О периодичности функционирования генераторов псевдослучайных чисел RC4, IA, IBAA

Citation preview

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И

ШЭ

RC4, IA, IBAA RC4 ( м.[1]). N v0

VN

ZN

ZN

SN

N,

, SN

–

ZN

–

N

).

n

2 (n=8 (i0 , j0 , s0 ) (0, 0, s ) -

t

(it , j t , s t ) ,

vt

vt

RC4. VN .

RC4

[1; ) : it 1 1 jt 1 st 1 (it ) s t 1  ( s t 1 (it ), s t 1 ( j t )) t

it jt st t

s t ( s (it )

s t ( j t ))

(

),

–

N, st , 

SN ,

st ( z ) –

z

–

( x, y) –

SN ,

y.

x

-

RC4: j

...0,N-1,N-2,…,2,1 s Тео ем 1.

Q =1,2,…N-1,0,1,2… st , t

2n

{1, 2,...}

s0 . n 1

s0 (1) 1 2 st , t

{1, 2,...}

s0

, N

IA ( м.[2]).

1

2n .

(i,q,S),

IA

S : Zm

m=2 K

2 , q K

2n

,

0, i

-

Z 2K

n

Z 2K

Zm .

IA IA [1; ) ( St , qt , it )

t

( St 1 , qt 1 , it 1 )

.

(i0

0, q0

,

0, S0 ) . qt

:

it

it

1

st [it ]

1 mod(m) , st 1[ st 1[it ](mod m)] qt

st [ j ]

,

n )(mod m)] st 1[it ] ,

st [( st [it ]

qt

1

2K, ( s[it ]

j

St ,

n) -

+ -

s[it ]

n

(

),

n

Zm .

-

IA: q ...0,N-1,N-2,…,2,1 s

Тео ем 2.

Q =1,2,…m-1,0,1,2… 0

(q0

0, S0 )

m=2n,

St , t {1, 2,...}

1, qt , t {1, 2,...}

m. Тео ем 3.

1,

St , t {1, 2,...}

(S )

,

:

S0 [1]

S0 [ S0 [1](mod m)]

S0 [2]

S0 [ S0 [2](mod m)] q1 = S0 [ S0 [2](mod m)] S0 [( S0 [1]

n )(mod m)] S0 [1]

S0 [3]

S0 [ S0 [3](mod m)] q 2 = S0 [ S0 [3](mod m)] S0 [( S0 [2]

n )(mod m)] S0 [2]

S0 [4]

S0 [ S0 [4](mod m)] q3 = S0 [ S0 [4](mod m)] S0 [( S0 [3]

n )(mod m)] S0 [3]

………………………………………………………………………………. n )(mod m)] S0 [m 1] S0 [0] S0 [ S0 [m 1](mod m)] q m 1 = S0 [ S0 [0](mod m)] S0 [( S0 [m 1] 0

St , t {1, 2,...}

IBAA. ( . . n=8), K=32. +-

xor (

m 2n , K 2n , p

:

2 ; m;

0, S0 ) ,

1. m=256 p+q= .

q 2);

K

(q0

p -

(p

p );

(q

q

q

).

i Z m , a Z K , q Z K , S (S[1], S[2],..., S[m]) –

(i ,a, S ,q) ,

i S[i]

n,

Zm

Z K , S[i ] -

S, S[S[i] nmod m] S[i]

n

modm

S. t. (i0

at

((at

p) (at

qt

St [ St [it ]

1

m ], 2

q)) St 1[it

1

0, a0 , S0 , q0 ) .

n mod m] St 1 [it ] ,

St [it ] St 1 [ St 1 [it ]mod m] at

qt 1 .

qt , t {1,2,...}

. Тео ем 4. (i0

m,

0, a0 , S0 , q0 ) (i0

(IBAA) ( S j , j {0,1,2,...}

IBAA . IBAA

0, a0 , S0 , q0 )

), m. (IBAA):

1) ,

*q1 ,*q2 ,...,*q j ,... ; *a2 ,*a3 ,...,*a j ,...

m. *at

F * at

*qt

S0 [ S0 [it ]

1

m ], 2 n mod m] S0 [it ] ,

(1)

S0 [it

(2) (3)

S0 [it ] S0 [ S0 [it ]mod m] *at *qt 1 ,

,

, 1,2,3,…, m-1,0,1.2.3…;

2) *q1

S0 [ S0 [1]

n mod m] S0 [1] ,

*q2

S0 [ S0 [2]

n mod m] S0 [2] ,

…………………………………………. *qm 1 S0 [ S0 [m 1] n mod m] S0 [m 1] , *qm S0 [ S0 [0] n mod m] S0 [0] , *a1 Fa0

S0 [1

m ], 2

it

m

*a2

m ], 2 m ], 2

F * a1 S0 [2

*a3 F * a2

S0 [3

…………………….. *a j

F * aj

m ], 2

S0 [ j

1

……………………… *am

F * am

1

*am F * am *am

1

2

S0 [m 1

1

S0 [0

F * am

S0 [1

:

m ], 2

m ], 2 m ] 2

*a1

S0 [1] S0 [S0 [1]mod m] q0 ,

*a2

S0 [2] S0 [S0 [2]mod m] *q1 ,

…………………………… *am

S0 [0] S0 [ S0 [0]mod m] *qm 1 , *am

. .,

1. 2012. 2.

. .,

1

S0 [1] S0 [ S0 [1]mod m] *qm .

. . RC-4. . . IA. . № 3 2013.

№2 .8, .