Almost Perfect Nonlinear functions

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Almost Perfect Nonlinear functions Thierry P. Berger — Anne Canteaut — Pascale Charpin — Yann Laigle-Chapuy

N° 5774 Décembre 2005

N 0249-6399

ISRN INRIA/RR--5774--FR+ENG

THÈME 2

apport de recherche


    
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¦ RSFijk±Fš›r‡aQV q¥r‡aQ|Ba,ks^_es^baQ]#‚j±y§—W!SjfÁ–,WNš›a,r{fg]Fm a [

gcd(d, 2n − 1) = 3 X 3 F 2 (Da fλ ) = ν(f1 ) + 2ν(fα ),

§ SjWr‡W ^hkfN|Ur‡^bV5^_es^b–QWW`_WV5W]@eKaQš RSU^hkda,V5W-k$š›rsa,V¨esSUWwšŸfQdeesSjfge ν(f ) = ν(f ) oOW-dfQijknW oBage‡S#š›iUα]jdœe‡^_a,]jk{fQrsWC`_^b]UW-fgr‡`_m­W-p@iU^_–fQF`_W],e ¦ ¦ zW-dfQ`_`OesSjfge^_š ^hkw˜Nq±F§—W{SjfÁ–QW žŸksWWNq¥rsa,|Oa@kn^_es^baQ]V,  ¦ W]jdWQ± f,ddaQr}xy^_]jce‡a>RSjWaQFr‡W:Vx 7→XF± Fx ^hk{˜Nq¨^_šfQ]jxa,]U`bm­gcd(d, ^£š 2 − 1) = 3 λ∈F2n

2n

α2

α

d

n

'

ν(f1 ) + 2ν(fα ) = 3 · 22n+1 .

 We g ± λ ∈ F ±yxyW]Uage‡WesSjWCdaQV>|Oa,]UW]@e}k¥agš G : x 7→ x a–,Wr F ¦  š F : x 7→ x ^hkw˜Nq0e‡SUW] ± ±y§^_esS gcd(d, 2 − 1) = 3 ¦ RSFijk§wW‰SjfÁ–,W [ d=3 k r>0 R®r X (−1) F(f ) = R$r X (−1) = R$r = F(g ). X = (−1) λ r

3

2n

d

2n

n

(λxd )

λ

x∈F2n

r

(λy 3 )

y∈F2n

(λz 3 )

λ

RSj^bk^hk{oBWdfgijksW x 7→ x ^hk 3¡*esaQ¡ 1 š›rsa,V F esa>e‡SUWksWe { z | z ∈ F } ±Bkn^b]jdW u = v ^£š¥fg]Ox a,]U`bm­N^£aš §ƒ(u/v) §SUWr‡W β ∈ FR®r ¦ RSUWw–ÁfQ`_ijWklaQš §wWrsWwxyWe‡Wr‡V5^b]UW-x u = vβ§^£e‡S §—WSOfÁ–Q=W 1F(f±Fe‡Sjf)e{=^hk F(g F(g ) oFm#v—fQrs`b^_es©GT_«ÁXF±jRSUWaQr‡WV «+W [ ) g (x)= (λx ) ¦ z∈F2n

d

λ

3

∗ 2n

3

λ

∗ 4

λ

d

2n

3

d

λ

(−1)t+1 2t+1 , (−1)t 2t

F(gλ ) =

¦ ” WxyWxyiOdWCesSjfge fQ]jx F(f ) = 2 ; F(f ) = −2 š›a,rW–,W] t fQ]jx fg]jx F(f ) = −2 F(f ) = 2 š›a,rayxUx t ¦ ÃaQr‡Wa–QWr±@§—WCSOfÁ–QWCš›aQr{fQ]@m t [ f,ddaQr}xy^_]jc5f,k λ ^bka,r^bk]jageNf>diUoBW!^b] 1

1

F∗2n

t+1

t

t+1

α

t

α

Λ(F ) ≥ |F(f1 )| = 2t+1 .

z W-dfQ`_` e‡SjfeNš›aQr{ayxUx ±U^_eN^hk§wW`_`_¡æ¯@]ja§]e‡Sjfe ) ≥ 2 š›aQr2fg]Fm­š›iU]Odœes^baQ] F aQ] F §Sja,ksW^bV>fQcQWwSjfQkKxy^bV5W]jksn^_a,] n ¦ RSUW]UW+MFedaQ]tuW-dœesijΛ(F rsWda,V5W-k®]Ofesijr‡fQ`_`bmCš›rsa,V e‡SUW|Ur‡W–F^_a,ijklr‡WksiU`_e ¦ ·
-³ ’gE® ³  F @# :.*, =#3N*`+š›a,r n ∈ {10, 20}
fg]jx
ν(f ) = 2 +2 2 +2 −1 ν(f ) = 2 −2 2 +2 −1 . WrsW,±U§wW‰SjfÁ–,W Λ(F ) > 2 ¦ .

2n

n 2

i

2n+2

1

4g

1

2n+1

n+2g+1

3g

2g

g

g

2n+1

α

n+2g

2g

g

n 2 +1







2g

2n

α

(  *

  



” W!|Ur‡W–F^baQijks`_m>š›aydijknW-x#aQ]™˜{qδ(F) °š›iU]O≥dœes4^baQ]Ok±U^ ¦ W ¦ ±yesSUW‰š›ij]jdœe‡^_a,]jkš›aQr§Sj^bd}S


δ(F ) =

max

δ(F ) = 2

§^_esS

#{x ∈ Fn2 , Da F (x) = b}.

Na§!±j§—W|BaQ^b],e2aQiye{e‡SjfeCkna,V5WrsW-kniU`_e‡kNaQ]#esSjW†ksiUV5¡*aQšÚ¡æk‡p@ijfgr‡W‰^_]jxU^bdfesa,r‡k{agšesSUW†daQV>|BaQ]UW]@e‡k{aQš dfQ]#oBWxyWrs^b–QW-x'š›r‡aQV Nm@oBWr‡c,4 kr‡WksiU`£e TÌX‚;W*±y§SUW] δ(F ) ≥ 4 ¦ F  · · Ž  Ÿ·
+ @H :i=#3N*`+0$2C.*R=#')C; F $9*,C F i$9^" #C.R>,C.*`+*,W‰]Uage}fe‡^_a,]#f,k^_]¤„2W¢j]U^_es^baQ]IX ¦  We a ∈ F oOWksijd}S#esSOfeesSjWr‡WCW+My^bkne‡k §^_esS §^_]Ub)c†š›=aQr‡Vδ(Fij`bf>) ¦ xyiU W‰We e‡a'A{mFxyoOWW]Ursagcce‡W†TÌXe‡‚.SUW W>š›]FaQiUr{VfQ]FoBmWr!aQš b ∈ F kniOd}SÃesSOfe δ(a, b) = i ¦ bz{W∈dfQF`_`7e‡SUW‰š›aQ`b`baδ(a, [ a∈F λ∈F2n




∗ 2n

2n

2n

i

∗ 2n

X

” WxyW-xyijdWCe‡Sjfe

λ∈F2n

F 2 (Da fλ ) = 2n

X

δ 2 (a, b) .

b∈F2n

δ(F )−2

2

−n

X

2

2

F (Da fλ ) = δ(F ) Aδ(F ) +

X i=2

λ∈F2n

δ(F )−2

≥ δ(F )2 Aδ(F ) + 2

X i=2

 

» ï




i2 A i

iAi ,

žn«Q«- 

«-Š

' e+'#d :.*SJ#:.3N`3N 











§^_esS™W-p,iOfg`b^£eum­^_š4fg]jxaQ]U`bm­^_š δ(a, b) ∈ {δ(F ), 2, 0} š›a,r{fg`b` b ∈ F ¦ ¤aQr‡Wa–,Wr-±,§—W‰SjfÁ–QW 2n

δ(F )−2

X

RSFijk±

iAi = 2n − δ(F )Aδ(F ) .



žn«-XQ 

i=2

X

F 2 (Da fλ ) ≥ 22n+1 + 2n δ(F )(δ(F ) − 2) ,

§^_esS™W-p,iOfg`b^£eum­^_š4fg]jxaQ]U`bm­^_š A = 1 fg]jx A = 0 š›aQr{fQ`_` 4 ≤ i ≤ δ(F ) − 2 ¦  ·  · o?C '>,'A3NJ:;,C.*]*S2 CQ= f a ]"?+*?dU:;B9B ν(f ) d λ ∈ F d :.')A#13,:;B:.*]G+:.0$j^=# 1) = 1 λ∈F2n

i

δ(F )



+

*



d 

2n

(

λ

λ

 C;')#C .'#d $ =mH1+3,:.BX$9RSj|BWa]l§w±7We‡r—SU|BW5Wr‡ksiUVV5iUe‡¡æagfšÚe‡¡¾^_ksa,p@]ijaQfQšrsWF^b]jxy^h§dSjfgesWa,] r‡kNn ag^hšk^_We‡–,k‰Wd] a,¦ V>|Oa,]UW]@e‡kCfQd}Sj^_W–QW!e‡SUW†`ba§—Wkne ν(fλ ) = 22n+1 + 2n+3

∗ 2n

2n

ë lë º  

«-¬

BXGC.#
8+'^=H+ C.*SBf$9*`#:;' =#3N*`+§SUWe‡SUWr—˜{qª|OWrsV†iye‡fges^baQ]jk4aQš WMy^hkuew§SUW] ^hk¥W–,W]l±QesSUWNijknW aQš4|OWrsV†iye‡fges^baQ]jk F §^_esS δ(F ) = 4 ^bk2knij^£e}fgoU`bW!^_],+'m3N:.0$2C.* F C.* F d n  .*,d i$99" #C;(>,C.*]*S2 f d λ ∈ F d 3,#"c^"?:; δ(F ) = 4 :.*] n

4






.



-

!

*



2n

X

λ

∗ 2n

ν(fλ ) = (2n − 1)22n+1 + A2n+3

=C;' +C. 9$ *SJ l' A < 2 − 1 a " ‰ $ ¨ "# $# ˜N]^b]y¢j]U^_esWd`hfQk‡klaQšjp@ijfQxyr}fe‡^bdw˜{qôš›iU]jdes^baQ]jk±§SU^hd}S†fQrsW¥]jage$W-p@iU^_–fQ`_W],e$esa‰fg]FmC|Oa§—Wr®š›iU]jdœe‡^_a,]l± §fQkd}Sjfgr}fQdesWr‡^b©Wxô–QWrsm rsW-dW]@e‡`_m°žŸksWW T ‚.WCfQ]jx Tb«Á W›  ¦ RSU^hkxy^hks|Ursa–,Wk'f daQ]tuW-dœesijrsW¤a,]˜{q λ∈F∗ 2n







 

» ï




n

XQˆ

' e+'#d :.*SJ#:.3N`3N 













š›ij]jdœe‡^_a,]jk#agš†xyWcQr‡WW 2 ±2k‡fÁmF^_]Uc esSjfge™ksijd}Sš›ij]jdœe‡^_a,]jk™fgr‡WIWp@iU^b–fg`bW]@ee‡aôe‡SUWƒ|Oa§—Wrš›iU]jdes^baQ]jk a–,Wr §^£e‡S fg]jx Wr‡WQ±¥§—W¤da,],e‡rs^boUiye‡We‡aƒesSUW x d`bf,7→ksks^_¢Oxdfges^baQ]aQš†˜{Fq p@ijf,xyr‡gcd(k, fges^hd¤š›iUn)]Odœ=es^baQ]O1k ¦ ” 1Wƒ≤|Ur‡ak–QWÃ≤e‡Sjn/2 fe#¦esSjWr‡WƒfQrsWƒ]Ua ˜{q p@ijfQxUr‡fges^hd š›ij]jdœe‡^_a,]jkaQ] F aQš®esSjW‰š›aQr‡V X žn«-€Q  F (x) = cx , c ∈F , W+MUdW|UeCe‡SUW'|Ur‡W–F^baQijks`_mIV5W]@es^baQ]UW-xI|Ba§wWr‰š›iU]jdes^baQ]jk ¦ ” W'§^b`b`¥ijksW†e‡SUW WrsV>^_esWe4 kdrs^_esWrs^baQ] ¦ ˜ |jrsaFagšKagš$esSUW!]UW+MFeesSUWaQr‡WV dfg]#oOW‰š›a,iU]jx#^_] TÌXQXF±ORSUWa,rsWV ¦ ‚;W ¦  ³F·  ³ T WrsV>^_esW4 kdrs^_esWrs^baQ]?W  F @H :;*, *,$9 UB mC = #"?:.'H:+J+'$j]jW'S@mF|BWr‡|U`hfg]UW ¦ vwrsaFa,¯QWxڛiU]jdes^baQ]jkfQrsW'˜ ¹ fg]OxIe‡SUW­a,]U`_mI¯@]ja§]h+')CEC e#5š›iU]jdes^baQ]jk fQrsW‰p@ijfQxUr‡fges^hd ¦ ³  · Œ o ³ C.*?0'3,_'HC C # =#3N*`+^hfg`hkNaQš rs`bmQ±esSjW—d`bf,ksks^£¢Bdfe‡^_a,]!agšj˜Nq p@ijf,xyr}fes^hdš›iU]Odœes^baQ]Ok®^hk®]jage$m,WefQd}SU^bW–,Wx ¦ F [x] §^£e‡S n = 3k fQvw]j`bWx fQgcd(3, k) = 1 ¦ 2n

a



n 2

2n

a




2n

.



-

n 2



a



2n

ë lë º  

XU«

BXGC.#
8+'^=H+ C.*SBf$9*`#:;' =#3N*`+,C +'=#3N*]0$2C.* =C.' n 6= 3k d ",'H a gcd(3, k) = 1 ]ÃesSUW5]UWMFe!knW-dœe‡^_a,]l±O§—W|jrsa–,W!esSjfge2e‡SUW†d`bf,ksk2agšp@ijf,xyr}fes^hd‰š›iU]Odœes^baQ]OkCxUW¢j]UW-xÃoFm žu«Á€g {^hkC˜{q a,]U`bmF0 :;*` j > 0 a


.



-

!



*



-

/.



 

*









+

i

(

2 +2

j

∗ 2n



     
(   )(.  *.




Nage‡W2e‡Sjfežn«-€Q w^hk]Uagee‡SUW!cQW]UWr}fg`BWMy|Ur‡Wk‡kn^baQ]agšp,iOfQxyr}fe‡^bdNš›iU]jdes^baQ]jka,] ks^b]jdW‰^_exUa@W-k]Uage b^ ]jd`_ijxUW{fg]Fm!esWrsV agšBesSUW{š›aQr‡V x ±,§^£e‡S i, j > 0 ¦  ]5esSjW{ag”e‡SUWr4SjfQ]jx ±,]UFage‡WesSjfge¥^£š F ^hk¥˜Nq e‡SUW] F + L ^bk˜Nq°esaFaj±F§SUWrsW L ^bkfQ]@m'f;F>]jWC|BaQ`bmF]UaQV>^hfg` ¦ WC¢jr}kue|Ur‡a–QWNf†ijksWš›iU` `bWV>V'f5^_] a,r‡xUWr¥e‡a5d}SOfgr}fQdœe‡Wr‡^_©WesSjWC˜Nq°|Ursa,|OWrneum5š›a,rwe‡SUW‰d`hfQk‡kaQš$p@ijfQxyr}fe‡^bd{š›iU]jdes^baQ]jkxyW¢j]UWxoFm¤žu«-€,  ¦ w³ ‘ 5+ H @HA:Z>,C.BX.*`C.A$2:.BiC.* F 3,#"^"?:. H(0) = 0 :;*`:;W]@e2± k‡fÁ−m 1β ∈ F ±B§Sj^bd}SÃ^hkC]UageC0^_] I ¦  WeCiOk2xyW¢j]UW†esSUW ¦ |BaQ`bmF]UaQV>^hfg` P aQ] F oFm  š›aQr x 6= e H(x) › š Q a r P (x) = . RSjW] e‡SUW^bV'fgcQWagš P Sjf,kdfQr‡xU^_]jfQ`_^_eum 2 ±4V>βWfg]j^_]UxcI=esSjefge P ^hk†fI|OWrsV†iye‡fges^baQ] ¦ Na§ §wW#fgr‡W c,aQ^b]Uc‰esaWMy|Ur‡Wk‡kesSUWN|Oa,`_mF]Ua,V5^hfg` W = H + P ¦ {aQesWe‡Sjfe-±Qš›r‡aQV esSjWNxyW¢O]U^£e‡^_a,]­agš P ± W (x) = 0 ij]U`_W-ksk x = e fQ]jx W (e) = P (e) = β ¦ ” Wd`bfQ^_V esSjfge—e‡SUW!iU]U^hp@iUW!r‡W|Ur‡WksW]@e‡fges^baQ]agš W V>ayxyiU`ba ^bk x +x [   W (x) = β (x + e) +1 . RSj^bk2^bk‰ks^_V>|U`bm™oOW-dfQijknWesSUW5r‡^_c,S,es¡*SOfg]jx™|BaQ`bmF]UaQV>^hfg`fgoBa–QWSjf,k2xUWcQr‡WW 2 − 1 fQ]jxÃ^hkNW-p@ijfg`$esa š›aQrW-fQd}S x ¦ RSFijk§—W‰|Ursa–,Wx­esSjfge W   2n

2i +2j








2n

(




n

n

2n

∗ 2n

2n

n

2n

2n −1

n

P (x) = H(x) + β (x + e)2

n

−1

+1 .

Py^_]jdW P ^hkf†|BWr‡ViUe‡fe‡^_a,]l±y^_e‡kxyWcQr‡WWC^bkfgeV>a@kue 2 − 2 žÚš›r‡aQV RSjWaQr‡WV€g  ¦ RSU^hk^bV5|j`_^bWke‡Sjfe ViOkue{SjfÁ–,WNe‡SUW‰esWrsV βx + ^_e‡k{xUWcQr‡WWC^hk 2 − 1 ¦ H  n

2n −1

 

» ï




n

X,X

' e+'#d :.*SJ#:.3N`3N 




 · · Ž  Ÿ· + F @# *`# @  
a ]"?+*?d F $j d $2a9EaDd Q : x 7→ F (x)/x -

/.



 



"

(



Q(x) =

j$  : >?+'m3N:.0$2C.* >?C;Bf.*`C;m$2:;BC;* F a
_')C C =a a,r{fg]Fm ±y§—WCSOfÁ–QW a∈F

c i x2

i






$ =:.*] C;*,Bf$ =^"? >,C.Bf.*]C.A$2:.B žn«Ág 




2

n−1 X



−1

i=1

2n

∗ 2n

Da F (x) =

n−1 X

 i  i ci x2 a + xa2 + F (a) .

RSjW]l±®esSUW­ksWe {D F (x), x ∈ F } Sjf,k!dfgr}xy^b]jfg`b^_eum SjfQkf5¯QWrs]jW` agš4xy^bV>W]jks^_a,] 1 ±y^ ¦ W ¦ ± D F i=1

n 2

a

2n−1

^£šfg]jx]jW'|Oa,`_mF]Ua,V5^hfg`

a

›š aQr{fg`b` x 6∈ {0, a} , a,rWp@iU^b–ÁfQ`_W]@es`bm^hfg` ¦ N   a ! § , ± j i s k b ^ U ] c  W ‡ r 5 V _ ^ s e e4 k—dr‡^_esWr‡^baQ]l±,aQ]UWCdaQV>|U`bWe‡W`bm5d}SOfgr}fQdœe‡Wr‡^_©Wke‡SUWN|BWr‡ViUe‡fe‡^_a,]­|BaQ`bm@]jaQV>^£¡ W fQ`bkagšle‡SUW{š›a,rsV žn«Á  RSU^bkw§fQkf,dœe‡ijfg`b`_m>|Ur‡a–QW-x>oFm5q4fÁmF]UW TÌX W7^b]fQ]Uage‡SUWr—daQ]@e‡WMFe±@esSjWNcQW]UWr}fg` |jrsa,oU`_WV agšZ9",#C.R¦ >`BD+ e'A$9*`:;,:.$9' (c , c ) $j E+')Ca ” W‰SjfÁ–,W ±
_')C C =a ” W!ijknW Wr‡V>^£e‡W4 kdr‡^£e‡Wr‡^_a,]ƒžHRSUWa,rsWV€g —§^_esS t = 2 + 1 k ∈ [1, n − 1] ¦ i=1

$

2n



k

n−k

k



Q2

 We

k

+1

(x) =

n−1 X n−1 X

k

c2i cj x2

i+k

+2j −2k −1

¦ RSUW!V>a,]UaQV>^hfg` m SjfQkxyWcQr‡WW

.

i=1 j=1

mi,j (x) = x2

i+k

+2j −2k −1

x2

i+k

i,j

+2j −2k −1

≡ x2

n

−1

2n − 1

^_šfQ]jx#aQ]U`bm­^_š

n

(mod x2 + x) .

Na§!±l§—W5iOknW  WV>V'f™X§^_esS µ = 1 ± (u , u ) = (0, k) fg]jx (a , a ) = (j, i) ±®^ ¦ W ¦ ± u = 2 + 1 fQ]jx v = 2 + 2 ¦ ” W†xyW-xyijdWesSjfge m Sjf,kNxyWcQr‡WW 2 − 1 ^£šfQ]jx™aQ]U`bm™^£š {0, k} = {j, i + k RSj^bk2kn^_esijfges^baQ]™ayddiUr‡ka,]U`bm§SUW] fg]Ox RSjWr‡Wš›aQr‡WQ±Fe‡SUWa,]U`_mesWrsV (mod n)} . aQš¥xyWc,rsWW 2 − 1 aQš Q Sjf,k{daFW+F'd^_W]@e c i +c k¦ =e{š›na,`_`ba§{kjesSO=fe{k ¦ e‡SU^bk{esWrsV–fg]U^hknSjWkš›a,r2fg]Fm §SjW] Q ^hk{f5|OWrsV†iye‡fges^baQ]#|Oa,`_mF]Ua,V>^bfQ` ¦ k ∈ [1, n − 1]  Na§!±y§wWdfg]#|Ursa–,WNe‡SUW!cQW]UWr}fg`Be‡SUWa,rsWV ¦ j

n

 

» ï




0

i+k

1

i,j

2k +1

2k n−k k

n

0

1

k

Xg‚

' e+'#d :.*SJ#:.3N`3N 


 ³F·  ³


>?C;Bf.*`C;m$2:;B C =

F2n [x]












CQ=-^"? =C.'

Q(x) =

n−1 X

c i x2

i

−1

, ci ∈ F 2n

#:;*,*`C;U@#A:Z>,'A3NJ:;,C.Bf.*]C.A$2:.B3N*,BD# Q(x) = cx i$9^" gcd(k, n) = 1 :.*` c ∈ F a C.*,+H1+3,+*,0Bfd :!1+3,:.'H:.0$2(=#3N*`+0$2C.* F C .' F C =A9",8=C.'  
$j
 $ =:.*]!C.*SBfc$ = i$9^" gcd(k, n) = 1 :;*` c ∈ F a F (x) = cx
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