅ א ܫ_௡ǡ ׊ߪ א ܵ_௡ǡ

ܵሺܥǡ ݅ሻ ൌ ܵ൫ߪሺܥሻǡ ߪሺ݅ሻ൯

NRúXOXVD÷ODQÕ\RUVD ܵG|QúPQH ܨ]HULQGHELULP]DGHQLU

gQHUPH. [11] ܸELULQYDU\DQWROPDN]HUH

ܵ௏ሺܥǡ ݅ሻ ൌ ܸሺܥ௜ሻ

RODUDNWDQÕPODQDQ ܵ_௏G|QúPELULP]DGÕU

øVSDW ׊ߪ א ܵ௡ǡ ܵ௏൫ߪሺܥሻǡ ߪሺ݅ሻ൯ ൌ ܸ൫ߪሺܥሻఙሺ௜ሻ൯ ൌ ܸሺߪሺܥ௜ሻ ൌ ܸሺܥ௜ሻ ൌ ܵ௏ሺܥǡ ݅ሻ.

7DQÕP. [11]ܵ, ܨ]HULQGHGH÷erler alan bir imza ve

ܲ ൌ ൛ܲ௙ൟ_௙אி , ܲ௙ ൌ ሼ݅ א ܫ௡ȁܵሺܥǡ ݅ሻ ൌ ݂ א ܨሽ

ROPDN]HUH ܲNPHVLQH ܫ௡LQGLVNPHVLQLQ ሺܥǡ ܵሻ ±SDUoDODQÕúÕYH ܲ௙ NPHOHULQHGH

SDUoDODQÕúÕQVÕQÕIODUÕGHQLU.

7DQÕP. [11] ܲ ൌ ൛ܲ௙ൟ_௙אிve ܲ^ᇱൌ ൛ܲ^ᇱ_௙ൟ_௙אிǡ ܫ_௡LQGLVNPHVLQLQLNLSDUoDODQÕúÕ

olmak ]HUH ׊݂ א ܨǡ หܲ௙ห ൌ หܲ^ᇱ௙ห oluyorsa ܲve ܲ^ᇱSDUoDODQÕúODUÕQDGHQNWLUOHUGHQLU

ve bu durumda ̱ܲܲ^ᇱ\D]ÕOÕU

(28)

7DQÕP. [11] Bir ܥ kodu ve ܵ LP]DVÕYHULOGL÷LQGHH÷HU

׌݅ǡ ݆ א ܫ௡ LoLQ ܵሺܥǡ ݅ሻ് ܵሺܥǡ ݆ሻ

\D]ÕODELOL\RUVD ܵLP]DVÕQD ܥkRGXLoLQELUGLVNULPLQDQW, e÷HU

׊݅ǡ ݆ א ܫ_௡LoLQ ܵሺܥǡ ݅ሻ ് ܵሺܥǡ ݆ሻ

oluyorsa tam diskriminant denir.

gQHUPH [11] ሺܥǡ ܵሻ±SDUoDODQÕúÕ ܲLOH J|VWHULOLUVH ሺߪሺܥሻǡ ܵሻ±SDUoDODQÕúÕ ߪሺܲሻ\HHúLWROXU

øVSDW ሺܥǡ ܵሻveሺߪሺܥሻǡ ܵሻSDUoDODQÕúODUÕVÕUDVÕ\OD ܲ ൌ ൛ܲ௙ൟ_௙אிve ܲ^ᇱൌ ሼܲ^ᇱ௙ሽ௙אி ROPDN]HUH

ሺ֜ሻǣ݆ א ߪ൫ܲ௙൯ve ׌݅ א ܲ௙ǡ ݆ ൌ ߪሺ݅ሻǡ ROPDN]HUH

݂ ൌ ܵሺܥǡ ݅ሻ ൌ ܵ൫ߪሺܥሻǡ ߪሺ݅ሻ൯ ൌ ܵሺߪሺܥሻǡ ݆ሻ ฺ ݆ א ܲ^ᇱ_௙ ฺ ߪ൫ܲ_௙൯ ك ܲԢ_௙

elde edilir.

ሺ֚ሻǣ݆ א ܲ^ᇱ௙ ฺ ݆ ൌ ߪሺ݅ሻǡ׌݅ א ܫ௡, ݂ ൌ ܵሺߪሺܥሻǡ ݆ሻ ൌ ܵ൫ߪሺܥሻǡ ߪሺ݅ሻ൯ ൌ ܵሺܥǡ ݅ሻ

ฺ ݅ א ܲ௙, ݆ א ߪ൫ܲ௙൯ ฺ ܲ^ᇱ௙ ك ߪ൫ܲ௙൯ \D]ÕODELOLU %XQDJ|UH

׊݂ א ܨLoLQ ܲ^ᇱ_௙ ൌ ߪ൫ܲ_௙൯ \D]ÕODELOHFH÷LQGHQ

ܲ^ᇱൌ ߪሺܲሻ

elde edilir.

(29)

6RQXo . ܥ ve ܥԢ NRGODUÕ YHULOGL÷LQGH ܲ ൌ ሺܥǡ ܵሻ ve ܲ^ᇱൌ ሺܥ^ᇱǡ ܵሻ ROPDN ]HUH

ܥ^ᇱൌ ߨሺܥሻǡ ߨ א ܵ௡ ise ܲ^ᇱൌ ߨሺܲሻ \D]ÕODELOLU

6RQXo 2.3. ܣݑݐ஼grubu biULPGHQIDUNOÕLVH ܥNRGXLoLQWDPGLVNULPLQDQW\RNWXU

gQHUPH . ܥ^ᇱ̱ܥROPDN ]HUH ܵLP]DVÕ ܥNRGX ]HULQGH WDP GLVNULPLQDQW LVH ܥile ܥ^ᇱDUDVÕQGDNLSHUPWDV\RQEXOXQDELOLU

øVSDW ܥ^ᇱൌ ߨሺܥሻROPDN]HUH ሺܥǡ ܵሻ ve ሺܥ^ᇱǡ ܵሻSDUoDODQÕúODUÕQÕVÕUDVÕ\OD ܲve ܲ^ᇱ ile J|VWHULOLUVH ܲ^ᇱ ൌ ߨሺܲሻ\D]ÕODELOLU YH ܵǡ ܥ]HULQGH WDP GLVNULPLQDQW ROGX÷XQGDQ GROD\Õ her bir VÕQÕI WHN ELU HOHPDQGDQ ROXúDFD÷ÕQGDQ ߨ SHUPWDV\RQX EHOLUOHQPLú

olur.

gUQHN 2. ܥ ൌ ሼͳͳͳͳǡͲͳͳͳǡͳͲͳͲǡͲͲͲͳሽǡ ܥ^ᇱൌ ሼͳͳͳͳǡͳͳͳͲǡͲͲͳͳǡͲͳͲͲሽdenk kodlar ve ܥ^ᇱൌ ߨሺܥሻROPDN]HUH

Tablo 2.1. ܥ ve ܥԢ NRGODUÕLoLQܥ௜ǡ ܥ_௜^ᇱǡ ܹ஼೔ሺݔሻ ve ܹ_஼_೔^ᇲሺݔሻ GH÷HUOHUL

࢏ ࡯_࢏ ࢃ࡯_࢏ሺ࢞ሻ ࡯Ԣ_࢏ ࢃ_࡯^ᇲ_࢏ሺ࢞ሻ

1 ͲͳͳͳǡͲͳͳͳǡͲͲͳͲǡͲͲͲͳ ʹݔ ൅ ʹݔ^ଷ ͲͳͳͳǡͲͳͳͲǡͲͲͳͳǡͲͳͲͲ ݔ ൅ ʹݔ^ଶ൅ ݔ^ଷ 2 ͳͲͳͳǡͲͲͳͳǡͳͲͳͲǡͲͲͲͳ ݔ ൅ ʹݔ^ଶ൅ ݔ^ଷ ͳͲͳͳǡͳͲͳͲǡͲͲͳͳǡͲͲͲͲ ͳ ൅ ʹݔ^ଶ൅ ݔ^ଷ 3 ͳͳͲͳǡͲͳͲͳǡͳͲͲͲǡͲͲͲͳ ʹݔ ൅ ݔ^ଶ൅ݔ^ଷ ͳͳͲͳǡͳͳͲͲǡͲͲͲͳǡͲͳͲͲ ʹݔ ൅ ݔ^ଶ൅ݔ^ଷ 4 ͳͳͳͲǡͲͳͳͲǡͳͲͳͲǡͲͲͲͲ ͳ ൅ ʹݔ^ଶ൅ ݔ^ଷ ͳͳͳͲǡͳͳͳͲǡͲͲͳͲǡͲͳͲͲ ʹݔ ൅ ʹݔ^ଷ

Tablo ¶H J|UH

ܹ_஼^ᇲ_భሺݔሻ ൌ ܹ஼_మሺݔሻ ฺ ͳ ൌ ߨሺʹሻ

ܹ_஼^ᇲ_మሺݔሻ ൌ ܹ_஼_రሺݔሻ ฺ ʹ ൌ ߨሺͶሻ

ܹ_஼^ᇲ_యሺݔሻ ൌ ܹ஼_యሺݔሻ ฺ ͵ ൌ ߨሺ͵ሻ

ܹ_஼^ᇲ_రሺݔሻ ൌ ܹ஼_భሺݔሻ ฺ Ͷ ൌ ߨሺͳሻ

\D]ÕODELOHFH÷LQGHQ ߨ ൌ ሺͳͶʹሻ elde edilir.

(30)

7DQÕP . [11] ܥkodu ve ܵim]DVÕ YHULOGL÷LQGH \XNDUÕGD DQODWÕOGÕ÷Õ JLEL ܥ kodunun supportunun bir ሺܥǡ ܵሻ±SDUoDODQÕúÕQÕQ HOGH HGLOGL÷L \|QWHPH 'HVWHN

$\ÕUPD AlgoritmaVÕ denir. Bu tezde 'HVWHN $\ÕUPD $OJRULWPDVÕ NÕVDFD ܦܣܣ úHNOLQGHJ|VWHULOPLúWLU.

Uygulamada ܦܣܣ('HVWHN $\ÕUPD $OJRULWPDVÕ X\JXODQÕUNHQ LQYDU\DQW RODUDN

JHQHOGH+DPPLQJD÷ÕUOÕNVD\DFÕNXOODQÕOÕUYHEXGXUXPGD HQE\N]RUOXN NRGODUÕQ

ER\XWX DUWWÕNoD D÷ÕUOÕN VD\DFÕQÕQ KHVDSODQPDVÕGÕU. Bu nedenle Hull kod NDYUDPÕ

WDQÕPODQPÕúWÕU[12]

7DQÕP. [13] ܥǡ ݊X]XQOX÷XQGDlineer bir kod ve ܥ^ୄǡ ܥ kodunun duali olmak

]HUH

࣢ሺܥሻ ൌ ܥ ת ܥ^ୄ

úHNOLQGH WDQÕPODQDQNRGD ܥ kodunun Hull kodu denir.

Lemma 2.1. [11]ܥve ܥ^ᇱ݊X]XQOX÷XQGD LNL NRG ROPDN ]HUH KHU ߪ א

ܵ௡SHUPWDV\RQXLoLQ

ߪሺܥ ת ܥ^ᇱሻ ൌ ߪሺܥሻ ת ߪሺܥ^ᇱሻ

\D]ÕODELOLU

Lemma 2.2. ܥǡ ݊X]XQOX÷XQGDOLQHHUELUNRGROPDN]HUH׊ݔǡ ݕ א ܥ LoLQ

ݔ ൌ ሺݔ_௜ሻ_௜אூ_೙ǡ ݕ ൌ ሺݕ_௜ሻ_௜אூ_೙ǡ ݔǤ ݕ ൌ ݔݕ^௧ൌ ෍ ݔ_௜

௜אூ_೙

ݕ_௜

úHNOLQGH WDQÕPOÕVNDOHUoDUSÕPSHUPWDV\RQLúOHPLDOWÕQGDGH÷LúPH]

(31)

øVSDW. ǣ ֜׊ݔ א ܥ ve ׊ݕ א ܥ^ୄLoLQ ݔ ൌ ߪሺݔሻǡ ݕ ൌ ߪሺݕሻ ROPDN ]HUH ߪ SHUPWDV\RQXQD NDUúÕOÕN JHOHQ PDWULV ܲ LOH J|VWHULOLUVH SHUPWDV\RQ PDWULVLQ

ortogonallikሺܲ^௧ ൌ ܲ^ିଵሻ [14] |]HOOL÷LQGHQGROD\Õ

ݔݕ ൌ ሺݔܲ^௧ሻሺݕܲ^௧ሻ^௧ ൌ ሺݔܲ^௧ሻሺܲݕ^௧ሻ ൌ ሺݔܲ^ିଵሻሺܲݕ^௧ሻ ൌ ݔݕ^௧ ൌ ݔǤ ݕ ൌ Ͳ

olarak istenen elde edilir.

Lemma 2.3. [11] ܥǡ ݊ X]XQOX÷XQGD OLQHHU ELU NRG YH ܥ^ୄǡ ܥ kodunun dualini J|VWHUPHN]HUHKHUߪ א ܵ௡ SHUPWDV\RQXLoLQ

ሾߪሺܥሻሿ^ୄ ൌ ߪሺܥ^ୄሻ

\D]ÕODELOLU

øVSDWLemma 2.2 den ߪሺܥ^ୄሻ ك ሾߪሺܥሻሿ^ୄ ROGX÷XDoÕNWÕU ܾ݋ݕሺܥሻ ൌ ݇ROPDN]HUH

ܾ݋ݕሺܥ^ୄሻ ൌ ݊ െ ݇ǡ ܾ݋ݕ൫ߪሺܥሻ൯ ൌ ݇ǡ ܾ݋ݕ൫ߪሺܥ^ୄሻ൯ ൌ ݊ െ ݇ǡ ܾ݋ݕሺሾߪሺܥሻሿ^ୄሻ ൌ ݊ െ ݇

ܾ݋ݕ൫ߪሺܥ^ୄሻ൯ ൌ ܾ݋ݕሺሾߪሺܥሻሿ^ୄሻ ise ሾߪሺܥሻሿ^ୄ ൌ ߪሺܥ^ୄሻ

elde edilir.

gQHUPH. [11] ܥve ܥ^ᇱǡ ݊X]XQOX÷XQGDLNLOLQHHUNRGROPDN]HUH ܥ̱ܥ^ᇱise

࣢ሺܥሻ̱࣢ሺܥԢሻ

\D]ÕODELOLU

øVSDW . ߪ൫࣢ሺܥሻ൯ ൌ ߪሺܥ ת ܥ^ୄሻ ൌ ߪሺܥሻ ת ߪሺܥ^ୄሻ ൌ ߪሺܥሻ ת ሾߪሺܥሻሿ^ୄൌ

࣢൫ߪሺܥሻ൯ ൌ ࣢ሺܥԢሻ.

(32)

%g/h0 MCELI(&(ùø)5(/(0(6ø67(0ø1'(=$<,)

ANAHTARLAR

gQHUPH e J|UH ܥ ve ܥԢ NRGODUÕ YHULOGL÷LQGH H÷HU EX NRGODU GHQN LVH EXQODUD

NDUúÕOÕNJHOHQܲ ൌ ሺܥǡ ܵሻ ve ܲ^ᇱൌ ሺܥ^ᇱǡ ܵሻ SDUoDODQÕúODUÕGDGHQNROXU$QFDNEXQXQ

WHUVLKHU]DPDQGR÷UXGH÷LOGLU'DKDDoÕNELULIDGH\OHܲ ve ܲԢ SDUoDODQÕúODUÕQÕQGHQN

ROPDVÕ ܥ ve ܥԢ NRGODUÕQÕQ GHQN ROGXNODUÕQÕ J|VWHUPH] $QFDN SUDWLNWH ܲ ve ܲԢ SDUoDODQÕúODUÕQÕQ VÕQÕIODUÕQÕQ KHSVL WHN ELU HOHPDQGDQ ROXúPDGÕ÷Õ ]DPDQ NRGODUÕQ

X]XQOXNODUÕ \HWHULQFH E\N ROGX÷X WDNGLUGH ݊ ൒ ͳͲʹͶሻ EXQXQ GR÷UX ROGX÷X

YDUVD\ÕODELOLU. <LQHgQHUPHGHQ ݅ ve ݆ HOHPDQODUÕH÷HUܣݑݐ஼ JUXEXQDJ|UHD\QÕ

orbitte ise ܵሺܥǡ ݅ሻ ൌ ܵሺܥǡ ݆ሻ yazabiliriz$QFDNEXQXQWHUVLKHU]DPDQGR÷UXGH÷LOGLU

'ROD\ÕVÕ\OD ܵሺܥǡ ݅ሻ ൌ ܵሺܥǡ ݆ሻ ROPDVÕJ|UH݅ ve ݆ nin ܣݑݐ_஼ JUXEXQDJ|UHD\QÕRUELWWH

ROGXNODUÕ DQODPÕQD JHOPH] $QFDN \LQH GH ܵሺܥǡ ݅ሻ ൌ ܵሺܥǡ ݆ሻ NRúXOXQX VD÷OD\DQ

HOHPDQODUÕQ ROXúWXUGX÷X VÕQÕIODU ܣݑݐ_஼ JUXEXQD J|UH RUELWOHULQ ELUOHúLPLQGHQ

ROXúDFD÷ÕQGDQ H÷HU ܣݑݐ_஼ JUXEXQD J|UH RUELWOHUL ELOL\RUVDN EXQGDQ KDUHNHWOH ED]Õ

VRQXoODUD YDUÕODELOLU [4].

7DQÕP ܨݎǣ ܨଶ^೘ ՜ ܨଶ^೘ ROPDN ]HUH ܨݎሺݖሻ ൌ ݖ^ଶ úHNOLQGH WDQÕPOÕ G|QúPH

)UREHQLXVG|QúP denir.

gQHUPH [2] ߁ሺܮǡ ݃ሻ*RSSD NRGX LoLQ ݃UHWHo SROLQRPX ܨଶ ]HULQGH

WDQÕPODQÕUVD ܣݑݐ_௰grubu Frobenius G|QúP WDUDIÕQGDQUHWLOLU

gQHUPH ¶ H J|UH ܮ ൌ ܨ_ଶ^೘ ROPDN ]HUH ܮ nin ܣݑݐ_௰grubuna g|UH RUELWOHULQH

D\UÕOPÕúúHNOL ࣪௅ LOHJ|VWHULOLUVH ࣪௅̱ܦܣܣሺܥሻ ROXSROPDGÕ÷ÕQDEDNÕODUDN݃UHWHo

polinomunun ܨ_ଶ ]HULQGH WDQÕPODQÕS WDQÕPODQPDGÕ÷Õ DQODúÕODELOLU 'ROD\ÕVÕ\OD

denenmesi gereken ݃ SROLQRPODUÕQÕQVD\ÕVÕD]DOPÕúROXUBu nedenle bu tip Goppa NRGODUÕ]D\ÕI DQDKWDUODURODUDNDGODQGÕUÕOÕU [4].

Destek ayırma algoritması ve McEliece şifreleme sisteminde zayıf anahtarlar

)(1%ø/ø0/(5ø(167ø7h6h

'(67(.$<,50$$/*25ø70$6,9(

0F(/,(&(ùø)5(/(0(6ø67(0ø1'(

ZAYIF ANAHTARLAR

<h.6(./ø6$167(=ø

7(ù(..h5

ødø1'(.ø/(5

6ø0*(/(59(.,6$/70$/$5/ø67(6ø

ù(.ø//(5/ø67(6ø

g=(7

SUPPORT SPLITTNG ALGORITHM AND THE WEAK KEYS IN THE MCELIECE CRYPTOSYSTEM

SUMMARY

%g/h0*ø5øù

%g/h0 DESTEK AYIRMA $/*25ø70$SI

%g/h0 MCELI(&(ùø)5(/(0(6ø67(0ø1'(=$<,)

ANAHTARLAR

**'(67(.$<,50$$/*25ø70$6,9(**

**6ø0*(/(59(.,6$/70$/$5/ø67(6ø**

**%g/h0*ø5øù**

**%g/h0 DESTEK AYIRMA $/*25ø70$SI**

%g/h0 MCELI(&(ùø)5(/(0(6ø67(0ø1'(=$<,)