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BY (11) 9864
(13) C1
(46) 2007.10.30
(19)
(12)
(51)
(2006)
G 06F 7/38
N
(54)
(21)
(22) 2005.11.04
(43) 2006.04.30
(71)
:
: a 20051067
(73)
(BY)
(BY)
(72)
:
:
;
;
(BY)
(56) BY 5079 C1, 2003.
BY 5224 C1, 2003.
BY 5472 C1, 2003.
RU 2018928 C1, 1994.
SU 1783516 A1, 1992.
US 4229802, 1980.
JP 4128922 A, 1992.
(57)
(N ≥ 2)
N
,
D-
-
,
,
-
,
, j- ( j = 1, N )
,
,
i-
,
,
, (N + 3)D-
j(i = 1, 2, 3) DD- (N + 1)-
,
i(N + 2)-
,
BY 9864 C1 2007.10.30
-
D(N + 4)-
,
(N + 7)- .
.1
BY 9864 C1 2007.10.30
,
.
,
,
[1].
.
N
-
.
,
D-
[2].
-
N n-
.
N n,
.
.
N
, j-
(N ≥ 2)
D( j = 1, N)
.
jD-
,
-
.
,
i-
.
(i = 1, 2, 3) DD-
,
i-
D-
.
(N + 1)-
.
-
(N + 2)(N + 3)-
-
D,
D-
(
.
(N + 4)-
. 1)
D(N + 7)- .
N
.
D-
2, 3
6,
1,
51 – 5N,
4, N
7
8, 9
10.
.
i
n-1
=2
n-2
xi,n + 2
N nxi,n-1 +…+ 2xi,2 + xi,1, xi,j ∈ {0,1}, i = 1, N, j = 1, n,
:
 N

R = 4r1 + 2r2 + r3 =  ∑ X i  mod 7,
 i =1 
(1)
r1, r2, r3 ∈ {0,1}.
,
n = 3k, k = 1, 2, 3,…
2
(1)
:
BY 9864 C1 2007.10.30
k
k
N k

R =  ∑  ∑ 23l − 3 x i ,3l − 2 + ∑ 23l − 2 x i , 3l −1 + ∑ 23l −1 x i , 3l   mod 7 =
l =1
l =1

 i =1 l =1
N
N
k N

=  ∑   ∑ 23l − 3 x i , 3l − 2  mod 7 +  ∑ 23l − 2 x i ,3l −1  mod 7 +  ∑ 23l −1 x i , 3l  mod 7   mod 7 =

 i =1

 i =1


 l =1  i =1
(2)
N
N
k N

=  ∑   ∑ x i , 3l − 2  mod 7 +  2 ∑ x i, 3l −1  mod 7 +  4 ∑ x i , 3l  mod 7   mod 7.

 i =1

 i =1


 l =1  i =1
(2)
,
xi,1, xi,4, …, xi,n-2
xi,2, xi,5, …, xi,n-1 21,
(2),
0
2,
) j, 1 ≤ j ≤ n,
(
xi,3, xi,6, …, xi,n 2 , i = 1, N .
j
j
j
j
S j = 4s1 + 2s 2 + s 3 , s1 , s 2j , s 3j ∈ {0,1}
2
(
)
:
1
j
N
S j =  ∑  2 (t mod 3−1)mod 3 ∑ x i , t  mod 7  mod 7.
i =1

 t =1

,
,
(3)
(- l)mod 3 = 2.
Sn = R s1n = r1 , s n2 = r2 , s 3n = r3 .
(3)
:
N
S j =  2( j mod 3−1) mod 3 ∑ x i , j + S j−1  mod 7 =
i =1


N
∑
j −1
j −1
j −1 
 x i , j + 4s1 + 2s 2 + s 3  mod 7,
 i =1

N
=  2 ∑ x i , j + 4s1j −1 + 2s 2j −1 + s3j−1  mod 7,
 i =1

N
 4 ∑ x + 4s j −1 + 2s j −1 + s j−1  mod 7,

1
2
3 
 i =1 i , j

j mod 3 = 1;
(4)
j mod 3 = 2;
j mod 3 = 0,
S0 ≡ 0.
,
(4)
 N x + s j−1 + s j −1 + s j−1 + s j−1 + s j−1 + s j−1 + s j −1  mod 7,
 ∑ i, j 1
1
1
1
2
2
3 
 i =1

:
j mod 3 = 1;
 N

S j =  2 ∑ x i , j + s1j−1 + s1j −1 + s 2j−1 + s 3j−1 + s 3j −1 + s 3j−1 + s 3j−1   mod 7,

  i =1
j mod 3 = 2;
 N
j −1
j −1
j −1
j −1
j −1
j −1
j −1  
 4 ∑ x i , j + s1 + s 2 + s 2 + s 2 + s 2 + s 3 + s 3   mod 7,
=
i
1



j mod 3 = 0.
(5)
:
2(s 3j−1 + s 3j−1 + s 3j−1 + s 3j−1 )mod 7 = (8s 3j−1 )mod 7 = s 3j−1 ;
4(s 2j−1 + s 2j−1 + s 2j−1 + s 2j−1 )mod 7 = (16s 2j−1 )mod 7 = 2s 2j−1 ;
4(s 3j−1 + s 3j−1 )mod 7 = (8s 3j−1 )mod 7 = s 3j−1 .
3
(5)
BY 9864 C1 2007.10.30
,
(5)
N n-
Xi
(1 ≤ j ≤ n)
x1,j, x2,j, …, xN,j
nN
,
j-
Sj-1:
s1j −1 , s1j −1 , s1j −1 , s1j −1 , s 2j −1 , s 2j −1 , s 3j −1
j mod 3 = 1;
s1j −1 , s1j −1 , s 2j −1 , s 3j −1 , s 3j −1 , s 3j −1 , s 3j −1
j mod 3 = 2;
s1j −1 , s 2j −1 , s 2j −1 , s 2j −1 , s 2j −1 , s 3j −1 , s 3j −1
j mod 3 = 0.
-
– 20
j mod 3 = l, 21
N+7
j mod 3 = 2, 22
j mod 3 = 0.
1,
s1j , s 2j
s 3j
Sj
2, 3
2
(
2)
4.
1
2,
21) 20 ) -
(
(
3,
-
4.
2, 3
1,
1,
4
2-
4-
1.
,
m
20):
(
S = 4sl + 2s2 + s3 = (y1 + y2 +…+ ym)mod7,
s1, s2, s3 ∈ {0,1} ,
,
.
21
,
(2
(6)
21
22.
,
+ 2 2 +…+ 2 m) mod7 = (2(4s1 + 2s2 + s3))mod7 =
= (8s1 + 4s2 + 2s3) mod7 = 4s2 + 2s3 + s1;
(6)
(4y1 + 4y2 +…+ 4ym) mod7 = (4(4s1 + 2s2 + s3))mod7 =
= (l6s1 + 8s2 + 4s3) mod7 = 4s3 + 2s1 + s2.
(7)
1
,
s2
,
s3
s1
-
,
-
(7)
22
,
s3
,
s1
s2
,
-
-
.
(5), (6)
j−1
1
: Q , Q
4
-
.
j−1
2
Q
j−1
3
-
(7).
2,
3
Q10 ≡ 0, Q 02 ≡ 0, Q ≡ 0, 1 ≤ j ≤ n.
:
0
3
j-
4
BY 9864 C1 2007.10.30
N

S j = 4s1j + 2s 2j + s 3j =  ∑ x i , j + Q1j −1 + Q1j −1 + Q 2j−1 + Q3j−1 + Q3j−1 + Q3j−1 + Q3j−1  mod 7;
 i =1


j
s1 ,
j mod 3 = 1;


j
j
Q1 = s 3 ,
j mod 3 = 2;

j

s2 ,
j mod 3 = 0;

j

s2 ,
j mod 3 = 1;

j
j

Q 2 = s1 ,
j mod 3 = 2;

s 3j ,
j mod 3 = 0;


s 3j ,
j mod 3 = 1;


Q3j = s 2j ,
j mod 3 = 2;

j

s1 ,
j mod 3 = 0.

(8)
(
n = 3k, k = 1, 2, 3,…).
j = 1:
N
S1 = 4s11 + 2s12 + s13 =  ∑ x i,1 + 0 + 0 + 0  mod 7;
 i=1


1
1
1
1
1
1

Q1 = s1 ; Q 2 = s 2 ; Q 3 = s 3 .
j = 2:
N
S 2 = 4s12 + 2s 22 + s 32 =  ∑ x i , 2 + s11 + s11 + s12 + s13 + s13 + s13 + s13  mod 7;
 i =1


2
2
2
2
2
2

Q1 = s 3 ; Q 2 = s1 ; Q 3 = s 2 .
j = 3:
N
S3 = 4s13 + 2s 32 + s 33 =  ∑ x i ,3 + s 23 + s 32 + s12 + s 22 + s 22 + s 22 + s 22  mod 7;
 i =1

.
3
3
3
3
3
3

Q1 = s 2 ; Q 2 = s 3 ; Q 3 = s1 .
j = 4:
N
S 4 = 4s14 + 2s 42 + s 34 =  ∑ x i , 4 + s 32 + s 32 + s 33 + s13 + s13 + s13 + s 13  mod 7;
 i =1

.
4
4
4
4
4
4

Q1 = s1 ; Q 2 = s 2 ; Q 3 = s 3 .
j = 5:
N
S5 = 4s15 + 2s 52 + s 53 =  ∑ x i ,5 + s14 + s14 + s 42 + s 34 + s 34 + s 34 + s 34  mod 7;
 i =1

.
5
5
5
5
5
5

Q1 = s 3 ; Q 2 = s1 ; Q 3 = s 2 .
j= 6:
N
S6 = 4s16 + 2s 62 + s 63 =  ∑ x i ,6 + s 53 + s 53 + s15 + s 52 + s 52 + s 52 + s 52  mod 7;
 i=1

.

Q16 = s 62 ; Q 62 = s 36 ; Q 63 = s16 .
5
(8)
BY 9864 C1 2007.10.30
j= n −2:
N
S n −2 = 4s1n −2 + 2s n2 −2 + s 3n −2 =  ∑ x i ,n −2 + s n2 −3 + s n2 −3 + s 3n −3 + s 1n −3 + s1n −3 + s1n −3 + s1n −3  mod 7;
 i =1

.
n −2
n −2
n −2
n −2
n −2
n −2

Q1 = s1 ; Q 2 = s 2 ; Q 3 = s 3 .
j = n −1:
N
S n −1 = 4s1n −1 + 2s n2 −1 + s 3n −1 =  ∑ x i ,n −1 + s1n −2 + s1n −2 + s n2 −2 + s 3n −2 + s 3n −2 + s 3n −2 + s 3n −2  mod 7;
 i =1

.
n −1
n −1
n −1
n −1
5
n −1

Q1 = s 3 ; Q 2 = s1 ; Q 3 = s 2 .
j= n:
N
S n = 4s1n + 2s n2 + s 3n =  ∑ x i ,n + s 3n −1 + s 3n −1 + s1n −1 + s n2 −1 + s n2 −1 + s n2 −1 + s n2 −1  mod 7;
 i =1

.
n
n
n
n
n
n

Q1 = s 2 ; Q 2 = s 3 ; Q 3 = s1 .
R = 4r1 + 2r2 + r3 = Sn r1 = s1n = Q n2 , r2 = s n2 = Q 3n , r3 = s 3n = Q1n .
,
n4
r1
,
2r2
,
3r3
N n.
N
.
7
,
2, 3
51, 52,…, 5N
x1,j, x2,j,…, xN,j
1, X2,…, XN (
,
,…,
),
n
,
1,1
2,1
N,1
6.
(n- )
2, 3 4
N
r1, r2 r3
R =  ∑ X i  mod 7 ,
 i =1 
8, 9 10
.
1
4
= 010010111,
(
5
= 100010110.
. 2)
= 010011011,
(
2
= 011011110,
3
(s
9
3
)
=1 ,
(
s =1 ,
(
4-
9
2
)
-
s 2j
s 3j
)
(s19 = 0) .
1
(
4)
(
s1j ,
.2
-
= 100111101,
1
2),
S j = 4s1j + 2s 2j + s 3j .
,
4.
)
(
3)
-
2
3 -
5
, R = 4r1 + 2r2 + r3 =  ∑ X i  mod 7 = S9 = 4s19 + 2s 92 + s 93 = 011 .
 i =1 
:
1
= 155,
2
= 222,
3
= 317,
4
= 151,
6
5
= 278.
BY 9864 C1 2007.10.30
5
R =  ∑ X i  mod 7 = 1123 mod 7 = 3 .
 i =1 
,
n,
.
2, 3 4.
,
n = 3k-1 (k = 1, 2, 3,…)
3,
r3 = s 3n -
,
-
r1 = s1n
r2 = s n2 -
4,
-
2.
r1 = s1n
n = 3k-2 (k = 1, 2, 3,…)
2,
r3 = s 3n -
r2 = s n
2
3,
4.
-
,
-
(
. 2).
1) n = 8:
X1 = 10011011, X2 = 11011110, X3 = 00111101, X4 = 10010111, X5 = 00010110.
.2
,
(
)
8
(s 3 = 0),
3s18 = 0 ,
2
(s
4-
8
2
(
)
)
=1 .
R = 4r1 + 2r2 + r3 =  ∑ X i  mod 7 = S8 = 4s18 + 2s 82 + s 83 = 010 .
 i =1 
5
: X1 = 155,
= 222,
= 151, 5 = 22.
5
R =  ∑ X i  mod 7 = 611 mod 7 = 2 .
 i =1 
2) n = 7:
X1 = 0011011, X2 = 1011110, X3 = 0111101, X4 = 0010111, X5 = 0010110.
.2
,
(
)
7
2
s1 = 0 ,
3 7
7
s2 = 1 ,
4(s 3 = 1) .
3
= 61,
2
(
4
(
)
)
5
R = 4r1 + 2r2 + r3 =  ∑ X i  mod 7 = S 7 = 4s 17 + 2s 72 + s 73 = 011 .
 i =1 
:
3
= 61,
= 27,
2
= 94,
= 23, 5 = 22.
5
R =  ∑ X i  mod 7 = 227 mod 7 = 3 .
 i =1 
4
N
,
-
.
:
1.
2.
1
3707,
5079,
G 06F 7/49, 2000.
G 06F 7/50, 2003 (
7
).
220034, .
,
1
Q1j−1
8
Q 2j−1
9
Q 3j−1
10
Q 3j−1
10
Q 3j−1
10
Q 3j−1
10
2
3
4
1
0
0
0
0
0
0
0
0
s11 = 0
s12 = 1
s13 = 1
0
1
1
0
0
1
1
1
1
1
s 32 = 0
s12 = 1
s 22 = 0
1
1
1
1
0
0
1
0
0
0
0
s 32 = 1
s 33 = 0
s13 = 1
1
1
1
0
0
1
1
0
1
1
1
1
s14 = 0
s 42 = 1
s 34 = 0
5
1
1
1
1
1
0
0
1
0
0
0
0
s 53 = 1
s15 = 1
s 52 = 0
6
0
0
1
0
0
1
1
1
0
0
0
0
s 62 = 1
s 36 = 0
s16 = 0
7
0
1
0
0
0
1
1
0
0
0
0
0
s17 = 0
s 72 = 1
s 37 = 1
8
1
1
0
1
0
0
0
1
1
1
1
1
s 83 = 0
s18 = 0
s 82 = 1
9
0
0
1
0
1
0
0
0
1
1
1
1
s 92 = 1
s 93 = 1
s19 = 0
x 2, j
x 3, j
x 4, j
x 5, j
51
52
53
54
1
1
0
1
2
1
1
3
0
4
, 20.
.
.2
BY 9864 C1 2007.10.30
.
55
Q1j−1
8
x 1, j
j
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