Algebraic Properties of Modular Addition Modulo a Power of Two

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1 Agebac Popees of Moda Addo Modo a Powe of Two S M Dehav Aeza Rahmpo Facy of Mahemaca ad Compe Sceces Khaazm Uvesy Teha Ia sd_dehavsm@hac Facy of Sceces Qom Uvesy Qom Ia aahmpo@sqomac Absac; Moda addo modo a powe of wo s oe of he mos appcabe opeaos symmec cypogaphy; heefoe vesgag cypogaphc popees of hs opeao has a sgfca oe desg ad aayss of symmec cphes Agebac popees of moda addo modo a powe of wo have bee sded fo wo opeads by Baee fse 05 Aso he ahos of hs pape have sded hs opeao some speca cases befoe I hs pape ag advaage of pevos eseaches hs aea we geeaze agebac popees of hs opeao fo moe ha wo smmads Moe pecsey we deeme he agebac degee of he compoe Booea fcos of moda addo of abay mbe of smmads modo a powe of wo as a vecoa Booea fco aog wh he mbe of ems ad vaabes hese compoe fcos As a es agebac degees of he compoe Booea fcos of Geeazed Psedo-Hadamad Tasfoms ae comped Keywods; Moda addo modo a powe of wo; Booea fco; Agebac Noma Fom; Agebac degee; Psedo-Hadamad Tasfom

2 I INTRODUCTION Moda addo modo s oe of he mos sed opeaos symmec cypogaphy Hee s a posve ege whch s say eqa o he sze of pca pocessos e o 64 Fo sace moda addo s sed Beooh [] ad RC4 [] seam cphes ad IDEA [3] RC6 [4] Twofsh [5] ad MARS [6] boc cphes I [7] agebac popees of hs opeao fo wo opeads have bee sded ad he ANF (Agebac Noma Fom) of s compoe Booea fcos ae deemed; aso we examed some agebac popees of moda addo modo a powe of wo speca cases [8] We oe ha fo a o of appcaos desg ad aayss of symmec cphes s mpoa ad sef o ow he agebac degee of he compoe fcos ad he mbe of ems ad vaabes hese fcos fo moda addo modo a powe of wo I [8] we obaed hese paamees fo moda addo modo a powe of wo wh a powe of wo smmads ad we poposed a agohm fo compg agebac degees of moda addo compoe Booea fcos geea case I hs pape we pese a expc foma fo agebac degees of he compoe Booea fcos of moda addo modo a powe of wo wh abay mbe of smmads aog wh he mbe of ems ad vaabes hese fcos The as a coseqece we deeme he agebac degee of he compoe Booea fcos of geeazed Psedo-Hadamad Tasfomaos; hese ypes of asfomaos ae sed some symmec cphes e Twofsh I Seco II we pese pemay defos ad heoems; Seco III we fd he agebac degee of compoe Booea fcos of moda addo some speca cases Seco IV peses a agohm fo fdg hese degees geea case Seco V s dedcaed o o ma ess fo fdg agebac degee of he compoe fcos of moda addo modo a powe of wo geea case Seco VI sdes he agebac degee of compoe Booea fcos of geeazed PHT s ad Seco VII s he cocso Le F II PRILIMINARY DEFINITIONS AND THEOREMS be he fed of ode Caesa podc of copes of Accodg o he defo of eges modo A paa ode as foows: o F F hee s a oe-o-oe coespodece bewee ( x x : F ca be defed as foows: 0 Z ) ( x) x 0 F ca be cosdeed as a veco space F ad Z g of I he above epeseao f he x s defed as x a x a 0 x ( x x0) ( 0) x x 0 0 x Each fco f : F F s caed a Booea fco Sppose ha f s a Booea fco; f ca be epeseed by s ANF as foows:

3 hee he coeffces ae deemed as f ( x) h x h F ; Z h h f (x) x () Agebac degee of a Booea fco s defed as he mbe of vaabes he oges em of s ANF o eqvaey he maxmm Hammg wegh wh m whee each w() s caed a vecoa Booea fco Obvosy f ( f m f0) f of ozeo s Each fco s eqvae o m f : F F 0 m s a Booea fco f : F F I s we-ow ha moda addo modo a powe of wo ca be epeseed Booea fom as foows: Sppose ha x x ) y y ) ad ( x0 ( y0 x y mod If ( 0 ) he x y c c x y c c x y x c y c 0 () Theoem [7]: Sppose ha he ANF of a Booea fco s x f : F F wh Z ; he ANF of he fco f ( x y) s of he fom f ( x y) c0 x ( c) y c ; (3) hee he sbaco s doe Z Theoem [7]: Sppose ha f ( y y y ) s of he fom y y y F ad f s defed as Theoem ; he ANF of f ( y y y ) y y y (4) 0 Regadg Theoem he agebac degee of he compoe Booea fcos of moda addo modo ca be deemed as foows: 0 Fo ad f ( x) x we ca acqe he agebac degees of he afoemeoed compoe fcos va eao (4) ad he fac ha a he ems he ghmos of hs eao ae dffee So f we fd he maxmm of he vaes w ( ) a ses } we ca fd he agebac degee of hese compoe Booea fcos {

4 III THE CASE The ess of hs seco have bee peseed [8]; b we evew hem hee becase o ew heoems ae poved wh he ad of he oaos ad coceps Le ad be hee oegave eges Accodg o he dscssos of Seco II we cosde he eqao (5) whee s ae Z ad 0 Now we asfom s o vecos wh maxmm Hammg wegh sm whch we ca hem opmm soos F Regadg Theoem ad eao (4) we see fo soos Theoem 3: Wh he above oaos fo Hammg wegh s eqa o Hammg wegh sm of he soos wh maxmm ad f w( ) ( ) (6) w( ) ( )( ) ; (7) he maxmm Hammg wegh sm s eqa o Poof: Cosde wo cases: Case ) coespods o he case : Cosde wo caegoes: Caegoy hese soos ae opmm soos coespods o he case ad caegoy A fs we pese a soo fo each caegoy ad he we pove ha Caegoy : 0 (8) ad fo Caegoy : ; (9) Hammg wegh sm of hese soos ae as (6) ad (7); ad ceay hese soos sasfy Eqao (5) Cosde a abe (Tabe A) fo each caegoy: A has coms The coms of A coespod o epeseao of s hs abe by F Each com s odeed boom-p fom dces p o We deoe ees A( ) A ( ) ad A () deoe he com ad he ow of A especvey Aso by oe (A) we efe o mbe of ' ' ees A If A ( ) ' ' he we defe he sgfcace of A( ) as ; ad he sgfcace

5 of A( ) s defed as zeo ohewse Fay ees A Now cosde he wo caegoes: sm( A) s defed as he sm of he sgfcaces of a a ) : Sppose ha ) ' ' soo 0 Yo ca see Tabe A fge Sce fo ees A have eas possbe sgfcaces so he peseed soo s opmm : I hs case Tabe A s modfed o fge Sppose ha a Tabe B coespods o aohe ad oe( B) oe( A) We defe epace he coespodg ey We defe B { Y Y Y Y } (0) sch a way ha each ow B() wh a A he same mae Yo ca see he Tabes Assme ha hee exs m Sce a ' ' We have ' ' sm( A ees B '0' ees A ( s ) fo s s s fo evey ozeo ey ey f he coespodg ey A ad B fge 3 ad m ' ' A() s a A() ' ' we ey ees B ( ) fo m ) sm( B ) () have dffee sgfcaces (we oe ha hee ae ees oy B ( ) ) afe smpfcao of he ef-had sde we ca om eqa ems wo sdes of Eqao () Now sm( A ) sm( B ) ad oe( B ) oe( A ) We oe ha oe of he s ad s ae eqa ad so wo sdes of () have dffee ems Sppose ha s he eas powe who oss of geeay We have s ss s s m s whch s a coadco; so (7) s opmm s s Case ) : We pese he wo foowg soos: Fo we have ad fo : 0 Fge

6 Sce a ees A have eas possbe sgfcaces so he peseed soo s opmm Yo ca see Tabe A fge 4 ' ' IV AN ALGORITHM FOR OPTIMUM SOLUTIONS I hs seco e [8] we pese a mehod fo effcey sovg he pobem of fdg he agebac degee of compoe Booea fcos of moda addo modo a powe of wo wh abay mbe of smmads Regadg he oaos ad assmpos of Seco IV we sa fom A() ad p ' ' ees he abe A ees A ( ) A() A() A() ad coe hs pocede seg a exa ey A yeds fom he pemae ow we deee sm( A) We e ohe ees be ' ' '0' ees fom A Now f ' ' sm( A) he covesey sag sm( A) ; ohewse we deee ' ' ees fom he owe ows ad coe hs pocede smay We oe ha hs soo s sch ha each ow ohe ha he as ow a mos oe of he ees s '0' sce ohewse hese wo ees ca be epaced by oe ey fom he above ow Now we cam ha hs soo has maxmm Hammg wegh sm Assme ha hee exss aohe soo wh geae Hammg wegh sm Aga eae a Tabe B o hs soo As befoe we cosc A ' ' ad B Sppose ha hee exs m Fom eqao (5) we have: ' ' ees A s ) ( m m ad ' ' ees A ) ( () Sma o he Caegoy powe; he we have of Case Theoem 33 who oss of geeay sppose ha s he eas m (3) whch s a coadco V THE MAIN THEOREMS I Seco IV we peseed a mehod fo effcey sovg he pobem of fdg he agebac degee of compoe Booea fcos of moda addo modo a powe of wo geea case I hs seco we oba he expc foma of hese degees Le ad be hee oegave eges Accodg o he dscssos of Seco II we cosde he eqao Fge

7 whee s ae Z (4) ad 0 As we ow hs eqao has oegave ege soos whch s eqa o he mbe of ems he ANF of he -h compoe fco of moda addo modo a powe of wo So we have: Fac 5: The mbe of ems he ANF of he -h compoe fco of moda addo modo wh smmads s eqa o Aso s easy o vefy ha he mbe of vaabes he ANF of he -h compoe fco s eqa o ( ) : Fac 5: The mbe of vaabes he ANF of he -h compoe fco of moda addo modo wh opeads s eqa o ( ) Accodg o he agohm poposed Seco IV we oba he opmm soos fo he eqao (5) geea case These soos ae eqa o agebac degees of he compoe Booea fcos of moda addo modo becase we have Noe 53: I eqao (5) f f ( y y y ) ( y y he some of y ) 0 y y y 's ae zeo ad so (5) s modfed o a eqao wh A ad B Fge 3

8 Case ad Case Fge 4 Theoem 54: The opmm soos fo (5) have Hammg wegh fo 0 ; hee w( ) og ( ) Poof: Accodg o he peseed agohm we cosc Tabe A Fge 5 Sce he sgfcace of each ey A () s ode o fd we have Smpfyg (6) we ge ad fo we have ( ) ( ) (6) og ( ) ( ) ( ) ( ) whch eads o Fge 5

9 Addg A ( ) '' o Tabe A eads o sm( A) ; so opmm Hammg wegh s obaed by sbacg he Hammg wegh of he exa amo fom w ( sm( A)) The exa amo s eqa o ( ( ) ) (7) Sbsg he vaes of (7) yeds If we deoe ad of wo ( Theoem 5) by og ( ) we have so we have he foowg es: coespodg o -h compoe fco of moda addo modo a powe ad w( he by some easy cacaos we ca vefy ha fo ) w( ) ; Res 55: The agebac degees of compoe Booea fcos of moda addo modo a powe of wo wh smmads fom a ahmec pogesso wh commo dffeece fo ) og ( VI GENERALIZED PSEUDO-HADAMARD TRANSFORMATIONS Psedo-Hadamad Tasfoms (PHT s) ae sed symmec cypogaphy Fo exampe boc cphe Twofsh PHT s sed hs fom: Le be he p wods ad x y be coespodg op wods The PHT s defed as x y x x y y x y Agebac degee of he secod eao was obaed pevos secos Now we wa o oba he agebac degee of he fs eao Accodg o eao (3) f (x y) (x y) Agebac degees of compoe Booea fcos of 0 (x) y x y ae deved by sbsg 0 he above foma Becase he eas sgfca b of x s zeo ode o fd hese degees ms be eve Maxmm Hammg wegh of opmm soo s acheved va ; 0 s ee I he seqe we sppose ha I hs case Tabe A he peseed agohm s modfed o Fge 6 Regadg Tabe A we have ( ) ( ) ;

10 ad afe smpfcao s obaed as foows: The vae of s aso eqa o ( 3) og ad s sch ha whch eads o Theefoe s chaged o 3 og ( ) ( ) ( ) ( ) 3 Sce 0 so ad ( 3 ) I he case ad we have 3 wh w ( ) ad fo wh w ( ) Ths Hammg wegh of opmm soo s eqa o fo w( Agebac degees of compoe Booea fcos of asfomaos of he fom ) we have 3 whch we ca hem geeazed PHT s ca be deved he same mae We oe ha o defo Accodg o (4) fo compoe Booea fcos of geeazed PHT s we have f ( ) ( ) 0 ( ) ( ) Fge 6

11 Fge 7 Agebac degees of compoe Booea fcos of s obaed fom opmm soos of wh cosas We sppose ha he eqao s smpfed befoe appyg he agohm Sce has A s modfed o Fge 7 So he vae of s obaed as foows whch s smpfed o Sce ( ) ( ) so we have og Theefoe 0 ( whch s smpfed o ad s eqa o ) ( ( zeo bs s eas sgfca posos Tabe ) og ( ) og ) ( ) Ths he maxmm Hammg wegh s eqa o Exampe 6: Cosde he asfomao [( If ( ) / he ) ] ( ) ( ) ( 0 ( ) ; ad fo ) we have w( ) (8)

12 o 4 ) ( 8 Y Y Y Y Z whch s a geeazed PHT We have comped he agebac degees of he compoe Booea fcos of he ops of hs asfomao Fo Y we have ( 30753) Fo Y we have Fo Y 3 we have ad fo Y 4 we have These ess ae compabe wh (8) ( ) ( ); ( 85964) VII CONCLUSION Moda addo modo a powe of wo s oe of he mos appcabe opeaos symmec cypogaphy; heefoe vesgag cypogaphc popees of hs opeao has a sgfca oe desg ad aayss of symmec cphes Agebac popees of moda addo modo a powe of wo have bee sded fo wo opeads by Baee fse 05 Aso he ahos of hs pape have sded hs opeao some speca cases befoe I hs pape ag advaage of pevos eseaches hs aea we geeazed agebac popees of hs opeao fo moe ha wo opeads Moe pecsey we deemed he agebac degee of he compoe Booea fcos of moda addo of abay mbe of smmads modo a powe of wo as a vecoa Booea fco aog wh he mbe of ems ad vaabes hese compoe fcos As a es agebac degees of he compoe Booea fcos of Geeazed Psedo-Hadamad Tasfoms wee dve REFERENCES [] Beoh SIG Specfcao of he Beooh Sysem Veso Febay 00 avaabe a hp://wwwbeoohcom [] RLRves The RC4 ecypo agohm RSA Daa Secy Ic Ma 99 [3] La ad J Massey A poposa fo a ew boc ecypo sadad I I Damg_ad edo Advaces Cypoogy Eocyp'90: Woshop o he Theoy ad Appcao of Cypogaphc Techqes

13 Aahs Dema May 990 Poceedgs vome 473 of LeceNoes Compe Scece pages Spge-Veag 99 [4] J Josso ad B S Kas J RC6 boc cphe Pmve sbmed o NESSIE by RSA Sep 000 [5] B Schee J Kesey D Whg D Wage C Ha N Fegso Twofsh: A 8-B Boc Cphe 998 Avaabe va hp://wwwcoepaecom/wofshhm [6] C Bwc D Coppesmh E D'Avgo R Geao S Haev C Ja SM Mayas J L O'Coo M Peyava D Saffod ad N Zc MARS: a caddae cphe fo AES Peseed he s AES cofeece CA USA Ags 998 [7] A Baee I Semaef The ANF of Composo of Addo ad Mpcao mod wh a Booea Fco FSE 03 LNCS 887 pp Spge-Veag 003 [8] Aeza Rahmpo S M Dehav ad Mehd Aaeya Agebac Popees of Moda Addo Modo Soheas Asa Be of Mahemacs 36:

Suppose we have observed values t 1, t 2, t n of a random variable T.

Suppose we have observed values t 1, t 2, t n of a random variable T. Sppose we have obseved vales, 2, of a adom vaable T. The dsbo of T s ow o belog o a cea ype (e.g., expoeal, omal, ec.) b he veco θ ( θ, θ2, θp ) of ow paamees assocaed wh s ow (whee p s he mbe of ow paamees).