The Properties of Orthomorphisms on the Galois Field

Transcription

1 Research Joural of Applied Scieces, Egieerig ad Techology 5(5): , 013 ISSN: ; e-issn: Maxwell Scieific Orgaizaio, 013 Submied: Augus 11, 01 Acceped: Sepember 03, 01 Published: February 11, 013 The Properies of Orhomorphisms o he Galois Field 1 Haiqig Ha, 1 Xiaofag Xu ad Siru Zhu 1 Deparme of Mahemaics ad Physics, Hubei Polyechic Uiversiy, Huagshi , Hubei, Chia Deparme of he Basics, AFRA, Wuha , Hubei, Chia Absrac: The orhomorphism o he Galois field is a kid of permuaios ha is he mos widely used i crosscuig issue, he orhomorphic polyomials over he fiie field is a effecive mehod o sudy i, his sudy has obaied he coefficies relaioship of he orhomorphisms over he Galois field by algebraic mehods. I addiio, his sudy have udersood he maximal subgroup srucure ad couig i he Abelia group. I is help o i-deph sudy he applicaio ad he aure of he orhomorphism qua he heoreical suppor. Keywords: Orhomorphic polyomials, orhomorphisms, he domai of algebraic ieger, he fiie field, he maximal subgroup INTRODUCTION Wih he populariy of he compuer ad he Iere, he gae of he ework is opeig a he iformaio age. Compuer ework ad iformaio securiy become more ad more criical, he crypography is oe of he key echologies i iformaio securiy. The permuaio plays a impora role i he cipher desig; a well permuaio ca be used o desig he cipher, he digial sigaure or auheicaio algorihms. I he cipher desig, he cryposysem based o mahemaical hard problems has bee usually divided io some cipher compoes o desig, which hese pars iclude he liear ad oliear permuaio. Ad he liear permuaio is kow as he P-permuaio ad he oliear permuaio is called S-box (Haiqig ad Huaguo, 010). I is proved ha he orhomorphisms have a good crypographic propery i Lohrop (1995): he complee balace. The orhomorphisms have bee researched widely from he perspecive of mahemaics ad crypography ad are also used i he desig of he cipher, digial sigaure ad auheicaio algorihms. The cryposysem SMS4 is commercial block cipher i Chia whose roud fucio is desiged i he oliear orhomorphisms (Shuwag e al., 008). I addiio o he commercial cipher, here are oher relaed applicaios o he orhomorphisms, icludig he research ad developme produc DSD (Lohrop, 1995) ehaced securiy (Qibi ad Cheg, 1996) ad he cosrucio of Boolea fucios i cipher (Dawu e al., 1999). I order o explore orhomorphisms o crypographic properies ad applicaios, people have sudied he orhomorphisms from differe perspecives: The Lai square agle: Lai square is used o sudy he orhomorphism over he Galois Field GF( ) ad which are obaied by he orhomorphic Lai square rasversal i Baoyua e al. (1997); i 006, i is poied ou ha here is he oe o oe correspodig relaios bewee he orhomorphisms ad he orhomorphic Lai square rasversal (Shuwag e al., 008), he couig boud of orhomorphisms have bee obaied by he orhomorphic Lai square i Qi e al. (008). The permuaio polyomials agle: The permuaio polyomials have bee sudied firs he orhomorphisms over he Galois field i Zhihui (00), icludig he disribuio of permuaio polyomials over GF( ), GF( 3 ), GF( 4 ) ad so o. The geeral coclusios have bee obaied ha a cerai class of permuaio polyomials do o exis i Yua ad Huaguo (007), he degree disribuio of he orhomorphic permuaio polyomials o GF( 4 ) become clear hrough he classificaio mehod ad he whole he orhomorphic permuaio polyomials o GF( 4 ) are geeraed i Yua ad Huaguo (007). The boolea fucio agle: Boolea fucios have is ow advaages i he cosrucio ad research o he permuaio, (Degguo ad Zhehua, 1996) have cosruced some orhomorphisms over he Galois field usig mulioupu Boolea fucio, YANG Yixia ad Gu Dawu ec., have also sudied he orhomorphisms ad obaied beer resuls usig he Boolea Correspodig Auhor: Xiaofag Xu, Deparme of Mahemaics ad Physics, Hubei Polyechic Uiversiy, Huagshi , Hubei, Chia 1853

2 Res. J. Appl. Sci. Eg. Techol., 5(5): , 013 fucios as he maor weapos. I is more coveie o cosruc he orhomorphisms from low order o high order usig he Boolea permuaio mehod (Degguo ad Zhehua, 1996; Degguo ad Zhehua, 1998; ad Yuse e al., 1999). The loop srucure agle of he permuaio: The Mahemaical kowledge kows us ha ay permuaio ca be wrie i he produc of circulaed facors which does o iersec. The circulaed facors are kow as he circle srucure. The circle srucure has bee used o sudy he circle srucure characerisics of orhomorphisms (Dawu ad Guozhe, 1997) ad he maximum liear orhomorphisms (Zhihui, 00; Ahua, 003). The agles of he vecor represeaio ad he permuaio marix: Dr. L. Miehal ad Xiao Guozhe have sudies he orhomorphisms from he agles of he vecor represeaio ad he permuaio marix. These differe mehods have heir ow advaages whe he orhomorphisms are sudied. If he example ad eumeraio of orhomorphisms eed o be give, he i is more effecive ad coveie ha we will uilize geerally Lai square o carry o. If i eeds o deermie ha he orhomorphism is liear or oliear, he we will use he muli-oupu Boolea permuaio or permuaio polyomial o udge i. Whe he maximum liear orhomorphisms are sudied, we will use he circle srucure of he permuaio. Afer he aalysis of he domesic ad ieraioal saus of orhomorphisms, i is clear ha he orhomorphisms are divided io he liear ad oliear from srucural poi of view. We have maily sudied he orhomorphisms issues icludig he srucure, eumeraio or couig upper ad lower bouds. This sudy will sudy he relaios of he orhomorphic permuaio polyomial coefficies ad he applicaios of he orhomorphisms i he maximal subgroups of he Galois field. PRELIMINARIES Le F = {0, 1} be a biary fiie field. F or GF( ) = The -degree exesio field of F, i also ca be cosidered ha he -dimesio liear space o F. Geerally, le F q be he fiie field wih a arbirary prime umber characerisic p, amely q = p k. Similarly, F q = The exesio field of F q wih degree. Le S be a biecio o GF( ), ha is saisfied: xy, GF( ) if x y he S(x) S(y) For he arbirary cosa a, x is he exisece ad uiqueess, so ha S(x) = a. We said S a permuaio Defiiio 1: Le S be a permuaio o GF ( ), l be he ideiy rasformaio ( I( x) x, x GF( )). S is called a orhomorphism, if S I is sill a permuaio o GF( ) ( is he addiio operaio of GF( )). Furher, S = A liear orhomorphism o GF( ), if S(X + Y) = S(X) + S(Y) se up XY, GF( ). By he defiiio 1, we have simply pu he Galois field GF( ) as a addiive group whe he orhomorphisms o he Galois field GF( ) are sudied. I has preseed he exisece heorem of he orhomorphisms i Hall ad Paige (1957): he ecessary ad sufficie codiios ha he orhomorphism exiss i a fiie Abelia group G are ha he Sylow- subgroup of he group G is o cyclic group or is rivial. I is idicaed ha a permuaio is he orhomorphism if ad oly if he sum of he permuaio ad he ideiy rasformaio is sill a permuaio by Defiiio 1. The orhomorphisms is a special kid of he permuaio ad o all he permuaios are he orhomorphisms. Example 1: Le S be he permuaio o GF( ) ad S saisfies: (0,0) (0,1),(0,1) (1,0) (1,0) (0,0),(1,1) (1,1) he S is a orhomorphism. Bu he ideiy rasformaio o GF( ) is o a orhomorphism. Defiiio : Le G be a fiie group, S be a biecio o G. If he mappig S : x xs( x) is sill he permuaio o G, he S is called he complee mappig. (xs(x) represes he muliplicaio bewee x ad S(x) i G). Defiiio 3: Le G be a fiie group, S be a biecio 1 o G. If he mappig S : x x S( x) is sill he permuaio o G, he S is called he orhogoal mappig. (x -1 S (x) = The muliplicaio bewee he iverse of x ad S(x) i G). Defiiio 4: Le S be a permuaio o GF( ), if V is a arbirary maximal subgroup i GF( ) (or a maximal subspace), ad he compleme se V GF( ) \ V saisfies: SV ( ) V SV ( ) V The S is kow as he perfecly balace mappig. By he above defiiios, he orhomorphism is he complee mappig, he orhogoal mappig ad he

3 Res. J. Appl. Sci. Eg. Techol., 5(5): , 013 perfecly balace mappig. The orhomorphisms have bee well applied i pracice because of is ihere crypographic properies. We have firs give he applicaio of he orhomorphisms i he sudy of he maximal subgroups srucure o he Galois Field GF( ). RESULTS The addiio operaio i he Galois field is deoed by, he Galois field is a group for he addiio operaio ad you ca sudy he maximal subgroups. A he same ime, he Galois field ca also be see as a -dimesioal vecor space, you ca sudy he subspace. We have kow ha here is he oe o oe correspodig bewee he maximal subgroups ad from he ( - 1) -dimesio subspace limied aure of he domai ad is maximal subgroups o GF( ) correspod o he dimesio of subspace. I is easy o obai he followig resuls ha we have researched he srucure of maximal subgroups usig he orhomorphism o GF( ). Theorem 1: Le α 1, α,, α be a arbirary basis of he Galois field GF( ) o F, akig ou arbirarily he ( - 1) vecors is spaig a subspace M o F, he M is a maximal subgroup o he addiio operaio o GF( ) ad he all maximal subgroups o addiio operaio o GF( ) ca be expressed as: am { am m M}, a GF( )\{0} so here are ( - 1) maximal subgroups o GF( ). Proof: Due o he Galois field GF( ) is a fiie Abelia group for he addiio operaio, he order of he maximal subgroups o GF( ) is -1 because of he cycle decomposiio o he fiie Abelia group. So he vecor space M is a maximal subgroup o GF( ). From he algebra, x GF( ) \{0}, N {0, x} is a miimal group, There is a group isomorphism M GF( ) / N. Ad here are he { - 1} groups N = {0, x}, hece he umber of he differe maximal subgroups are ( - 1) o GF( ). Nex, we will give he evidece of he all maximal subgroups o GF( ) ca be expressed as: am { am m M}, a GF( )\{0} beig a orhomorphism o GF( ), he complee balace ells us M am because he half elemes of f ( M) { f( c) cm} am are i M ad he oher half are i M GF( ) \ M. A same reaso, if, 1 /0 ad a b, he ab M M am bm ad am { am m M} us has he ( - 1) o-zero elemes. I goes o show whe we have ake over all o-zero elemes a i GF( ), am { am mm} is ergodic o he maximal subgroups o GF( ). The orhomorphisms have a good effec o he sudy of algebraic srucure as a special kid of mappig by he heorem 1. The orhomorphisms ca also be used o he block desig, saisical aalysis, chael codig ad he orhogoal Lai squares ad so o. From he agle of he orhomorphic permuaio polyomials o research he orhomorphisms i Zhihui (00) ad Yua ad Huaguo (007), i is ells us ha ca sudy he orhomorphisms srucure by he permuaio polyomials. Le F GF( ), f F[ X] be he polyomial, he f ( c) F for c F. The polyomial f(x) is a rasformaio o GF( ). If f(x) is he oe o oe rasformaio, he f(x) is a permuaio o GF( ), f(x) is called he permuaio polyomial o GF( ). I has he followig facs: ha f(x) is he permuaio polyomial o GF( ) is equivale o oe of he followig codiios: f : c f( c) is iecive o GF( ) f is a subecive o GF( ) a F he equaio f(x) = a has soluios a F he equaio f(x) = a has a uique soluio Defiiio 5: Le f F [ ] X ad f(x) = a 0 + a 1 X + + a X, if a 0 he f(x) is said he polyomial of degree, deoed deg (f(x)) =. Le S be a permuaio o GF( ) ad S: c S( c), cgf( ), he correspodig permuaio polyomial ca be derived from he ierpolaio formula: f X S c x c = q 1 ( ) ( )(1 ( ) ) cgf( ) cgf( ) a x [ Sc ( ) ] a c, where q ac I is easy o udersad by he defiiio of he group, if a GF( ) \{0} he am { am mm} is a subgroup o GF( ). We ca udge ha am { am m M} is a maximal subgroup by he order. For a GF( ) \{0}, if a 1 he am { am mm} M ; if a 1 he f (x) = ax is 1855 f(x) is simplified o he degree deg (f(x)) q - 1. I idicaed ha he arbirary permuaio ca be used he polyomial wih he degree is o more ha (q - 1) o represe. I is easy o kow ha if f(x) is he permuaio polyomial o GF( ) he GF( ) { f ( c) c GF( )}, for GF( ) saisfies GF( ) { f ( c) c GF( )}. I shows also ha

4 Res. J. Appl. Sci. Eg. Techol., 5(5): , 013 f(x) is he permuaio o GF( ) if ad oly if f( X) is he permuaio o GF( ). Le q 1 f ( X) a0 a1x aq 1X be he permuaio o GF( ) if ad oly if f(x) + a 0 = a 1 X + + a q-1 X q-1 is he permuaio o GF( ). So we ca assume he cosa is 0 overall. If f(x) is he orhomorphic permuaio polyomial, he he regular of coefficies is as follow: ( q = e 1) (mod 4) q1 = q1 e 1 1 ( e 1) (mod4) = 1 (mod 4) (where 4 e q-1-1) Theorem : Le f(x) = a 0 + a 1 X + + a q-1 X q-1 be a orhomorphic permuaio polyomial o he Galois field GF( ), he he coefficies have he followig relaioships: i i q1, i aa 0 ad a i aa 0 q i i i q1, i Proof: Le q =, deoed GF( ) by F q. A Galois field wih he characer ca be isomorphism o a residue class rig of he algebraic ieger rig (Lag, 1994), amely i exiss a algebraic ieger rig E ad is ideal E, such ha Fq E/E. The isomorphic field is regarded as he same field, simply remember F q = E/E. I is very easy o ge he aural rig homomorphism : EFq E/E ad ker η = E. The homomorphism ca be lif o : E[ x] Fq [ x] ad he ideermiae me wih ( x) x, he homomorphism is resriced E o E. Le g be he muliplicaio geeraor of he Galois field, he origial image of g is e i E, we have he followig relaioship: () e g, 1 1 ( q q e ) g 1, ( i i e ) g 1,1iq 1. There are exacly he q cosses of he algebraic ieger rig E uder is ideal E, he cosse decomposiio of E is as follows E = {0+E}. Ad q1 q1 ( e ) g 1 (1), so hey are belog o he same cosse bewee 1 ad e q-1 q 1. Hece, ( e 1) E, ha is e q-1 = 1 (mod ). Similarly, e i = 0 (mod ), 1 i<q - 1 (if e i = 1 (mod ), 1 i<q - 1, we ca ge ( i i e ) g 1(1i q 1), which is he coradicio o he muliplicaio geeraor g) Nex, we have proved ha e q-1 = 1 (mod 4), if e q-1 = 0 (mod 4) 1, he: e 1 [(1 e ( )] q1 q1 = e 1 e 1 e C e e C e e q1 q1 q1 q1 1 q11 i q1i i q1 q1 [ ] i q 1 q1 q1 e 1 = e ( q 1) e (mod 4) 1856 Owig also o 1, i is idicaed ha e q ad e are i he same cosse ad he represeaive eleme i he cosse q ca be selec radomly, so we ca selec e e as he represeaive eleme o saisfy: q1 ( e) 1(mod4), wihou loss of geeraliy: e q-1 = 1 (mod 4) (1) i Le S {0, e 0 iq1} E, i is obvious o { ( x) x S} F. The power sum of he elemes has q he followig relaioship: 3( mod 4) ( q 1) x () xs 0(mod 4) ( q1) Because of: xs q1 i q x ( e) 1 e ( e) ( e) i0 if (q - 1), he he above equaio is: q1 ( e ) 1 e 1 q x 1 e ( e ) ( e ) q1 13(mod4) xs if (q - 1), he he above equaio is: xs q1 i x ( e ) i0 q1 ( e ) 1 e 1 Due o e 1(mod ), 1 q - 1, if is he odd, sice 4 (e q-1-1) ad (e ) q-1-1 = (e q-1 ) - 1, he 4 ((e q-1 ) q1-1) ha is ( e ) 1 4 [ ]; If is he eve, he e 1 (e (q-1) - 1) ((e (q-1) - 1)), i addiio o 4 (e q-1-1), so (e q-1-1), ha is (e q-1 +1). Hece 8 (e (q-1) - 1), (e (q-1) - 1), ((e q-1 ) - 1), Tha is 8 ((e q-1 ) - 1). Bu q1 e 1(mod ), we have he equaio ( e ) 1 4 [ ]. e 1 Assume: F ( x) b bxb x b x E[ x] q q1

5 Res. J. Appl. Sci. Eg. Techol., 5(5): , 013 ad, f ( x) a a xa x a x F [ x] ( F( x)) f( x) q q1 Because f(x) is a permuaio, so 0, 0,,, = 4 (by he defiiio of he cosse) = 0(mod 4) (by he Eq. () O he oher had: q1 q1 i i i F () x ( bx i ) ( bi x ) bb i x xs xs i0 xs i0 xs i q1 i i bix bb i x i0 xs i xs By () ad i (q - 1), he above equaio saisfies: So, q1 i F x bx i bb i xs xs i0 i q1, i ( ) ( ) (mod4) 0 bb i (mod4) i q1, i From he aure of he cogruece: 0 bb (mod) ( bb ) E ker ker So, i i iq1, i iq1, i aa ( bb ) 0 i i iq1, i iq1, i If f(x) is a orhomorphism, so is f(x) + x, he is coefficie mus also saisfy he above propery. The secod equaio i he heorem is he coefficies relaioship of f(x) + x, i mus be esablished. The heorem has bee proved. For he orhomorphic permuaio polyomials, he polyomials of degree (q - ) mus o exis (Daqig, 1986). I is ecessary o a q- = 0, he wo formulas i he heorem are equivale. We ca also ge more iformaio o he relaioship o he coefficies i he orhomorphic permuaio polyomial, which eeds furher sudy. The proof of heorem has oly used he map ( F( x)) f( x) ad he square relaioships bewee he origial image ad he image. We ca furher q research he cubed, he fourh power relaioships bewee he origial image ad he image ad so o. We ca ge more relaioship o he coefficies of he orhomorphic permuaio polyomial. These relaed equaios reveal ha he coefficies of he orhomorphic permuaio polyomials exis i he cosrai relaios. If his research is clear, he i ca help us o obai he couig formula of he orhomorphisms by solvig he equaio sysem. CONCLUSION This sudy has maily sudied he properies of he orhomorphisms over he Galois field GF( ). Theorem 1 ells us, he orhomorphisms are a special kid of mappigs, such as isomorphism ad homomorphism, which has a uexpeced effec o sudy he algebraic srucure iself. I is maily o sudy he coefficies relaioships of he orhomorphisms i Theorem. The coefficies of he orhomorphisms mus have he resricive relaioships as a special kid of permuaio polyomial. I is a impora way o sudy couig formula of he orhomorphisms ha such cosrai relaioships are search for. All i all, if we wa o ge more coclusios o he orhomorphisms o he Galois field, heir applicaios ad srucure eed furher sudy. ACKNOWLEDGMENT This sudy is suppored by he aioal aural sciece foudaio of Hubei Polyechic Uiversiy (11YJZ10R) ( ). REFERENCES Ahua, Z., 003. The research ad cosrucio of orhomorphic permuaio. MA Thesis, Naioal Uiversiy of Defese Techology, Chagsha. Baoyua, K., T. Jiabo ad W. Yumi, Two resuls abou orhomorphic permuaio ad orhomorphic lai square. J. XiDia Uiv., 4: Daqig, W., O a problem of iederreier ad robiso abou fiie fields. J. Asrual. Mah. Soc. Ser A, 41: Dawu, G. ad X. Guozhe, A improved mehod of cosrucig oliear orhomorphic permuaio wih he aalysis of is propery. J. XiDia Uiv., 4: Dawu, G., L. Jihog ad X. Guozhe, Cosrucio of crypographic fucios based o orhomorphic permuaio. J. XiDia Uiv., 6: Degguo, F. ad L. Zhehua, O he cosrucio of he orhomorphic permuaio. Secure Commu., :

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