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Almost Perfect Nonlinear functions

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HAL Id: inria-00070246

https://hal.inria.fr/inria-00070246

Submitted on 19 May 2006

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Almost Perfect Nonlinear functions

Thierry Pierre Berger, Anne Canteaut, Pascale Charpin, Yann Laigle-Chapuy

To cite this version:

Thierry Pierre Berger, Anne Canteaut, Pascale Charpin, Yann Laigle-Chapuy. Almost Perfect Non- linear functions. [Research Report] RR-5774, INRIA. 2005, pp.28. �inria-00070246�

(2)

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

a p p o r t

d e r e c h e r c h e

THÈME 2

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

(3)
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n :.*` m @# CG*`C;*E'HCG$9*,e+'#+a

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a

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a $jR^"? =#3N*` +0$2C.*

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e#" *`#@+

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a $j +:.$2JC @#A:!`b^_]jWfgr{knesr‡ijdesiUr‡WCQ=

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F @# : =#3N*` +<$2C;* =#')C;

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δ(a, b) = #{xFn2, DaF(x) =b},

",'H

#E $j-^"?A H:;').$9*]:.Bf$90C= :.*SG+'&(a(]"?+*?d) "?:+*

δ(F) = max

a6=0, b∈Fn2δ(a, b)2

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[

deg

2n−1

X

i=0

λixi

!

= max {wt(i)|λi6= 0},

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λiF2n

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(β) =β+β2+· · ·+β2n−1¦

RSUWwš›iU]jdes^baQ]

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F @# :c=#3N*` +<$2C;* =#')C;

F2n

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F2n

a (]", Bf$9*`#:;' #C.@+$9*`:.0$2C.*, C = ^"?

#CEC.');$9*`:;J CQ=

F :;') ^"? CC;BD#:.*G=#3N*` +<$2C;*?

fλ:xF2n7→R$r (λF(x)), λF2n,

",'H

f0

$j-9", *S3NB9B;=#3N*` +<$2C;*`a (]"?U=#3N*` +<$2C;*?

fλ

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Bn

esSUWknWeKaQš ¹ a@a,`_W-fg]!š›ij]jdœe‡^_a,]jkKaQ]

F2n¦

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aQš Bn

fgr‡WNe‡SUW‰š›iU]jdes^baQ]jk

ϕa

±UxyW¢j]jWx#oFm

ϕa:xF2n7→R®r (ax), aF2n. žu«- 

RSjWNš›a,`_`ba§^_]jc]UaQe‡fe‡^_a,]§^b`_`7oBW‰WMFesW]jkn^b–QW`_m>ijksWx^_]esSjW2|jfQ|OWr

¦ aQrfg]Fm

f ∈ Bn

±F§wW‰xyW]Uage‡W2oFm

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f[

F(f) = X

x∈F2n

(−1)f(x)= 2n2wt(f), ž*XQ 

§SjWr‡W

wt(f) ^hk2esSUW fgV>V>^_]Uc™§—W^bcQS@e2aQš

f±^¦W ¦±e‡SUW>]FiUVoBWr‰agš

x Fn2

kniOd}SÃesSOfe

f(x) = 1¦

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f ^bkk‡fg^hx>esa†oOW@#:.BD:;*` ##N^_š$fQ]jx'aQ]j`_m>^_š

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wt(f) = 2n−1¦

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Ÿ· (]",

”

fg`hknS¤ks|OW-dœe‡rsiUV C=

f ∈ Bn

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@+

2n−11

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a∈F2n

|F(f +ϕa)| .

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a (]"?G=#3N*` +0$2C.*

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#CE +$2*S2-:em:; C ^"%'HH *.:.Bf3,#d *`:.+Bf

0,±L(f)a(]"?+*?d

L(f) = 2s i$99" sn/2a

$2= s =n/2 Q:.*] n *+* !^"?+* f $jG+:.$2 C @#†oOW]@ec:.*]!$9<A:;B)" #CE $2+*,< :e C

*.:.BX3, C.*SBfd *`:;G+Bf

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s= (n+ 1)/2=C;' Ce n

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F @#:-=#3N*` +<$2C;* =#')C.

F2n

$9*SJC

F2n

i$99" #C;(>,C.*]*S2

fλ

d

λ F2n

a (]"?

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fλ

a$ $j H1+3,:.B5JC

N(F) = 2n−1Λ(F) 2

",')

Λ(F) = max

λ∈F2n

L(fλ).

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RSUWm­WMy^hkueš›a,rayxUx

naQ]j`_m ¦

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F @#: =#3N*` +<$2C;*=#')C;

F2n

$9*,C

F2n

i$99" #C.R>?C;*`+*,<

fλ

d

λ F2n

a

(]",*,d

Λ(F)2n+12 .

(]", =#3N*] 0$2C.*,

F "N$2 H"+:.0$j^=#

Λ(F) = 2n+12

:;')m+:.$2cCc@#‰fg`bV5a@kueCoOW]@e>žH˜ ¹  +a (]"?+ e$j "?+*

n $jGCEC;*,Bfa!C;')#C *.'#d($=

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