THE McELIECE CRYPTOSYSTEM WITH ARRAY CODES. MATRİS KODLAR İLE McELIECE ŞİFRELEME SİSTEMİ

Transcription

1 SAÜ e Blmler Dergs, 5 Clt, 2 Sayı, THE McELIECE CRYPTOSYSTEM WITH ARRAY CODES Vedat ŞİAP* *Departmet of Mathematcs, aculty of Scece ad Art, Sakarya Uversty, 5487, Serdva, Sakarya-TURKEY vedatsap@gmalcom Abstract: Publc-key cryptosystems form a mportat part of cryptography I these systems, every user has a publc ad a prvate key The publc key allows other users to ecrypt messages, whch ca oly be decoded usg the secret prvate key I that way, publc-key cryptosystems allow easy ad secure commucato betwee all users wthout the eed to actually meet ad exchage keys Oe such system s the McElece Publc-Key cryptosystem, sometmes also called McElece Scheme However, as we lve the formato age, codg s used order to protect or correct the messages the trasferrg or the storg processes So, lear codes are mportat the trasferrg or the storg Due to rchess of ther structure array codes whch are lear are also a mportat codes However, the formato s the trasferred to the source more securely by creasg the error correcto capablty wth array codes I ths paper, we combe two terestg topcs, McElece cryptosystem ad array codes Key words: Publc-Key cryptosystem, Codg theory, Lear codes, Array codes, McElece cryptosystem AMS Classfcato: T7 MATRİS KODLAR İLE McELIECE ŞİRELEME SİSTEMİ Özet: Açık aahtarlı şfreleme sstemler krptograf öeml br parçasıı oluşturmaktadır Bu sstemlerde, her kullaıcı açık ve gzl aahtar adıı ala k tür aahtara sahp olup açık aahtar, sadece gzl aahtar kullaılarak şfres çözüleble mesajları şfrelemek ç dğer kullaıcılara z vermektedr Bu şeklde, açık aahtarlı şfreleme sstemler aahtar değşm ve br oktada bağlatıya gerek duymada bütü kullaıcılar arasıda güvel ve kolay br letşme olaak sağlamaktadır McElece şeması olarak ta adladırıla McElece açık aahtarlı şfreleme sstemler bu tp şfreleme sstemlere br örek teşkl etmektedr Buula brlkte, blg çağıı yaşadığımız bu gülerde blg trasfer ya da depolaması aşamasıda meydaa geleblecek blg zedelemeler koruma ve düzeltme amacıyla kodlama kullaılmaktadır Bu alamda kullaıla kodlar çde leer kodlar öeml br yer tutmaktadır Leer kodlar alesde ola matrs kodlar zeg br yapıya sahp olup bu kodlar le hata düzeltme kablyetler artmakta ve buu soucuda blg daha güvelr br şeklde letlmektedr Bu bağlamda, makalede güvelrlğ arttırma adıa McElece şfreleme sstem şasıda matrs kodları göz öüe alıacaktır Aahtar kelmeler: Açık aahtarlı şfreleme sstemler, Kodlama teors, Leer kodlar, Matrs kodlar, McElece şfreleme sstemler I INTRODUCTION The McElece publc-key ecrypto scheme s based o error-correctg codes The dea behd ths scheme s to frst select a partcular (lear) code for whch a effcet decodg algorthm s kow, ad the to use a trapdoor fucto to dsguse the code as a geeral lear code Sce the problem of decodg a arbtrary lear code s NP-hard, a descrpto of the orgal code ca serve as the prvate key, whle a descrpto of the trasformed code serves as the publc key [] 46

2 SAÜ e Blmler Dergs, 5 Clt, 2 Sayı, The McElece ecrypto scheme (whe used wth Goppa codes, as orgally proposed by McElece 978) has ressted cryptaalyss to date It s also otable as beg the frst publc-key ecrypto scheme to use radomzato the ecrypto process Although very effcet, the McElece ecrypto scheme has receved lttle atteto practce because of the very large publc keys [2], [3] Ths paper vestgates the applcato of array codes cryptography, wth specal atteto to the applcato the McElece cryptosystem The rest of paper s orgazed as follows: Secto II gves a troducto to cryptography It explas the terms ecessary to uderstad the rest of the paper ad cludes lear ad array codes I Secto III, we troduce the McElece cryptosystem We descrbe the way the system works I Secto IV, we use array codes to costruct a code that ca be used wth the McElece cryptosystem II CRYPTOGRAPHIC BACKGROUND I ths secto, we preset some cryptographc backgroud eeded to uderstad array codes I geeral, we cosder words of fxed legth wth letters from a fte alphabet Q Thus words are elemets of Q A code s a subset of Q ad the elemets of the code are called codewords The atural umber s the legth of the code A mportat class of codes s lear codes Ths wll be the oly class of codes cosdered ths paper 2 Lear rom ow o let the alphabet Q be a fte feld Q s a vector space, so Defto 2 [4] (Hammg dstace, weght) To gve the dfferece of two codewords a precse meag the (Hammg) dstace betwee two words s troduced Let x, y, the d x, y : x y (2) The (Hammg) weght of a codeword s the umber of ozero etres ad therefore the dstace from the zero vector: Defto 22 [4] A lear code C of dmeso k s a k - dmesoal lear subspace of ad s ofte called a, k code The thrd mportat parameter of a code C, besdes the legth ad dmeso, s the mmum dstace betwee ts codewords Defto 23 [4] The mmum Hammg dstace d of a lear code s d m d u, v m w u (23) uv u0 It s ofte called the mmum dstace or smply the dstace of the code; ay two codewords dffer at least d places A code of dmeso k, legth ad mmum dstace d s ofte called a, k, d code Two types of matrces play a mportat role for lear codes: geerator ad (party) check matrces They are defed as follows k Defto 24 [4] If the ecodg : from message m to codeword c s doe by the matrx multplcato c m mg, (24) where G s a k matrx wth etres, the G s called geerator matrx of the code The rows of G form a bass of C Defto 25 [4] A party check matrx of a lear, k code C s a k matrx H, such that T C x : xh 0 (25) Thus the rows of a check matrx geerate the orthogoal complemet of C Example 2 The bary code 0000, 00,0,0 C ca be defed by a geerator matrx G, where : 0,0 w x x d x (22) 47

3 SAÜ e Blmler Dergs, 5 Clt, 2 Sayı, 0 G (25) 0 0 It ca also be defed by a check matrx H wth 0 0 H (26) 0 I may cases we trasmt the ecoded message Because of the chael ose the receved word may cota some errors, so we wat to be able to at least detect or better correct these errors Usually we do ths by choosg the codeword whch s closest (wth respect to the Hammg metrc) to the receved word to mmze the probablty of makg a mstake Lemma 2 [4] or a, k, d -code C the spheres Sc x : d x, c d 2, c C, (27) do ot overlap, so every receved word S c wll be 2 d corrected to c Hece ths code corrects up to errors Proof Assume two of the spheres overlap, e they both cota a pot x The the dstace betwee the two ceters of the spheres s ot greater tha twce the dstace to x, thus ot greater tha d Ths cotradcts the assumpto that C s a code wth mmum dstace d or ay gve lear code we ca costruct ts dual code Defto 26 [4] If C s a, k lear code over, ts dual or orthogoal code C s the set of vectors whch are orthogoal to all codewords of C : t C u : uv 0 for all v C (28) Now, we defe matrx space ad array codes as follows: Defto 27 [5] Let Let Matm The, Matm be a fte feld wth elemets deote the set of matrces wth etres s a vector space over Defto 28 [5] A array code s a subset of Matm ad a lear array code s a Mat m -lear subspace of Example 22 The elemets of the lear array code 0 C Mat 3 spaed by v ad v2 0 0 are as follows: ,,,,, C ,,, Note that the space Matm m space Every matrx Matm s detfable wth the ca be represeted as a m vector by wrtg the frst row of matrx followed by secod row ad so o Smlarly, every vector ca be represeted as a m s matrx Mat m m by separatg the co-ordates of the vector to m groups of s -coordates [5] Defto 29 [5] The mappg wth parameters m ad, deoted by M m,, s the oe that maps the vector a0,0 a0, v v, v2,, vm to matrx A a a so that a, j v j, for 0,,, m, j 0,,, m,0 m, m Example 23 Let m 3, 4 ad e 0,0,,0,4,0,0,0,0,0,0,3 The, A Defto 20 [6] Gve a array a 0 m horzotal sydrome h h0, h,, hm are defed by v v, v,, v 0, j 0 j, ts ad 48

4 SAÜ e Blmler Dergs, 5 Clt, 2 Sayı, h a, 0 m, l 0 m l 0, l v a, 0 j l, j Clearly, a array s Matm f ad oly f both the horzotal ad the vertcal sydromes are eual to zero As a example, cosder the array A Its horzotal sydrome s h 0, whle ts vertcal sydrome s v It follows that ths array s ot Mat 3 7 The related work about array codes ca be foud [7] III THE McELIECE CRYPTOSYSTEM Let C be a, k lear code wth a fast decodg algorthm that ca correct up to t errors Let G be a geerator matrx for C To create the dsguse, let S be a radom k k vertble matrx (also called the scrambler) ad let P be a radom permutato matrx The matrx G SGP (29) s made publc whle S, G ad P form the prvate key Ecrypto: Represet the message as a strg m of legth k, choose a radom error vector e of weght at most t ad compute the cpher text c mg e ad zp has weght at most t, the decodg algorthm for the code geerated by G corrects c to m ms ally, ms m ad, hece, decrypto works [2] 3 Practcalty of the McElece Scheme As poted out by Rao ad Nam, the McElece scheme reures rather large block legth They suggested 024, but today ths s ot eough aymore, so 2048 should be chose Therefore ths scheme produces too much computatoal overhead for ecrypto ad decrypto for most practcal applcatos [8] IV A McELIECE CRYPTOSYSTEM USING ARRAY CODES I ths secto, we wll dscuss how to costruct a McElece cryptosystem usg array codes [7] Example 4 Let C be a code over 5 Cosder the matrx G whch s the geerator matrx for the code Suppose Alce wshes to sed a message m 3, 2,0,, 4, 2 to Bob I order to do so, Bob must create a vertble matrx S ad a radom permutato matrx P that he wll keep secret If Bob chooses Decrypto: To recover the platext m from c, we compute c cp, use the decodg algorthm for the code geerated by G to decode c to m ad compute m ms Proof that decrypto works Sce c cp mg z P msgp z P ms G zp ad S

5 SAÜ e Blmler Dergs, 5 Clt, 2 Sayı, P Usg these, Bob geerates the publc ecrypto matrx G I order to ecrypt, Alce geerates her ow radom error vector e ad calculates the cpher text c mg e I the case of the array code the error vector has weght Suppose Alce chooses e 0,,0,0,0,0,0,0,0,0,0,0 cotas oe error It s clear that the error vector s 0,,0,0,0,0,0,0,0,0,0,0 Therefore, the errors ca be foud ad removed V CONCLUSION Array codes have the potetal to mprove the use of the McElece cryptosystem REERENCES Stso, DR, Cryptography theory ad practce, CRC Press LLC, USA (995) 2 McElece, RJ, A publc-key cryptosystem based o algebrac codg theory, DSN Progress Report 42-44, 4-6 (978) 3 Golay, MJE, Notes o dgtal codg, Proc IRE, 37, 657 (949) 4 Roma, Codg ad Iformato Theory, Graduate Text Mathematcs, Sprger Verlag (992) 5 Sapa, J, Campopao-type bouds o- Hammg array codg, Lear Algebra ad ts Applcatos, 420, (2007) 6 Pless, VS, Hufma, WC, Hadbook of codg theory, Elsever BV, The Netherlads (998) 7 Şap, V, Matrs Kodlar le McElece Şfreleme Sstem, Yüksek Lsas, Sakarya Üverstes, (2008) 8 Rao, TRN, Nam, KH, Prvate-key algebraccode ecryptos, IEEE Tras o Iform Theory, 35 (989) Thus the receved word s c mg e 2,0,,2,4,2,,3,4,3,3,0 0,,0,0,0,0,0,0,0,0,0,0 2,,,2,4,2,,3,4,3,3,0 The decodg procedure eed oly c to the matrx c from Defto 29 ad fd the errors by checkg M 3,4 these rows ad colums, ad remove the errors va horzotal ad vertcal sydromes The procedure of fdg errors s demostrated the followg 2 2 rom c, we kow that M 3,4 c Its horzotal ad vertcal sydromes are h 00 ad v 000, respectvely That s, by checkg bts, we fd that the frst row cotas oe error ad the secod colum 50