1_Mikayel_Evoyan.DVI


Mathematical Problems of Computer Science 39, 5{12, 2013.

On k-Switching of M appings on Finite Fields

Mika ye l G. E vo ya n 1, Go h a r M. K yu r e g ya n 2 a n d Me ls ik K . K yu r e g ya n 1

Institute for Informatics and Automation Problems of NAS RA1

Institute of Algebra and Geometry, Otto-von-Guericke University Magdeburg2

e-mail: michael.evoyan@gmail.com, gohar.kyureghyan@ovgu.de, melsik@ipia.sci.am

Abstract

The switching construction was used in several recent papers to construct special
mappings on ¯nite ¯elds. In this paper we generalize the concept of switching to a
k-switching with 1 · k · n. We present some general properties of k-switching and
describe permutations produced using k-switching.

Keywords: Mappings of ¯nite ¯elds, switching, permutation.

1 . In t r o d u c t io n

L e t F : Fqn ! Fqn a n d ( °1; : : : ; °n ) b e a n Fq-b a s is o f Fqn . Th e u n iqu e ly d e t e r m in e d fu n c t io n s
fi : Fqn ! Fq; 1 · i · n; s u c h t h a t

F ( x ) = f1 ( x ) ¢ °1 + : : : + fn ( x ) ¢ °n;

a r e c a lle d t h e coordinate functions o f F wit h r e s p e c t t o t h e b a s is ( °1; : : : ; °n ) . Th e component
functions o f F o ve r t h e s u b ¯ e ld Fq a r e t h e fu n c t io n s T rqn=q ( ®F ( x ) ) wit h ® 2 F¤qn , wh e r e
T rqn=q is a t r a c e m a p p in g o f Fqn in t o Fq g ive n b y

T rqn=q = x + x
q + : : : + xq

n¡1
:

Th e s e t o f c o m p o n e n t fu n c t io n s o f a m a p p in g c o in c id e s wit h o n e o f it s c o o r d in a t e fu n c t io n s :

P r oposition 1: Any component function over Fq of a mapping F : Fqn ! Fqn is a coordi-
nate function with respect to some Fq-basis, and vice versa.

P r oof. R e c a ll t h a t a n y b a s is ( °1; : : : ; °n ) h a s a u n iqu e d u a l b a s is ( ¹°1; : : : ; ¹°n ) d e ¯ n e d b y

T rqn=q ( °i ¹°j ) =

(
1 if i = j;
0 if i 6= j:

fo r a ll 1 · i; j · n. In p a r t ic u la r fo r a n y a 2 Fqn t h e c o e ± c ie n t s ai in t h e lin e a r c o m b in a t io n
a =

Pn
i=1 ai°i a r e g ive n b y

ai = T rqn=q ( ¹°ia ) :

5



6 On k-Switching of Mappings on Finite Fields

Co n s e qu e n t ly, t h e c o o r d in a t e fu n c t io n fi ( x ) o f F ( x ) wit h r e s p e c t t o ( °1; : : : ; °n ) is t h e c o m -
p o n e n t fu n c t io n T rqn=q ( ¹°iF ( x ) ) . On t h e o t h e r h a n d , a c o m p o n e n t fu n c t io n T rqn=q ( ®F ( x) ) is
a c o o r d in a t e fu n c t io n o f F ( x ) wit h r e s p e c t t o t h e d u a l b a s is o f a n y b a s is c o n t a in in g ®.

A m a p p in g F : Fqn ! Fqn is c a lle d a s wit c h in g o f G : Fqn ! Fqn if t h e r e is a n Fq-b a s is
( °1; : : : ; °n ) s u c h t h a t a ll b u t t h e ¯ r s t c o o r d in a t e fu n c t io n s o f F a n d G a r e e qu a l, i.e .

F ( x ) = f1 ( x ) ¢ °1 + : : : + fn ( x) ¢ °n;

a n d
G ( x ) = g1 ( x ) ¢ °1 + : : : + gn ( x ) ¢ °n;

wit h f1 ( x ) 6= g1 ( x ) a n d fi ( x ) = gi ( x ) fo r a ll 2 · i · n. S wit c h in g wa s u s e d t o p r o d u c e
in t e r e s t in g c la s s e s o f s p e c ia l m a p p in g s o f ¯ n it e ¯ e ld s in [2 ]{ [8 ]. Co n s t r u c t io n o f b ije c t ive
m a p p in g s b y c h a n g in g t wo c o o r d in a t e fu n c t io n s o f t h e id e n t it y m a p p in g we r e s t u d ie d in
[5 , 7 ]. In t h is p a p e r we g e n e r a liz e t h e c o n c e p t o f s wit c h in g t o a k-s wit c h in g wit h 1 · k · n
a n d d e s c r ib e p e r m u t a t io n s p r o d u c e d u s in g k-s wit c h in g .

2 . k-S wit c h in g

De¯nition 1: L et 1 · k · n be an integer. A mapping F : Fqn ! Fqn is called a k-
switching of G : Fqn ! Fqn over Fq if k is the minimal integer such that there is an Fq-basis
( °1; : : : ; °n ) with respect to which all but the ¯rst k coordinate functions of F and G are
equal, i.e.

F ( x ) = f1 ( x ) ¢ °1 + : : : + fn ( x) ¢ °n;
and

G ( x ) = g1 ( x ) ¢ °1 + : : : + gn ( x ) ¢ °n;
with fj ( x ) 6= gj ( x ) for all 1 · j · k and fi ( x ) = gi ( x) for all k + 1 · i · n.

N o t e t h a t 1 -s wit c h in g r e d u c e s t o a s wit c h in g d e ¯ n e d a b o ve . Cle a r ly, fo r a n y t wo d i®e r e n t
m a p p in g s F a n d G t h e r e is a n in t e g e r 1 · k · n s u c h t h a t F is a k-s wit c h in g o f G. Mo r e o ve r ,
if F is a k-s wit c h in g o f G, t h e n a ls o G is a k-s wit c h in g o f F .

Remar k 1: L et 1 · k < k0 · n. If the mapping F : Fqn ! Fqn is a k-switching of
G : Fqn ! Fqn, then there is an Fq-basis of Fqn with respect to which exactly k0 coordinate
functions of F and G di®er. Indeed, let

F ( x ) =
kX

i=1

fi ( x) °i +
nX

i=k+1

ai ( x ) °i;

and

G( x ) =
kX

i=1

gi ( x ) °i +
nX

i=k+1

ai ( x ) °i;

with respect to an Fq-basis ( °1; : : : ; °n ) and fi ( x ) 6= gi ( x ) for all 1 · i · k. Then with respect
to the Fq-basis

( °1; : : : ; °k¡1;
k0X

j=k

°j; ; °k+1; : : : ; °n ) ;



M. Evoyan, G. Kyureghyan and M. Kyureghyan 7

the coordinate functions of F and G are as follows:

F ( x) =
k¡1X

i=1

fi ( x) °i + fk ( x )

0
@

k0X

j=k

°j

1
A +

k0X

i=k+1

( ai ( x ) ¡ fk ( x ) ) °i +
nX

i=k0+1

ai ( x ) °i;

and

G ( x) =
k¡1X

i=1

gi ( x ) °i + gk ( x )

0
@

k0X

j=k

°j

1
A +

k0X

i=k+1

( ai ( x) ¡ gk ( x ) ) °i +
nX

i=k0+1

ai ( x ) °i:

In t h e fo llo win g fo r a s u b s e t S µ Fqn we u s e hSi t o d e n o t e t h e Fq-s u b s p a c e s p a n n e d b y
S.

T heor em 1: L et 1 · k · n and F; G : Fqn ! Fqn. Then the following statements are
equivalent:

(i) F is a k-switching of G.

(ii) The image set of the mapping F ¡ G : x 7! F ( x ) ¡ G( x ) spans a k-dimensional vector
space over Fq.

P r oof. L e t F : Fqn ! Fqn b e a k-s wit c h in g o f G : Fqn ! Fqn . Th e n t h e r e is a n Fq-b a s is
( °1; : : : ; °n ) o f Fqn s u c h t h a t

F ( x ) =
kX

i=1

fi ( x) °i +
nX

i=k+1

ai ( x) °i;

a n d

G( x ) =
kX

i=1

gi ( x ) °i +
nX

i=k+1

ai ( x ) °i;

wh e r e fi ( x ) 6= gi ( x) fo r a ll 1 · i · k. H e n c e

F ( x ) ¡ G( x ) =
kX

i=1

( fi ( x) ¡ gi ( x ) ) °i;

s h o win g t h a t t h e d im e n s io n l o f hImage ( F ¡ G) i is le s s o r e qu a l t o k. N o w le t ( ±1; : : : ; ±` )
b e a b a s is fo r hImage( F ¡ G ) i, a n d ( ±1; : : : ; ±`; : : : ±n ) a b a s is fo r Fqn . L e t hi; uj : Fqn ! Fq
b e s u c h t h a t

F ( x ) ¡ G ( x ) =
X̀

i=1

hi ( x) ±i;

a n d

G ( x) =
nX

i=1

ui ( x ) ±i:

Th e n

F ( x ) =
X̀

i=1

( ui ( x ) + hi ( x ) ) ±i +
nX

i=`+1

ui ( x ) ±i;

a n d t h u s k · ` b y t h e d e ¯ n it io n o f k-s wit c h in g , c o m p le t in g t h e p r o o f.
P r oposition 2: L et 1 · k · n and F : Fqn ! Fqn be a k-switching of G : Fqn ! Fqn :
Then F and G have exactly qn¡k ¡ 1 equal component functions over Fq .



8 On k-Switching of Mappings on Finite Fields

P r oof. L e t ® 2 Fqn ; ® 6= 0 . Th e n T rqn=q ( ®F ( x ) ) = T rqn=q ( ®G ( x ) ) h o ld s if a n d o n ly if
T rqn=q ( ®( F ( x) ¡ G ( x ) ) ) is t h e c o n s t a n t z e r o fu n c t io n , o r , t h e e qu iva le n t , im a g e s e t o f t h e
m a p p in g F ¡ G is c o n t a in e d in t h e h yp e r p la n e H® = fy 2 Fqn : T rqn=q ( ®y ) = 0 g. N o t e
t h a t Image( F ¡ G) µ H® if a n d o n ly if t h e lin e a r s p a n hImage ( F ¡ G ) i is a s u b s p a c e o f
H®. B y Th e o r e m 1 t h e d im e n s io n o f hImage ( F ¡ G ) i is k. H e n c e , t h e r e a r e q

n¡k¡1
q¡1 d i®e r e n t

h yp e r p la n e s c o n t a in in g hImage ( F ¡ G ) i. To c o m p le t e t h e p r o o f it r e m a in s t o r e c a ll t h a t
H® = H®0 if a n d o n ly if ®0 = ® ¢ u wit h a n o n -z e r o u 2 Fq.

3 . k-S wit c h in g o f t h e Id e n t it y Ma p p in g

In t h is s e c t io n we c o n s id e r b ije c t ive k-s wit c h in g o f t h e id e n t it y m a p p in g . A s t h e n e xt o b s e r -
va t io n s h o ws t h e s t u d y o f k-s wit c h in g o f a n a r b it r a r y p e r m u t a t io n o n Fqn c a n b e r e d u c e d t o
o n e o f t h e id e n t it y m a p p in g s . L e t F : Fqn ! Fqn b e a k-s wit c h in g o f G : Fqn ! Fqn , a n d
e it h e r F o r G b e a p e r m u t a t io n o n Fqn . W it h o u t lo s s o f g e n e r a lit y, s a y F is a p e r m u t a t io n
a n d d e n o t e it s in ve r s e m a p p in g b y F ¡1. Th e n

G( x ) = F ( x ) +
kX

i=0

fi ( x ) ¢ °i; ( 1 )

wit h r e s p e c t t o s o m e b a s is ( °1; : : : ; °n ) . N o t e t h a t ( 1 ) h o ld s if a n d o n ly if

G ± F ¡1 ( x ) = x +
kX

i=0

fi ± F ¡1 ( x) ¢ °i;

t h a t is wh e n G ± F ¡1 is a k-s wit c h in g o f t h e id e n t it y m a p p in g . H e n c e , u n d e r s t a n d in g o f
t h e b e h a vio u r o f k-s wit c h in g o f t h e id e n t it y m a p p in g is a n im p o r t a n t s t e p fo r t h e g e n e r a l
p r o b le m .

Th e r e m a in in g p a r t o f t h is s e c t io n is d e vo t e d t o a c la s s o f p e r m u t a t io n s o b t a in e d b y
k-s wit c h in g u s in g t h e s o -c a lle d fu n c t io n s wit h a lin e a r t r a n s la t o r . Ou r r e s u lt s g e n e r a liz e
s e ve r a l c o n s t r u c t io n s o f p e r m u t a t io n s g ive n in [1 , 3 , 5 , 7 ].

A n o n -z e r o e le m e n t ® 2 Fqn is c a lle d a n a-lin e a r t r a n s la t o r ( o r a-lin e a r s t r u c t u r e , c f. [3 ])
fo r t h e m a p p in g f : Fqn ! Fq if

f ( x + u®) ¡ f ( x ) = ua; ( 2 )

fo r a ll x 2 Fqn ; u 2 Fq a n d s o m e ¯ xe d a 2 Fq.
Th e fo llo win g t h e o r e m fr o m [3 ] a llo ws t o c o n s t r u c t fu n c t io n s wit h lin e a r t r a n s la t o r s e x-

p lic it ly.

T heor em 2: L et G : Fqn ! Fqn and f ( x) = T rqn=q ( G ( x ) ) . Then f has a linear translator
if and only if there is a non-bijective Fq-linear mapping L : Fqn ! Fqn such that

f ( x ) = T rqn=q ( G( x ) ) = T rqn=q ( H ± L( x ) + ¯x ) ;

for some H : Fqn ! Fqn and ¯ 2 Fqn. In this case, any element from the kernel of L is a
linear translator for f .



M. Evoyan, G. Kyureghyan and M. Kyureghyan 9

T heor em 3: L et 1 · k · n, ¸1; ¸2; ¢ ¢ ¢ ; ¸k 2 Fqn be linearly independent over Fq and
fj : Fqn ! Fq, j = 1 ; : : : ; k. F urther, suppose ¸i is a bj;i-linear translator for fj, where
i; j 2 f 1 ; 2 ; ¢ ¢ ¢ ; kg. Set

B :=

0
BBBB@

1 + b1;1 b1;2 ¢ ¢ ¢ b1;k
b2;1 1 + b2;2 ¢ ¢ ¢ b2;k

. ..

bk;1 bk;2 ¢ ¢ ¢ 1 + bk;k

1
CCCCA

;

and let F : Fqn ! Fqn be de¯ned as

F ( x ) = x + ¸1f1 ( x ) + ¸2f2 ( x ) + ¢ ¢ ¢ + ¸kfk ( x ) :

Then F ( x ) = F ( y ) for some x; y 2 Fqn if and only if

x = y + ¸1a1 + ¸2a2 + : : : + ¸kak;

and

0
BB@

a1
...
ak

1
CCA 2 Fq

n belongs to the kernel of B. In particular, the mapping F is a qn¡r-to-1

on Fqn where r is the rank of the matrix B.

P r oof. L e t x; y 2 Fqn b e s u c h t h a t F ( x) = F ( y ) . Th e n , b y t h e d e ¯ n it io n o f F

x + ¸1f1 ( x) + ¸2f2 ( x ) + ¢ ¢ ¢ + ¸kfk ( x ) = y + ¸1f1 ( y ) + ¸2f2 ( y ) + ¢ ¢ ¢ + ¸kfk ( y ) ;

a n d t h u s
x = y + ¸1a1 + ¸2a2 + : : : + ¸kak;

fo r s o m e e le m e n t s ai 2 Fq. Ob s e r ve , t h a t wh e n ai 2 Fq, t h e n

F

Ã
y +

kX

i=1

¸iai

!
= y +

kX

i=1

¸iai +
kX

j=1

¸j fj

Ã
y +

kX

i=1

¸iai

!

= y +
kX

i=1

¸iai +
kX

j=1

¸j

Ã
fj ( y ) +

kX

i=1

bj;iai

!

= y +
kX

j=1

¸j fj ( y ) +
kX

i=1

¸iai +
kX

j=1

¸j

Ã
kX

i=1

bj;iai

!

= F ( y ) +
kX

j=1

¸j

0
@( bj;j + 1 ) aj +

kX

i=1;i 6=j
bj;iai

1
A :

To c o m p le t e t h e p r o o f it r e m a in s t o n o t e t h a t t h e lin e a r in d e p e n d e n c e o f ¸j im p lie s

F ( y ) +
kX

j=1

¸j

0
@( bj;j + 1 ) aj +

kX

i 6=j;i=1
bj;iai

1
A = F ( y ) ;

if a n d o n ly if a ll c o e ± c ie n t s

( bj;j + 1 ) aj +
kX

i 6=j;i=1
bj;iai = 0 ;



1 0 On k-Switching of Mappings on Finite Fields

o r e qu iva le n t ly if

0
BB@

a1
...

ak

1
CCA is in t h e ke r n e l o f t h e m a t r ix B.

Th e o r e m 1 r e d u c e s t o Th e o r e m 8 fr o m [5 ] wh e n k = 1 . Fo r k = 2 ; 3 , it e xt e n d s Th e o r e m
1 0 fr o m [5 ] a n d Th e o r e m 1 , 2 fr o m [7 ] c o n s id e r a b ly.

Remar k 2: L et ( °1; : : : ; °k ) be linear independent over Fq and f1 ( x ) ; : : : ; fk ( x ) non-zero
functions from Fqn into Fq. Suppose the k-switching of the identity mapping

x +
kX

j=1

fj ( x) °j;

is a permutation on Fqn. Note that in this case the ( k ¡ 1 ) -switching

x +
k¡1X

j=1

fj ( x ) °j;

must not be a permutation on Fqn . This follows easily from Theorem 3 . Indeed, there are
non-singular matrices over Fq whose leading principal ( n ¡ 1 ) £ ( n ¡ 1 ) minor is 0.
Cor ollar y 1: W ith the notations of Theorem 3 , the mapping F is bijective on Fqn if and
only if the matrix B has a full rank. L et B¡1 be the inverses matrix of B. D e¯ne the
functions hj : Fqn ! Fq, j = 1 ; : : : ; k by

0
BB@

h1 ( x )
...

hk ( x)

1
CCA := B

¡1 ¢

0
BB@

f1 ( x )
...

fk ( x)

1
CCA :

Then the inverse mapping of F ( x) is given by

F ¡1 ( x ) = x ¡
kX

j=1

¸jhj ( x) :

P r oof. L e t Bj b e t h e j t h r o w in t h e m a t r ix B. Th e c a lc u la t io n s in t h e p r o o f o f Th e o r e m
3 s h o w t h a t

F

0
@x ¡

kX

j=1

¸jhj ( x )

1
A = x +

kX

j=1

¸j fj ( x ) ¡
kX

j=1

¸jBj ¢

0
BB@

h1 ( x )
...

hk ( x)

1
CCA

= x +
kX

j=1

¸j fj ( x ) ¡
kX

j=1

¸jBj ¢ B¡1 ¢

0
BB@

f1 ( x )
...

fk ( x )

1
CCA

= x +
kX

j=1

¸j fj ( x ) ¡
kX

j=1

¸jfj ( x )

= x:

Th e in ve r s e m a p p in g o f t h e p e r m u t a t io n F o b t a in e d in Co r o lla r y 1 is a k-s wit c h in g o f
t h e id e n t it y m a p p in g a s we ll. Th e n e xt p r o p o s it io n s h o ws t h a t t h is p r o p e r t y h o ld s fo r a ll
p e r m u t a t io n s o b t a in e d via s wit c h in g fr o m t h e id e n t it y m a p p in g .



M. Evoyan, G. Kyureghyan and M. Kyureghyan 1 1

P r oposition 3: If a permutation F : Fqn ! Fqn is a k-switching of the identity mapping
of Fqn, then its inverse is a k-switching of the identity mapping as well.

P r oof. L e t ( °1; : : : ; °k ) b e lin e a r in d e p e n d e n t o ve r Fq a n d f1 ( x ) ; : : : ; fk ( x ) : Fqn ! Fq
b e n o n -z e r o fu n c t io n s . Fu r t h e r s u p p o s e t h a t t h e m a p p in g

F ( x) = x +
kX

j=1

fj ( x) °j;

is a p e r m u t a t io n o f Fqn . If F
¡1 is t h e in ve r s e m a p p in g o f F , t h e n

x = F ± F ¡1 ( x ) = F ¡1 ( x) +
kX

j=1

( fj ± F ¡1 ) ( x ) °j :

Th u s

F ¡1 ( x ) = x ¡
kX

j=1

( fj ± F ¡1 ( x ) ) °j;

im p lyin g t h e r e s u lt .

Refer ences

[1 ] A . A kb a r y, D . Gh io c a a n d Q. W a n g , \ On c o n s t r u c t in g p e r m u t a t io n s o f ¯ n it e ¯ e ld s " ,
F inite F ields Appl., vo l. 1 7 , p p . 5 1 -6 7 , 2 0 1 1 .

[2 ] L . B u d a g h ya n , C. Ca r le t a n d G. L e a n d e r , \ Co n s t r u c t in g n e w A P N fu n c t io n s fr o m
kn o wn o n e s " , F inite F ields Appl., vo l. 1 5 , p p . 1 5 0 -1 5 9 , 2 0 0 9 .

[3 ] P . Ch a r p in a n d G. K yu r e g h ya n , \ W h e n d o e s G ( x) + °Tr ( H ( x) ) p e r m u t e Fpn ?" , F inite
F ields Appl., vo l. 1 5 , p p . 6 1 5 -6 3 2 , 2 0 0 9 .

[4 ] Y . E d e l a n d A . P o t t , \ A n e w a lm o s t p e r fe c t n o n lin e a r fu n c t io n wh ic h is n o t qu a d r a t ic " ,
Adv. M ath. Commun., vo l. 3 , p p . 5 9 -8 1 , 2 0 0 9 .

[5 ] G. K yu r e g h ya n , \ Co n s t r u c t in g p e r m u t a t io n s o f ¯ n it e ¯ e ld s via lin e a r t r a n s la t o r s " , J .
Combin. Theory Ser. A, vo l. 1 1 8 , p p . 1 0 5 2 -1 0 6 1 , 2 0 1 1 .

[6 ] G. K yu r e g h ya n a n d Y in Ta n , \ On a fa m ily o f p la n a r m a p p in g s " , E nhancing crypto-
graphic primitives with techniques from error correcting codes, NATO Sci. P eace Secur.
Ser. D Inf. Commun. Secur. 23, IOS , A m s t e r d a m , p p . 1 7 5 -1 7 8 , 2 0 0 9 .

[7 ] M. K yu r e g h ya n a n d S . A b r a h a m ya n , \ A m e t h o d o f c o n s t r u c t in g p e r m u t a t io n p o lyn o -
m ia ls o ve r ¯ n it e ¯ e ld s " , Int. J . Information Theories and Applications, vo l. 1 7 , p p .
3 2 8 -3 3 4 , 2 0 1 0 .

[8 ] A . P o t t a n d Y . Zh o u , \ S wit c h in g c o n s t r u c t io n o f p la n a r fu n c t io n s o n ¯ n it e ¯ e ld s " ,
Arithmetic of ¯nite ¯elds, L ecture Notes in Comput. Sci. 6087, S p r in g e r , B e r lin , p p . 1 3 5 -
1 5 0 , 2 0 1 0 .

Submitted 14.12.2012, accepted 11.02.2013.



1 2 On k-Switching of Mappings on Finite Fields

ì»ñç³íáñ ¹³ßï»ñÇ íñ³ ³ñï³å³ïÏ»ñáõÙÝ»ñÇ k-÷á˳ñÏáõÙÝ»ñÇ Ù³ëÇÝ

Ø. ¾íáÛ³Ý, ¶. ÎÛáõñ»ÕÛ³Ý ¨ Ø. ÎÛáõñ»ÕÛ³Ý

²Ù÷á÷áõÙ

öá˳ñÏáõÙÝ»ñÁ ÏÇñ³éíáõÙ »Ý í»ñç»ñë ÉáõÛë ï»ë³Í í»ñç³íáñ ¹³ßï»ñÇ íñ³
Ûáõñ³Ñ³ïáõÏ ³ñï³å³ïÏ»ñáõÙÝ»ñÇ Ï³éáõóÙ³ÝÁ ÝíÇñí³Í ³ß˳ï³ÝùÝ»ñáõÙ: ²Ûë
³ß˳ï³ÝùáõÙ ÁݹѳÝñ³óíáõÙ ¿ ÷á˳ñÏÙ³Ý Ñ³ëϳóáõÃÛáõÝÁ ÙÇÝ㨠k -÷á˳ñÏáõÙ,
áñï»Õ · k · n , ÇÝãå»ë ݳ¨ Ý»ñϳ۳óíáõÙ »Ý k-÷á˳ñÏáõÙÝ»ñÇ ÁݹѳÝáõñ
ѳïÏáõÃÛáõÝÝ»ñ ¨ Ýϳñ³·ñíáõÙ ¿ k-÷á˳ñÏáõÙÝ»ñÇ ÙÇçáóáí ï»Õ³÷áËáõÃÛáõÝÝ»ñÇ
ϳéáõóÙ³Ý Ù»Ãá¹:

Î k-îáìåíàõ îòîáðàæåíèé íà êîíå÷íûõ ïîëåé

Ì. Ýâîÿí, Ã. Êþðåãÿí è Ì. Êþðåãÿí

Àííîòàöèÿ

Êîíñòðóêöèÿ îáìåíà èñïîëüçóåòñÿ â íåñêîëüêèõ íåäàâíèõ ðàáîòàõ ïî
ïîñòðîåíèþ ñïåöèôè÷åñêèõ îòîáðàæåíèé íà êîíå÷íûõ ïîëåé.  ýòîé ñòàòüå
ïîíÿòèå îá îáìåíå îáîáùåíî äî k-îáìåíà ñ · k · n . Òàêæå ïðåäñòàâëåíû
íåêîòîðûå îáùèå ñâîéñòâà k-îáìåíà è îïèñàí ìåòîä ïîëó÷åíèÿ ïåðåñòàíîâîê ñ
èñïîëüçîâàíèåì k-îáìåíà.