PRACTICAL BIJECTIVE S-BOX DESIGN

Transcription

1 Proceegs o the 5th Asa Matheatcal Coerece, Malaysa 009 PRACTICAL BIJECTIVE S-BOX DESIG Aburash Maaolov, Hera Isa, Moesa Soehela Mohaa,, Iorato Securty Cluster, Malaysa Isttute o Mcroelectroc Systes, Techology Park Malaysa, 57000, Kuala Lupur, Malaysa e-al: rasholov@osy, herasa@osy, oesa@osy Abstract Vectoral Boolea uctos are usually calle Substtuto Boxes (S-Boxes) a are use as basc copoet o block cphers Cryptography The cphers that are Substtuto-perutato etworks use bjectve S-Boxes e, Boolea perutatos S-Boxes wth low eretal uorty a hgh olearty are cosere as cryptographcally strog I ths paper we stuy soe propertes o S-Boxes We costruct bjectve cryptographcally strog S-Boxes Our costructo s base o usg o-bjectve power uctos over the te el Itroucto A vectoral Boolea ucto s a ap ro to, where s te el wth two eleets Vectoral Boolea uctos are usually calle S-Boxes a are use as a basc copoet o block cphers Cryptography The cphers that are Substtuto-Perutato etworks use bjectve S-Boxes whch eect are Boolea perutatos The bjectve S-Boxes are Boolea perutatos o A portat coto o S-Boxes s hgh resstace to eretal a lear cryptaalyss [], whch are the a attacks o block cphers The eretal cryptaalyss s base o the stuy o how erece a put ca aect the resultat erece at the output The uctos wth low eretal uorty [] Δ ax { x : ( x + a) + ( x) b} a, b, a 0 has goo resstace to the eretal attack The lear cryptaalyss s base o g ae approxatos to the acto o a cpher The uctos wth hgh olearty [] b ( x) + a x ax ( ) a, b, b 0 x possess a goo resstace to the lear attack S-Boxes wth low eretal uorty a hgh olearty are cosere as cryptographcally strog S- Boxes wth sze ca be cosere as strog they have eretal uorty at ost 0 a olearty at least 00 ote that kow best par o these paraeters s, a S-Boxes wth eretal uorty below 0 a olearty above 00 are very rare Several ethos to geerate cryptographcally strog S-Boxes exst [5-0], such as rao geerato, the use o te el operatos, as well as heurstc algorths The power ap x x, where x has bee systeatcally stue [,] The power ap x, x 0 x, where x, 0, otherwse whch act s the verse ucto, has the kow hghest olearty a lowest eretal uorty Iverse appg was rst stue by LCarltz a SUchyaa 957 a t s uque best bjectve S-Box sce A exaple o such S-Box s Rjael s S-Box [], whch possess eretal uorty a olearty I ths paper, we the uber o -varable o-ae Boolea perutatos up to ultple We stuy power a boal uctos There s o bjectve power ucto whch coul be use as strog S-Box, except verse ucto However, there are o-bjectve uctos wth hghest olearty a lowest eretal uorty We obta strog bjectve S-Boxes usg o-bjectve power uctos Our S- Boxes have eretal uorty, olearty 0 a aely equvalet to ay su o a power uctos a a ae uctos I ths paper we preset the costructo o x S-boxes, however, the results are prove or ay sze 5

2 Prelares Vectoral Boolea uctos Let a be the te els wth a eleets, respectvely Let (,+) be the vector space over, where + s use to eote the ato operator over both a the vector space Boolea ucto (lter) s a ap ( x,, x ) : A -varable The Hag weght wt() o a Boolea ucto o varables s the weght o ths strg, that s, the sze o the support sp( ) { x : ( x) } o the ucto The ucto s sa to be balace wt ( ) The Hag stace betwee two Boolea uctos a g s (, g) { x : ( x) g( x)} Clearly, the stace betwee Boolea uctos a g s equal to the weght o su o these uctos (, g) wt( + g) A (,) vectoral Boolea ucto (S-Box) s a ap x,, x ) ( ( x,, x ),, ( x,, x )) : ( Clearly, each copoet ucto,,,, s a -varable Boolea ucto A (,) vectoral Boolea ucto s calle -varable Boolea trasorato A (,) vectoral Boolea ucto s calle balace t takes every value o the sae uber o oe te A balace -varable tes I a Boolea trasorato s balace the t takes every value o Boolea trasorato s calle -varable Boolea perutato Clearly, -varable Boolea perutato s bjectve ucto ro to tsel The uque represetato o -varable Boolea ucto as a polyoal over varables o the or α ( x,, x ) c( α )( x ) s calle the algebrac oral or (A) o The egree o the A o s α eote by ( ) a s calle the algebrac egree o the ucto We call Boolea ucto as ae ( ) a lear t s ae a (0)0 A (,) vectoral Boolea ucto,, ) s calle ae (lear) each copoet ucto (,, s ae (lear) We cocetrate o o-ae Boolea perutatos Deretal Uorty The eretal uorty (DU) Δ o (,) vectoral Boolea ucto s ee as Δ ax { x : ( x + a) + ( x) b} a, a 0, b or ay (,) vectoral Boolea ucto, the DU Δ o satses ax{, } Δ I s lear the Δ I the Δ All kow perutatos wth Δ are ee wth o It s cojecture that or ay Boolea perutato wth eve olearty Let A() be the set o all -varable ae Boolea uctos The olearty (L) o a -varable Boolea ucto s ee as ( ) (, g) g A I other wors, the L o ucto s a stace betwee ucto a the set A() o all -varable ae Boolea uctos Clearly, 0 a oly s a ae ucto It s kow that or ay -varable Boolea ucto, the L satses the ollowg relato: uctos achevg the equalty are calle bet uctos whch exst whe s eve However, bet uctos are ot balace Let be a balace -varable Boolea ucto ( ) The the L o ucto s gve by, eve, o, where x eotes the largest eve teger less tha or equal to x The L o a (,) vectoral Boolea ucto, s ee as Δ c c 55

3 I other wors, the L o ucto s a stace betwee the set o all o-costat lear cobatos o copoet uctos o a the set A() o all -varable ae Boolea uctos Ths shows that 0 s ae However, the coto 0 oes ot expla the aty o It s kow that or ay (,) vectoral Boolea ucto, the L, satses uctos achevg the equalty are calle perectly olear a ca exst oly whe s eve a I s o a the we have uctos wth L are kow or eve a, a t s cojecture that ths value s the hghest possble L The uber o o-ae Boolea Perutatos I we eote the set o all (,) vectoral Boolea uctos, -varable Boolea trasoratos a - varable Boolea perutatos as B(,), BT() a BP() respectvely, the we have (, ) B, BT( ) a BP ( )! We eote the set o all o-ae -varable Boolea perutatos by ABP() ote that ABP() BP() BT() + Theore Let μ ( )! (!) ( ) The the uber o o-ae Boolea perutatos satses μ( ) ABP( ) μ( ) Proo: or provg the let se o equalty, t s eough to show that we ca costruct μ() eret oae -varable Boolea perutatos Clearly, a -varable Boolea perutato s just perutato o vectors Our etho o costructo cotas two steps: () Choose balace o-ae -varable Boolea ucto as rst copoet ucto o Boolea perutato () Choose two perutatos o vectors a set the perute vectors as values o (0,,, ) a (,,, ), respectvely The resultg ucto,,, ) s a o-ae Boolea perutato ( Ay o-costat ae ucto s balace Sce, A ( ) + a the uber o costat ae uctos + s, the uber o balace ae Boolea uctos s whle the uber o -varable balace Boolea uctos s Thereore the uber o balace -varable o-ae Boolea ucto s + ( ) The uber o perutatos step ) s (! ) Thus, we have + + (!) ( )! (!) ( ) μ( ) stct o-ae Boolea perutatos To prove the rght se o equalty, we rst costruct μ() o-ae Boolea perutatos The we show that each o-ae Boolea perutato ca be obtae by our costructo I the above costructo we take -th copoet as balace o-ae xe ucto or each,,, the we have μ() oae Boolea perutatos Let (,, ) be ay o-ae Boolea perutato The has at least oe o-ae copoet ucto Clearly, the Boolea perutato (,, ) ca be obtae by perutg the vectors o such that the obtae Boolea perutatos -th copoet ucto s sae wth Table shows the uber o uctos o three classes or soe sall 56

4 Table The uber o uctos the three classes BT () BP () ABP() ,777,6 0,0, Soe propertes o power a boal uctos over the te el Let be te el o teger We ote that power ucto Theore Let ucto, where q gc(, ) Proo: It s kow that the group eleets We coser x x power uctos, where x x s bjectve a oly, ) gc( x be power appg over the ultplcatve group Sce the group Z s also cyclc, s cyclc Let we coser the group the the ultplcatve group a s postve /{0} The t s a q-to-oe Z o resue classes oulo s soorphc to the atve group Z Thereore t s sucet to show that the appg x over the group Z s a q-to-oe ucto, where q gc(, ) Ths ples ro the ollowg well kow theore o uber Theory: The cogruece x r(o ) has a soluto Z a oly gc(,) ves r I ths case, the uber o solutos Z s exactly gc(,) [] Coser the boal ucto Theore DU( Proo: L( x x c, 0 g ' c A( ) x x x x + over the te el + )DU( x ) a L( x x + )L( x ) DU ( x ) ax { x :( x + a) + ( x + a) b} (sce a, b, a 0 a, b, a 0 ax { x :( x + a) + a b} x s lear) ax { x :( x + a) + a b} a, b, a 0 ax { x :( x + a) a, b, a 0 + ) wt( c c ( x ) + + g') c g ' A( ) b} (sce ( c ( x wt( c g A( ) 5 Costructo bjectve strog S-Boxes + g) a ), g') s a costat) DU( x ) wt( c ( x c g ' A( ) (, g) c g A( ) c ) + g') L ( x ) It s well kow soe power uctos wth lowest DU or hghest L [,] or exaple, power uctos x, where + ( + ) a gc(, ), are calle Gol (Kasa) uctos a they have lowest DU a s o, the they have hghest L too I (verse ucto) or k a 0 gc(, ), 0 o the power perutatos x have the hghest kow L It s ot cult to copute DU a L or all power uctos over the te el ote that sze S-Boxes ca be cosere as strog they have DU 0 a L 00 The oly bjectve power uctos (perutatos) satsy the cotos are x, x, x, x, x, x, x, x They have DU a L Actually, all these uctos are cycloatc cosets, whch eas they are equvalet uctos Thereore there s oly oe strog bjectve power ucto up to equvalece Rjael have use ths ucto [] We coser strog o-bjectve power uctos There are o-bjectve power uctos wth best paraeters (DU, L)(,) Actually, they are ro three o-equvalece classes: { x, x, x, x, x, x, x, x }, { x, x, x, x, x, x, x, x } a { x, x, x, x, x, x, x, x } We ca choose ay represetatve o these uctos or ext stuy or the costructo exaple, we choose x k 57

5 The power ucto x s Gol a Kasa ucto thereore t has lowest DU The L o power ucto x s Let I be age o ths ucto, the I 6 By Theore, the ucto x, as appg, where /{ 0}, s three-to-oe ucto We coser α x + β x ucto, where α, β ro Theore, DU( α x + β x )DU( x ) a L( α x + β x )L( x ) Let J eotes the age o α x + β x the J 9 Thereore we have bee brought earer to bjectve ucto wthout chagg DU a L ally, the reag uplcate eleets the age are replace by eleets ro \J such a way that the DU a L are ot coprose sgcatly There are 56 uctos o the or α x + β x, α, β Our best result s (DU, L)(, 0) a oly three uctos gve ths result The coecets ( α, β ) o these uctos are (50, 9), (67, 6) a (, 7) 6 Coclusos I ths paper, we prove soe propertes o power a boal uctos over the te el We the uber o -varable o-ae Boolea perutatos up to ultple o-ae Boolea perutatos are ot rare However, o-aty property s ot sucet or strog S-Boxes We propose a sple schee whch prouces a ew cryptographcally strog bjectve S-Boxes Costructo s base o usg o-bjectve strog power uctos over the te el The resultg S-boxes have DU a L 0 whch are cryptographcally strog or use block cphers Ackowleget We woul lke to thak Pro Dr Mohae Rza Wah (MIMOS Bh) or geeral guace We also thak Dr Isa Rakhov (UPM) or useul coets Reereces [] L Buaghya The equvalece o Alost Bet a Alost Perect olear uctos a ther geeralzato, PhD Dssertato, Otto-vo-Guercke-Uversty, Mageburg, Geray, 005 [] M Maxwell Alost Perect olear uctos a relate cobatoral structures, PhD Dssertato, Iowa State Uversty, Aes, Iowa, 005 [] J Daee, V Rje The Desg o Rjael Sprger-Verlag Berl Heelberg, 00 [] MErckso, AVazzaa Itroucto to uber Theory Chapa&Hall/CRC, 00 [5] L Cu, Y Cao A ew S-Box structure ae ae-power-ae, Iteratoal Joural o Iovatve Coputg, Iorato a Cotrol, vol, o, pp , Jue 007 [6] S-Y J, J-M Baek, H-Y Sog Iprove Rjael-lke S-Box a ts trasor oa aalyss, Sequeces a Ther Applcatos SETA 006, LCS, vol 06, pp 5-67, 006 [7] E Sakalauskas, K Luksys Matrx power S-Box costructo, Cryptology eprt Archve: Report, o (007) Avalable: [] J Clark, J Jacob, S Stepey The esg o S-Boxes by Sulate Aealg, ew Geerato Coputg, vol, ssue, Septeber 005, pp 9- [9] Hua Che, Deg-guo eg A Eectve Evolutoary Strategy or Bjectve S-Boxes, CEC00, Orego, 00 Vol, P 0- [0] Xu Y, Sh X Cheg, Xao Hu You, Kwok Ya La A Metho or Obtag Cryptographcally Strog x S- boxes, IEEE Global Telecoucatos Coerece, Phoex, Arzoa, ov -, 997, vol, ov, 997, pp 69-69, Isttute o Electrcal a Electrocs Egeers 5