Research Article A New Construction of Multisender Authentication Codes from Polynomials over Finite Fields

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1 Hdaw Publhg Corporato Joural of Appled Mathematc Volume 03, Artcle ID 3039, 7 page Reearch Artcle A New Cotructo of Multeder Authetcato Code from Polyomal over Fte Feld Xul Wag College of Scece, Cvl Avato Uverty of Cha, Taj , Cha Correpodece hould be addreed to Xul Wag; xlwag@cauceduc Receved 3 February 03; Accepted 7 Aprl 03 Academc Edtor: Yag Zhag Copyrght 03 Xul Wag Th a ope acce artcle dtrbuted uder the Creatve Commo Attrbuto Lcee, whch permt uretrcted ue, dtrbuto, ad reproducto ay medum, provded the orgal work properly cted Multeder authetcato code allow a group of eder to cotruct a authetcated meage for a recever uch that the recever ca verfy the authetcty of the receved meage I th paper, we cotruct oe multeder authetcato code from polyomal over fte feld Some parameter ad the probablte of decepto of th code are alo computed Itroducto Multeder authetcato code wa frtly cotructed by Glbert et al[] 974 Multeder authetcato ytem refer to who a group of eder, cooperatvely ed a meage to a recever; the the recever hould be able to acerta that the meage authetc About th cae, may cholar adreearcherhadmadegreatcotrbutotomulteder authetcato code, uch a [ 6] I the actual computer etwork commucato, multeder authetcato code clude equetal model ad multaeou model Sequetal model that each eder ue h ow ecodg rule to ecode a ource tate orderly, the lat eder ed the ecoded meage to the recever, ad the recever receve the meage ad verfe whether the meage legal or ot Smultaeou model that all eder ue ther ow ecodg rule to ecode a ource tate, ad each eder ed the ecoded meage to the ythezer, repectvely; the the ythezer form a authetcated meage ad verfe whether the meage legal or ot I th paper, we wll adopt the ecod model I a multaeou model, there are four partcpat: a group of eder U{U,U,,U },thekeydtrbuto ceter, he repoble for the key dtrbuto to eder ad recever, cludg olvg the dpute betwee them, a recever R, ad a ythezer, where he oly ru the truted ythe algorthm The code work a follow: each eder ad recever ha ther ow Cartea authetcato code, repectvely Let (S, E,T ;f ) (,,,)be the eder Cartea authetcato code, (S, E R,T;g) be the recever Cartea authetcato code, h:t T T T be the ythe algorthm, ad π : E E be a ubkey geerato algorthm, where E the key et of the key dtrbuto ceter Whe authetcatg a meage, the eder ad the recever hould comply wth the protocol The key dtrbuto ceter radomly elect a ecodg rule e E ad ed e π (e) to the th eder U (,,,), ecretly; the he calculate e R by e accordg to a effectve algorthm ad ecretly ed e R to the recever R If the eder would lke to ed a ource tate to the recever R, U compute t f (, e ) (,,,) ad ed m (,t ) (,,,)to the ythezer through a ope chael The ythezer receve the meage m (, t ) (,,,)ad calculate th(t,t,,t ) by the ythe algorthm h ad the ed meage m (, t) to the recever; he check the authetcty by verfyg whether t g(,e R ) or ot If the equalty hold, the meage authetc ad accepted Otherwe, the meage rejected We aume that the key dtrbuto ceter credble, ad though he kow the eder ad recever ecodg rule, he wll ot partcpate ay commucato actvte Whe tramtter ad recever are dputg, the key dtrbuto ceter ettle t At the ame tme, we aume that the ytem follow the Kerckhoff prcple whch, except the actual ued key, the other formato of the whole ytem publc

2 Joural of Appled Mathematc I a multeder authetcato ytem, we aume that the whole eder are cooperatve to form a vald meage; that, all eder a a whole ad recever are relable But there are ome malcou eder who together cheat the recever; the part of eder ad recever are ot credble, ad they ca take mperoato attack ad ubttuto attack I the whole ytem, we aume that {U,U,,U } are eder, R a recever, E the ecodg rule et of the eder U,ad E R the decodg rule et of the recever R If the ource tate pace S ad the key pace E R of recever R are accordg to a uform dtrbuto, the the meage pace M ad the tag pace T are determed by the probablty dtrbuto of S ad E R L {,,, l } {,,,}, l <, U L {U,U,,U l }, E L {E U,E U,,E Ul } Now coder that let u coder the attack from malcou group of eder Here, there are two kd of attack The oppoet mperoato attack to recever: U L,after recevg ther ecret key, ecode a meage ad ed t to the recever U L are ucceful f the recever accept t a legtmate meage Deote by P I the larget probablty of ome oppoet ucceful mperoato attack to recever; t ca be expreed a P I max { {e R E R e R m} m M E } () R The oppoet ubttuto attack to the recever: U L replace m wth aother meage m,aftertheyobervea legtmate meage m U L are ucceful f the recever accept t a legtmate meage; t ca be expreed a m P S max {max m M {e R E R e R m,m } {e } () R E R e R m} m M There mght be l malcou eder who together cheat the recever; that, the part of eder ad the recever are ot credble, ad they ca take mperoato attack Let L {,,, l } {,,,}, l < ad E L {E U,E U,,E Ul } Aume that U L {U,U,,U l }, after recevg ther ecret key, ed a meage m to the recever R; U L are ucceful f the recever accept t a legtmate meage Deote by P U (L) the maxmum probablty of ucce of the mperoato attack to the recever It ca be expreed a P U (L) max max{ max m M {e R E R e R m,p(e R,e P ) 0} e L E L e L e U {e } R E R p(e R,e P ) 0} (3) Note p(e R,e P ) 0mple that ay formato ecoded by e T ca be authetcated by e R I [],DemedtetalgavetwocotructoforMRAcode baed o polyomal ad fte geometre, repectvely To cotruct multeder or multrecever authetcato by polyomal over fte feld, may reearcher have doe much work, for example, [7 9] There are other cotructo of multeder authetcato code that are gve[3 6] The cotructo of authetcato code combatoal deg t ature We kow that the polyomal over fte feld ca provde a better algebra tructure ad eay to cout I th paper, we cotruct oe multeder authetcato code from the polyomal over fte feld Some parameter ad the probablte of decepto of th code are alo computed We realze the geeralzato ad the applcato of the mlar dea ad method of the paper [7 9] Some Reult about Fte Feld Let F q be the fte feld wth q elemet, where q a power of a prme p ad F a feld cotag F q ; deote by F q be the ozero elemet et of F q Ithpaper,wewlluethe followg cocluo over fte feld Cocluo A geerator α of F q elemet of F q called a prmtve Cocluo Let α F q ; f ome polyomal cota α a ther root ad ther leadg coeffcet are over F q, the the polyomal havg leat degree amog all uch polyomal called a mmal polyomal over F q Cocluo 3 Let F q,thef a -dmeoal vector pace over F q Letαbeaprmtve elemet of F q ad g(x) the mmal polyomal about α over F q ;the dm g(x) ad, α, α,,α a ba of F Furthermore,, α, α,,α lear depedet, ad t equal to α, α,,α,α (α a prmtve elemet, α 0) alo lear depedet; moreover, α p,α p,,α p,α p alo lear depedet Cocluo 4 Coder (x +x + +x ) m (x ) m + (x ) m + + (x ) m,wherex F q,( )ad m a oegatve power of character p of F q Cocluo 5 Let m The,theumberofm matrce of rak m over F q q m(m )/ m+ (q ) More reult about fte feld ca be foud [0 ] 3 Cotructo Let the polyomal p j (x) a j x p +a j x p( ) + + a j x p ( j k), wherethecoeffceta l F q, ( l ), ad thee vector by the compoto of ther coeffcet are learly depedet The et of ource tate S F q ;theetofth tramtter ecodg rule E U {p (x ), p (x ),,p k (x ), x F q }( );theet of recever ecodg rule E R {p (α), p (α),,p k (α), where α a prmtve elemet of F q };theetofth tramtter tag T {t t F q }( );theetofrecever tag T{t t F q }

3 Joural of Appled Mathematc 3 Defe the ecodg map f :S E U T, f (, e U ) p (x )+ p (x )+ + k p k (x ), The decodg map f:s E R T, f(, e R )p (α) + p (α)+ + k p k (α) The ythezg map h : T T T T, h(t,t,,t )t +t + +t The code work a follow Aume that q larger tha, or equal to, the umber of the poble meage ad q 3 Key Dtrbuto The key dtrbuto ceter radomly geerate k(k )polyomal p (x), p (x),,p k (x), where p j (x) a j x p +a j x p( ) + +a j x p ( j k), ad make thee vector by compoed of ther coeffcet learly depedet, t equvalet to the colum vector a a a k a a a k of the matrx ( ) learly depedet He a a a k elect dtct ozero elemet x,x,,x F q aga ad make x ( ) ecret;theheedprvately p (x ), p (x ),,p k (x ) to the eder U ( )The key dtrbuto ceter alo radomly chooe a prmtve elemet α of F q atfyg x +x + +x αad ed p (α), p (α),,p k (α) to the recever R It follow that ( ( Deote x p x p x p x p x p x p x p ) x p x p ) a a a k t a a a k t )( )( ) a a a k k t a a a k a a a k A( ), a a a k (5) 3 Broadcat If the eder wat to ed a ource tate S to the recever R, the eder U calculate t f (, e U ) A (x )p (x )+ p (x )+ + k p k (x ), ad the ed A (x )t to the ythezer 33 Sythe After the ythezer receve t,t,,t,he calculate h(t,t,,t )t +t + +t ad the ed m (, t) to the recever R 34 Verfcato Whe the recever R receve m (, t),he calculate t g(,e R )A (α) p (α) + p (α)+ + k p k (α)iftt, he accept t; otherwe, he reject t Next, we wll how that the above cotructo a well defed multeder authetcato code wth arbtrato Lemma Let C (S,E P,T,f ); the the code a A-code, Proof () For ay e U E U, S,becaueE U {p (x ), p (x ),,p k (x ), x F q },ot p (x )+ p (x )+ + k p k (x ) T F q Coverely,forayt T,chooee U {p (x ), p (x ),,p k (x ), x F q },wherep j(x) a j x p + a j x p( ) + +a j x p ( j k),adlett f (, e U ) p (x )+ p (x )+ + k p k (x );tequvaletto a a a k (x p,x p,,x p )( a a a k )( )t a a a k k (4) X( x p x p x p x p x p x p x p ), x p x p t S( ), t( ) k The above lear equato equvalet to XAS t, becaue the colum vector of A are learly depedet, X equvalet to a Vadermode matrx, ad X vere; therefore, the above lear equato ha a uque oluto, o oly defed; that, f ( )aurjecto () If S aother ource tate atfyg p (x )+ p (x )+ + k p k (x ) p (x )+ p (x )+ + k p k (x ) t,adtequvaletto( )p (x )+( )p (x )+ + ( k k )p k (x )0,the (x p,x p,,x p ) a a a k a a a k )( )0 a a a k k k t t (6) (7)

4 4 Joural of Appled Mathematc Thu ( ( x p x p x p x p x p x p x p ) x p x p ) a a a k 0 a a a k 0 )( )( ) a a a k k k 0 (8) Smlar to (), we kow that the homogeeou lear equato XAS 0 ha a uque oluto; that, there oly zero oluto, o So, theuqueourcetatedetermed by e U ad t ;thu,c ( ) a A-code Lemma Let C(S,E R,T,g); the the code a A-code Proof () For ay S, e R E R, from the defto of e R, we aume that E R {p (α), p (α),,p k (α), whereα a prmtve elemet of F q }, g(, e R )p (α) + p (α) + + k p k (α) T F q ;otheotherhad,forayt T,chooe e R {p (α), p (α),,p k (α),whereα a prmtve elemet of F q }, g(, e R )p (α) + p (α) + + k p k (α) t; t equvalet to (α p,α p,,α p ) a a a k a a a k )( )t, a a a k a a a k k a a a k A( ); a a a k that, (α p,α p,,α p )A ( ) tfromcocluo k 3, wekowthat(α p,α p,,α p ) learly depedet ad the colum vector of A are alo learly depedet; therefore, the above lear equato ha uque oluto, o oly defed; that, g a urjecto (9) () If aother ource tate atfyg tg(,e R ),the (α p,α p,,α p )A( ) k (α p,α p,,α p )A( ); k (0) that, (α p,α p,,α p )A ( [ ] [ ] ) 0Smlar to (), wegetthatthehomogeeoulearequato [ k ] [ k ] (α p,α p,,α p )A(S S) 0ha a uque oluto; that, there oly zero oluto, o SS ;that, So, theuqueourcetatedetermedbye R ad t; thu, C(S,E R,T,g) a A-code At the ame tme, for ay vald m (, t),wehavekow that αx +x + +x,adtfollowthatt p (α) + p (α)+ + k p k (α) p (x +x + +x )+ p (x +x + +x )+ + k p k (x +x + +x )Wealohavekow that p j (x) a j x p +a j x p( ) + +a j x p ( j k);from Cocluo 4, (x +x + +x ) pm (x ) pm +(x ) pm + + (x ) pm,wherem a oegatve power of character p of F q, adwegetp j (x +x + +x )p j (x )+p j (x )+ +p j (x ); therefore, t p (α) + p (α)+ + k p k (α) (p (x )+ p (x )+ + k p k (x ))+(p (x )+ p (x )+ + k p k (x ))+ +(p (x )+ p (x )+ + k p k (x )) t +t + +t t, ad the recever R accept m From Lemma ad,wekowthatuchcotructoof multeder authetcato code reaoable ad there are eder th ytem Next, we compute the parameter of th code ad the maxmum probablty of ucce mperoato attack ad ubttuto attack by the group of eder Theorem 3 Some parameter of th cotructo are S q, E U [q k(k )/ k+ (q )] ( q ) [q k(k )/ k+ (q )](q ) ( ), T q( ), E R [q k(k )/ k+ (q )]φ(q ), T q Where φ(q ) the Euler fucto of q,trepreet the umber of prmtve elemet of F q here Proof For S q, T q,ad T q, thereult are traghtforward For E U,becaueE U {p (x ), p (x ),, p k (x ), x F q },wherep j(x) a j x p +a j x p( ) + + a j x p ( j k), ad thee vector by the compoto of ther coeffcet are learly depedet, t equvalet to a a a k a a a k the colum of A ( ) lear depedet a a a k

5 Joural of Appled Mathematc 5 From Cocluo 5, we ca coclude that the umber of A atfyg the codto q k(k )/ k+ (q )Othe otherhad,theumberofdtctozeroelemetx ( ) F q ( q )q,o E U [q k(k )/ k+ (q )](q ) ForE R, E R {p (α), p (α),,p k (α), whereα a prmtve elemet of F q }Forα, fromcocluo, a geerator of F q called a prmtve elemet of F q, F q q ; by the theory of the group, we kow that the umber of geerator of F q φ(q );that,theumberofα φ(q ) For p (x), p (x),,p k (x) Fromabove,wehavecofrmed that the umber of thee polyomal q k(k )/ k+ (q ); therefore, E R [q k(k )/ k+ (q )]φ(q ) Lemma 4 For ay m M,theumberofe R cotaed m φ(q ) Proof Let m (, t) M, e R {p (α), p (α),,p k (α), where α a prmtve elemet of F q } E R Ife R m, the p (α) + p (α) + + k p k (α) t (α p,α p,,α p )A ( k ) t For ay α, uppoe that there aother A uch that (α p,α p,,α p )A ( ) t, k the (α p,α p,,α p )(A A )( ) 0, becaue k α p,α p,,α p learly depedet, o (A A )( ) k 0, but( k ) arbtrarly; therefore, A A 0;that, A A,adtfollowthatA oly determed by α Therefore, a α E R,foraygve ad t, theumberof e R cotaed m φ(q ) Lemma 5 For ay m (, t) M ad m (,t ) Mwth,theumberofe R cotaed m ad m Proof Aume that e R {p (α), p (α),,p k (α), where α a prmtve elemet of F q } E R Ife R m ad m,thep (α) + p (α) + + k p k (α) t e R (α p,α p,,α p )A ( k ) t, p (α) + p (α) + + k p k (α) t (α p,α p,,α p )A ( ) tit k equvalet to (α p,α p,,α p )A ( )t t becaue k k, o t t ; otherwe, we aume that t t ad ce α p,α p,,α p ad the colum vector of A both are learly depedet, t force that ;tha cotradcto Therefore, we get [ (t t ) [[ (α p,α p,,α p )A ( ), ] [ k k ] ( ) ce t, t gve, (t t ) uque, by equato ( ), for ay gve, ad t, t,weobtathat(α p,α p,,α p )A oly determed; thu, the umber of e R cotaed m ad m Lemma 6 For ay fxed e U {p (x ), p (x ),,p k (x ), x F q }( )cotag a gve e L, the the umber of e R whch cdece wth e U φ(q ) Proof For ay fxed e U {p (x ), p (x ),,p k (x ), x F q }( )cotag a gve e L, we aume that p j (x )a j x p +a j x p( ) + +a j x p ( j k, ), e R {p (α), p (α),,p k (α),whereα a prmtve elemet of F q } From the defto of e R ad e U ad Cocluo 4, we ca coclude that e R cdece wth e U f ad oly f x +x + +x αforayα, cerak(,,,) Rak(,,,,α)<,otheequatox +x + +x α alway ha a oluto From the proof of Theorem 3, we kow the umber of e R whch cdet wth e U (e, the umber of all E R )[q k(k )/ k+ (q )]φ(q ) Lemma 7 For ay fxed e U {p (x ), p (x ),,p k (x ), x F q }( )cotag a gve e L ad m (, t),theumber of e R whch cdece wth e U ad cotaed m Proof For ay S, e R E R, we aume that e R {p (α), p (α),,p k (α), where α a prmtve elemet of F q } Smlar to Lemma 6, for ay fxed e U {p (x ), p (x ),,p k (x ), x F q }, ( )cotag agvee L,wehavekowthate R cdet wth e U f ad oly f Aga, wth e R m,wecaget x +x + +x α () p (α) + p (α) + + k p k (α) t () By () ad() adthepropertyofp j (x) ( j k), we have the followg cocluo: p ( x )+ p ( t (p ( x )+ + k p k ( x ),p ( x ) x ),,p k ( x ))

6 6 Joural of Appled Mathematc ) k φ(q ) [q k(k )/ k+ (q )] φ(q ) q k(k )/ k+ (q ) (5) t ( p (x ), t ( ( x ),( p (x ),, x ),,( p k (x ))( ) k x ))A( ) k By Lemma 4 ad 5,weget m P S max {max m M φ(q ) m M By Lemma 6 ad 7,weget {e R E R e R m,m } {e } R E R e R m} (6) t [( p (x ), (( )0, k x ),( p (x ),, x ),,( p k (x )) x ))A] (3) becaue ay gve Smlar to the proof of Lemma 4, we ca get ( p (x ), p (x ),, p k(x )) (( x ),( x ),,( x ))A0;that,(( x ), ( x ),,( x ))A ( p (x ), p (x ),, p k(x )),butp (x ), p (x ),,p k (x ) ad x ( ) alo are fxed; thu, α ad Aare oly determed, o the umber of e R whch cdet wth e U ad cotaed m Theorem 8 I the cotructed multeder authetcato code, f the eder ecodg rule ad the recever decodg rule are choe accordg to a uform probablty dtrbuto,thethelargetprobablteofuccefordfferettype of decepto, repectvely, are P U (L) P I q k(k )/ k+ (q ), P S φ(q ), [q k(k )/ k+ (q )]φ(q ) Proof By Theorem 3 ad Lemma 4,weget P I max { {e R E R e R m} m M E } R (4) P U (L) max max{ max m M {e R E R e R m,p(e R,e P ) 0} e L E L e L e U {e } R E R p(e R,e P ) 0} [q k(k )/ k+ (q )]φ(q ) (7) Ackowledgmet Th paper upported by the NSF of Cha (67906) ad the Fudametal Reearch of the Cetral Uverte of Cha Cvl Avato Uverty of Scece pecal (ZXH0k003) Referece [] E N Glbert, F J MacWllam, ad N J A Sloae, Code whch detect decepto, The Bell Sytem Techcal Joural,vol 53, pp , 974 []YDemedt,YFrakel,adMYug, Mult-recever/multeder etwork ecurty: effcet authetcated multcat/ feedback, Proceedg of the the th Aual Coferece of the IEEE Computer ad Commucato Socete (Ifocom 9),pp , May 99 [3] K Mart ad R Safav-Na, Multeder authetcato cheme wth ucodtoal ecurty, Iformato ad Commucato Securty, vol334oflecture Note Computer Scece, pp 30 43, Sprger, Berl, Germay, 997 [4] W Ma ad X Wag, Several ew cotructo of mult tramtter authetcato code, Acta Electroca Sca, vol 8,o4,pp7 9,000 [5] G J Smmo, Meage authetcato wth arbtrato of tramtter/recever dpute, Advace Cryptology EUROCRYPT 87,WorkhopotheTheoryadApplcatoofof Cryptographc Techque,vol304ofLectureNoteComputer Scece, pp 5 65, Sprger, 988 [6] S Cheg ad L Chag, Two cotructo of mult-eder authetcato code wth arbtrato baed lear code to be publhed, WSEAS Traacto o Mathematc, vol, o, 0

7 Joural of Appled Mathematc 7 [7] R Safav-Na ad H Wag, New reult o multrecever authetcato code, Advace Cryptology EUROCRYPT 98 (Epoo),vol403ofLectureNote Comput Sc,pp57 54,Sprger,Berl,Germay,998 [8] R Apara ad B B Amberker, Mult-eder mult-recever authetcato for dyamc ecure group commucato, Iteratoal Joural of Computer Scece ad Network Securty, vol7,o0,pp47 63,007 [9] R Safav-Na ad H Wag, Boud ad cotructo for multrecever authetcato code, Advace cryptology ASIACRYPT 98 (Bejg), vol54oflecture Note Comput Sc, pp 4 56, Sprger, Berl, Germay, 998 [0] S She ad L Che, Iformato ad Codg Theory, Scece pre Cha, 00 [] J J Rotma, Advaced Moder Algebra, Hgh Educato Pre Cha, 004 [] Z Wa, Geometry of Clacal Group over Fte Feld, Scece Pre Bejg, New York, NY, USA, 00

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