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ɩɟɪɜɚɹ
ܲ
ଵ
= ݍ
ܳ
ଵ
ɜɬɨɪɚɹ
ݍ
ܲ
ଶ
ଶܲଵ
=
ܳ
ݍଶܳଵ+ ܳ
ଶ
ɬɪɟɬɶɹ
ݍ
ܲ
ଷ
ଷܲଶ
=
ܳ
ݍଷܳଶ+ ܳ
ଷ
ɱɟɬɜɺɪɬɚɹ
ݍ
ܲ
ସ
ସܲଷ
=
ܳ
ݍସܳଷ+ ܳ
ସ
ɩɹɬɚɹ
ܲ
ݍ
ହ
ହܲସ
=
ܳ
ݍହܳସ+ ܳ
ହ
ɉɨɥɭɱɟɧɚɢɫɯɨɞɧɚɹɞɪɨɛɶ
Ⱦɨɩɨɥɧɢɬɟɥɶɧɵɟɩɪɢɦɟɪɵ
ɉɪɢɦɟɪ 23 Ɋɚɡɥɨɠɢɬɶɩɪɨɫɬɭɸɞɪɨɛɶ
ɞɹɳɢɟɞɪɨɛɢ
ɇɚɨɫɧɨɜɚɧɢɢɉɪɢɦɟɪɚ
87
[
=
ݍ; ݍଵ, ݍଶ, ݍଷ, ݍସ, ݍହ, ݍ
55
]=[
+ ܲ
+ ܲ
+ ܲ
+ ܲ
1
+
ݍ
ଵ
ଵ
ଵ
ଶ
=
ଶ
ଷ
=
ଷ
1
=0+
1 ή1+0
=
1 ή1+1
2 ή1+1
=
2 ή2+1
2 ή3+1
2 ή5+2
2 ή7+3
2 ή12 + 5
1
=
;
1
1
=
=
=
12
=
଼
ɜɰɟɩɧɭɸɢɧɚɣɬɢɞɥɹɧɟɺɩɨɞɯɨ
ହହ
1; 1,1, 2, 1, 1, 4]=1+
1
2
3
5
7
17
29
,
,
,
.
1+
1+
1
2+
1
1
1+
1
1
1+
.
1
4
ɉɨɞɯɨɞɹɳɢɟɞɪɨɛɢ
ݍ
1
=
,
1
1
ݍ
+ ܲ
ଶܲଵ
ܲ
ହ
=
ܳ
ହ
ܲ
ସ
ܳ
ସ
ݍ
ସܲଷ
=
ݍସܳଷ+ ܳ
+ ܲ
ܲ
=
ܳ
ܲ
ଶ
=
ܳ
ݍଶܳଵ+ ܳ
ଶ
11
ଶ
=
,
7
ଶ
ɉɨɥɭɱɟɧɚɢɫɯɨɞɧɚɹɞɪɨɛɶ
ɉɪɢɦɟɪ 24 Ɋɚɡɥɨɠɢɬɶɩɪɨɫɬɭɸɞɪɨɛɶ
ɯɨɞɹɳɢɟɞɪɨɛɢ
ɇɚɨɫɧɨɜɚɧɢɢɉɪɢɦɟɪɚ
589
[
=
343
=1+
ݍ; ݍଵ, ݍଶ, ݍଷ, ݍସ, ݍହ, ݍ, ݍ
1
1+
1
2+
1+
1+
ɉɨɞɯɨɞɹɳɢɟɞɪɨɛɢ
ݍ
1
=
,
1
1
ݍ
+ ܲ
ଶܲଵ
ܲ
ହ
=
ܳ
ହ
ܲ
=
ܳ
ܲ
ସ
ܳ
ସ
ݍ
ସܲଷ
=
ݍସܳଷ+ ܳ
+ ܲ
ܲ
=
ܳ
ܲ
ଶ
=
ܳ
ݍଶܳଵ+ ܳ
ଶ
12
ଶ
=
,
7
ଶ
ɉɨɥɭɱɟɧɚɢɫɯɨɞɧɚɹɞɪɨɛɶ
ܲ
ଵ
ܳ
ଵ
3
=
,
2
ݍ
ହܲସ
ݍହܳସ+ ܳ
1
1
1
1
6+
2+
ܲ
ଵ
ܳ
ଵ
5
=
,
3
ݍ
+ ܲ
ହܲସ
ݍହܳସ+ ܳ
ݍ
+ ܲ
ܲ
ݍܳ+ ܳ
= ݍ
+ ܲ
1
3
= ݍ
1
+
ݍ
ଵ
ݍ
ܲ
ଷ
=
ܳ
ݍଷܳଶ+ ܳ
ଷ
19
ଷ
=
,
12
ଷ
ହ଼ଽ
ɜɰɟɩɧɭɸɢɧɚɣɬɢɞɥɹɧɟɺɩɨɞ
ଷସଷ
]=[
1; 1,2, 1,1, 6, 2, 3
=1+
+ ܲ
ଷܲଶ
ܲ
ܳ
1
=
1
ଵ
ଵ
=
2
,
1
8
=
,
5
ݍ
ܲହ
ݍܳହ+ ܳ
+ ܲ
]
87
ସ
=
.
55
ସ
.
1
+
ݍ
ଵ
ݍ
ܲ
ଷ
=
ܳ
ݍଷܳଶ+ ܳ
ଷ
79
ଷ
=
,
46
ଷ
589
ହ
=
343
ହ
=1+
+ ܲ
ଷܲଶ
ܲ
ܳ
.
1
=
1
ଵ
ଵ
=
2
,
1
7
=
,
4
ݍ
+ ܲ
ܲହ
ݍܳହ+ ܳ
170
ସ
=
,
99
ସ
ɋɪɚɜɧɟɧɢɹɩɟɪɜɨɣɫɬɟɩɟɧɢ
ɗɬɨ ɥɢɧɟɣɧɵɟɭɪɚɜɧɟɧɢɹɜɢɞɚ
ܽݔ ؠ ܾ(mod ݉
)
,
ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɵɟɜɬɟɪɦɢɧɚɯɫɪɚɜɧɟɧɢɣ Ɂɞɟɫɶܽ, ݔ, ܾ, ݉ – ɰɟɥɵɟɱɢɫɥɚɂɫɤɨ
ɦɵɦɹɜɥɹɟɬɫɹݔɚɨɫɬɚɥɶɧɵɟɱɢɫɥɚɢɡɜɟɫɬɧɵ
(
ȿɫɥɢ
)
ܽ, ݉
=1, ɬ ɟɱɢɫɥɚɜɡɚɢɦɧɨɩɪɨɫɬɵɟɬɨɫɭɳɟɫɬɜɭɟɬɟɞɢɧɫɬɜɟɧɧɨɟ
ɪɟɲɟɧɢɟɫɪɚɜɧɟɧɢɹɤɨɬɨɪɨɟɦɨɠɧɨɧɚɣɬɢɧɚɩɪɢɦɟɪɩɨɮɨɪɦɭɥɟ
ݔؠ(െ1
)
ܾܲ
௦ିଵ
(
mod ݉
)
,
௦
ɝɞɟ ܲ
– ɱɢɫɥɢɬɟɥɶ ɩɪɟɞɩɨɫɥɟɞɧɟɣ ɩɨɞɯɨɞɹɳɟɣ ɞɪɨɛɢ ɩɪɢ ɪɚɡɥɨɠɟɧɢɢ
௦ିଵ
ɜ
ɰɟɩɧɭɸɞɪɨɛɶ
ȿɫɥɢ
ܽήܿؠ1(mod ݉
)
,
ɬɨɱɢɫɥɚa ɢc ɧɚɡɵɜɚɸɬɫɹ ɦɭɥɶɬɢɩɥɢɤɚɬɢɜɧɨɨɛɪɚɬɧɵɦɢɩɨɦɨɞɭɥɸm
ɉɨɧɹɬɢɟ ɦɭɥɶɬɢɩɥɢɤɚɬɢɜɧɨɣ ɨɛɪɚɬɧɨɫɬɢɦɨɠɧɨ ɫɱɢɬɚɬɶɨɛɨɛɳɟɧɢɟɦ ɩɨɧɹɬɢɹ
ɜɡɚɢɦɧɨɨɛɪɚɬɧɵɯɱɢɫɟɥ
3 ή
1
3
=1, 5ή
1
=1,…
5
ɫɬɟɦɨɬɥɢɱɢɟɦɱɬɨ ɪɚɜ ɟ ɧɫɬ ɜɨ ɟɞɢɧɢɰɟɡɚɦɟɧɹɟɬɫɹ ɫɪɚɜɧɢɦɨɫɬɶɸɫɟɞɢ
ɧɢɰɟɣɩɨɞɚɧɧɨɦɭɦɨɞɭɥɸɫɪɚɜɧɟɧɢɹ
ɉɪɢɦɟɪ 25 Ɋɟɲɢɬɶɫɪɚɜɧɟɧɢɟ
3ݔؠ20
(
mod 161).
ଵଵ
ȼɨ-ɩɟɪɜɵɯɧɭɠɧɨɪɚɡɥɨɠɢɬɶɩɪɨɫɬɭɸɞɪɨɛɶ
ɜ ɰɟɩɧɭɸɇɚɨɫɧɨɜɚɧɢɢ
ଷ
ɉɪɢɦɟɪɚ
161
3
=[53; 1,2
]
=53+
1
1+
.
1
2
ɉɨɞɯɨɞɹɳɢɟɞɪɨɛɢ
ݍ
ܲ
=
ܳ
53
=
1
1
,
ܲ
ଶ
ܳ
ଶ
ܲ
ଵ
ܳ
ଵ
ݍ
ଶܲଵ
=
ݍଶܳଵ+ ܳ
= ݍ
+ ܲ
1
+
=53+
ݍ
ଵ
161
=
3
1
=
1
,
1
.
ହସ
భ
=
ɉɪɟɞɩɨɫɥɟɞɧɹɹɩɨɞɯɨɞɹɳɚɹɞɪɨɛɶ– ɷɬɨ
ɫɥɟɞɧɟɣɩɨɞɯɨɞɹɳɟɣɞɪɨɛɢܲ
= ܲଵ=54ɨɬɤɭɞɚݏ =2.
௦ିଵ
Ɍɨɝɞɚɱɢɫɥɢɬɟɥɶɩɪɟɞɩɨ
ଵ
ொ
భ
Ɋɟɲɟɧɢɟɫɪɚɜɧɟɧɢɹ
ݔؠ(െ1
௦
)
(
mod ݉)ؠ(െ1
ܾܲ
௦ିଵ
ؠ(1080 െ161 ή6
ଶ
)
ή20 ή 54
)(
mod 161)=114(mod 161).
(
mod 161)= 1080(mod 161
ɉɨɤɚɠɟɦɤɚɤ ɦɨɠɧɨ ɫɞɟɥɚɬɶɩɪɨɜɟɪɤɭɷɬɨɝɨ ɪɟɡɭɥɶɬɚɬɚɉɨ ɨɩɪɟɞɟɥɟɧɢɸ
ɫɪɚɜɧɟɧɢɹɩɨɦɨɞɭɥɸ
െ114
(
ݔെ114)ڭ 161, ˑ˕ˍ˖ˇ˃
ݔ
161
= ݇אԺ.
Ɍɚɤɢɦɨɛɪɚɡɨɦɧɚɣɞɟɧɧɵɟɡɧɚɱɟɧɢɹɯ ɩɪɢɧɚɞɥɟɠɚɬɫɟɪɢɢɪɟɲɟɧɢɣ
ݔ
=114+161݇.
ɉɪɢɞɚɜɚɹk ɤɨɧɤɪɟɬɧɨɟɰɟɥɨɟɡɧɚɱɟɧɢɟɢɩɨɞɫɬɚɜɥɹɹɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣɯ ɜ
ɢɫɯɨɞɧɨɟɫɪɚɜɧɟɧɢɟɦɨɠɧɨɩɪɨɜɟɪɢɬɶɭɞɨɜɥɟɬɜɨɪɹɟɬɫɹɥɢɫɪɚɜɧɟɧɢɟ
ɉɭɫɬɶɧɚɩɪɢɦɟɪ݇ = െ1. Ɍɨɝɞɚݔ
=114െ161 = െ47, ɢɢɦɟɟɦ
ିଵ
3 ή(െ47)ؠ 20
(
mod 161).
ɗɬɨɜɟɪɧɨɟɫɪɚɜɧɟɧɢɟɱɬɨɢɞɨɤɚɡɵɜɚɟɬɩɪɚɜɢɥɶɧɨɫɬɶɪɟɲɟɧɢɹ
Ⱦɨɩɨɥɧɢɬɟɥɶɧɵɟɩɪɢɦɟɪɵ
ɉɪɢɦɟɪ 26 Ɋɟɲɢɬɶɫɪɚɜɧɟɧɢɟ
55ݔؠ7
(
mod 87).
)
Ɋɚɡɥɨɠɢɦɞɪɨɛɶ
଼
ɜɰɟɩɧɭɸɇɚɨɫɧɨɜɚɧɢɢɉɪɢɦɟɪɚ
ହହ
87
[
=
ݍ; ݍଵ, ݍଶ, ݍଷ, ݍସ, ݍହ, ݍ
55
]=[
1; 1,1, 2, 1, 1, 4]=1+
ɉɨɞɯɨɞɹɳɢɟɞɪɨɛɢ
ݍ
1
=
=
1
,
1
1 ή2+1
1 ή1+1
ܲ
ହ
=
ܳ
ହ
ܲ
ଶ
ܳ
ଶ
ܲ
ܳ
ݍ
ଶܲଵ
=
ݍଶܳଵ+ ܳ
=
+ ܲ
ɉɪɟɞɩɨɫɥɟɞɧɹɹɩɨɞɯɨɞɹɳɚɹɞɪɨɛɶ– ɷɬɨ
ɩɨɞɯɨɞɹɳɟɣɞɪɨɛɢܲ
= ܲହ=19ɨɬɤɭɞɚݏ =6.
௦ିଵ
Ɋɟɲɟɧɢɟɫɪɚɜɧɟɧɢɹ
௦
ݔؠ(െ1
)
(
mod ݉)ؠ(െ1
ܾܲ
௦ିଵ
ؠ(133 െ87
)(
mod 87)=46(mod 87).
ૢ
12
1
1+
1+
1
1
2+
1+
ܲ
ଵ
= ݍ
ܳ
ଵ
=
,
)
+
3
,
2
ܲ
ܳ
ொ
ή7 ή 19
1
ݍ
ଵ
ܲ
ܳ
87
=
55
ఱ
=
ఱ
(
mod 87)=133(mod 87
1
=1+
ଷ
=
ଷ
2
=
,
1
1
8
5
ܲ
11
ସ
,
=
,
7
ܳ
ସ
.
ଵଽ
ɑɢɫɥɢɬɟɥɶɩɪɟɞɩɨɫɥɟɞɧɟɣ
ଵଶ
1
1
1+
)
.
1
4
ɉɪɨɜɟɪɤɚ
ݔ
(
ݔെ46)ڭ 87, ˑ˕ˍ˖ˇ˃
െ46
87
= ݇אԺ.
ɉɨɥɭɱɚɟɦɫɟɪɢɸɪɟɲɟɧɢɣ
ݔ=46+87݇.
ɉɭɫɬɶɧɚɩɪɢɦɟɪ݇ =2. Ɍɨɝɞɚݔଶ=46+87ή2=220, ɢɢɦɟɟɦ
55 ή220 ؠ 7
(
mod 87).
ɗɬɨ ɜ ɟɪ ɧɨɟ ɫɪɚɜɧɟɧɢɟɱɬɨɢɞɨɤɚɡɵɜɚɟɬɩɪɚɜɢɥɶɧɨɫɬɶɪɟɲɟɧɢɹ
ɉɪɢɦɟɪ ɇɚɣɬɢɱɢɫɥɚɦɭɥɶɬɢɩɥɢɤɚɬɢɜɧɨɨɛɪɚɬɧɵɟɱɢɫɥɭ ɩɨɦɨɞɭ
ɥɸ
Ɍɪɟɛɭɟɬɫɹɪɟɲɢɬɶɫɪɚɜɧɟɧɢɟ
29ݔؠ1
(
mod 17).
ଵ
Ɋɚɡɥɨɠɢɦɞɪɨɛɶ
ɜɰɟɩɧɭɸɇɚɨɫɧɨɜɚɧɢɢɉɪɢɦɟɪɚ22
ଶଽ
1+
1
1
1
17
[
=
ݍ; ݍଵ, ݍଶ, ݍଷ, ݍସ, ݍ
29
]=[
0; 1,1, 2, 2, 2]=0+
ହ
1+
2+
2+
ɉɨɞɯɨɞɹɳɢɟɞɪɨɛɢ
ݍ
ܲ
ଶ
ܳ
ଶ
ܲ
ܳ
ݍ
ଶܲଵ
=
ݍଶܳଵ+ ܳ
=
+ ܲ
0
=
1
1
=
ܲ
,
ܳ
1 ή1+0
1 ή1+1
ܲ
ܳ
ଵ
= ݍ
ଵ
=
ହ
=
ହ
1
2
17
29
1
+
ݍ
ଵ
ܲ
,
ܳ
=0+
3
ଷ
=
5
ଷ
1
1
=
,
1
1
ܲ
ૠ
ସ
,
=
12
ܳ
ସ
.
ɉɪɟɞɩɨɫɥɟɞɧɹɹɩɨɞɯɨɞɹɳɚɹɞɪɨɛɶ– ɷɬɨ
ɧɟɣɩɨɞɯɨɞɹɳɟɣɞɪɨɛɢܲ
= ܲସ=7ɨɬɤɭɞɚݏ =5.
௦ିଵ
ర
=
ɑɢɫɥɢɬɟɥɶɩɪɟɞɩɨɫɥɟɞ
ଵଶ
ொ
ర
Ɋɟɲɟɧɢɟɫɪɚɜɧɟɧɢɹ
ݔؠ(െ1
௦
)
ܾܲ
௦ିଵ
ؠ(െ7+17
(
mod ݉)ؠ(െ1
)(
mod 17)=10(mod 17).
)
ହ
ή1 ή 7
(
mod 17
)
= െ7
(
mod 17
.
1
1
2
,
)
ɉɪɨɜɟɪɤɚ
(
ݔെ10)ڭ 17,
ݔ
െ10
17
= ݇אԺ.
ɉɨɥɭɱɚɟɦɫɟɪɢɸɪɟɲɟɧɢɣ
ݔ=10+17݇.
ɉɭɫɬɶɧɚɩɪɢɦɟɪ݇ = െ2. Ɍɨɝɞɚݔିଶ=10െ17 ή2=െ24, ɢɢɦɟɟɦ
29 ή(െ24)ؠ 1
(
mod 17).
ɗɬɨ ɜɟɪɧɨɟɫɪɚɜɧɟɧɢɟ; ɪɟɲɟɧɢɟɜɟɪɧɨ
ɉɪɢɦɟɪ 28 ɇɚɣɬɢɱɢɫɥɚɦɭɥɶɬɢɩɥɢɤɚɬɢɜɧɨɨɛɪɚɬɧɵɟɱɢɫɥɭ ɩɨ ɦɨɞɭ
ɥɸ
Ɋɟɲɢɦɫɪɚɜɧɟɧɢɟ
23ݔؠ1
(
mod 40).
ସ
Ɋɚɡɥɨɠɢɦɞɪɨɛɶ
ɜɰɟɩɧɭɸɇɚɨɫɧɨɜɚɧɢɢɉɪɢɦɟɪɨɜɢ2
ଶଷ
1+
1
2+
1
1
1+
1
5
40
[
=
ݍ; ݍଵ, ݍଶ, ݍଷ, ݍ
23
]=[
1; 1,2, 1, 5]=1+
ସ
ɉɨɞɯɨɞɹɳɢɟɞɪɨɛɢ
ܲ
ݍ
ܲ
ଶ
ܳ
ଶ
ܳ
ݍ
ଶܲଵ
=
ݍଶܳଵ+ ܳ
=
+ ܲ
1
=
1
1
=
ܲ
,
ܳ
2 ή2+1
2 ή1+1
ଵ
ଵ
= ݍ
=
1
+
ݍ
ଵ
5
3
ܲ
,
ܳ
=1+
ૠ
ଷ
=
4
ଷ
1
2
=
,
1
1
ܲ
40
ସ
,
=
.
23
ܳ
ସ
య
=
ɉɪɟɞɩɨɫɥɟɞɧɹɹɩɨɞɯɨɞɹɳɚɹɞɪɨɛɶ– ɷɬɨ
ɩɨɞɯɨɞɹɳɟɣɞɪɨɛɢܲ
= ܲଷ=7ɨɬɤɭɞɚݏ =4.
௦ିଵ
ɑɢɫɥɢɬɟɥɶɩɪɟɞɩɨɫɥɟɞɧɟɣ
ସ
ொ
య
Ɋɟɲɟɧɢɟɫɪɚɜɧɟɧɢɹ
ݔؠ(െ1
௦
)
(
mod ݉)ؠ(െ1
ܾܲ
௦ିଵ
)
ସ
ή1 ή 7
(
mod 40)=7(mod 40).
ɉɪɨɜɟɪɤɚ
ݔ
(
ݔെ7)ڭ 40,
െ7
40
= ݇אԺ,
ݔ=7+40݇.
ɉɭɫɬɶ݇ = െ1. Ɍɨɝɞɚݔ
=7െ40 = െ33;
ିଵ
23 ή(െ33)ؠ 1
(
mod 40)
ɟɫɬɶ ɜ ɟɪ ɧ ɨɟ ɫɪɚɜɧɟɧɢɟɱɬɨɞɨɤɚɡɵɜɚɟɬɩɪɚɜɢɥɶɧɨɫɬɶɪɟɲɟɧɢɹ
Ɏɭɧɤɰɢɹɗɣɥɟɪɚɔ(ݖ
)
ɉɭɫɬɶɞɥɹɫɨɫɬɚɜɧɨɝɨɰɟɥɨɝɨɱɢɫɥɚz ɢɡɜɟɫɬɧɨɟɝɨɤɚɧɨɧɢɱɟɫɤɨɟɪɚɡɥɨɠɟɧɢɟ
ݖ =
ן
ן
భ
ଵ
ଶ
మ
…
ן
ೖ
.
Ɍɨɝɞɚɡɧɚɱɟɧɢɟɮɭɧɤɰɢɢɗɣɥɟɪɚɜɵɱɢɫɥɹɟɬɫɹɤɚɤ
െ1
ɔ(ݖ
ן1െ1
)
=
1
1െ1൯
൫
ן2െ1
2
2െ1൯…
൫
ן
݇
݇െ1
൫
݇
.
൯
ɑɚɫɬɧɵɟɫɥɭɱɚɢ
ɚ߮()= െ1, ɟɫɥɢɪ – ɩɪɨɫɬɨɟɱɢɫɥɨ
ɛ߮(ήݍ
)=(
െ1)(ݍെ1), ɟɫɥɢ, ݍ – ɩɪɨɫɬɵɟɱɢɫɥɚ
ɉɪɢɦɟɪɵ ɜɵɱɢɫɥɟɧɢɹ
(2)
߮
߮
߮
(12)
(
1000
=1;
= ߮
)
(
= ߮
ଶή3ଵ
2
(
ଷή5ଷ
2
)
=2
)
ଶିଵ
=2
(
2 െ1)ή3
ଷିଵ
(
2 െ1)ή5
ଵିଵ
(
3 െ1
ଷିଵ
(
5 െ1
)
=4;
)
=400.
ɋɦɵɫɥɮɭɧɤɰɢɢɗɣɥɟɪɚ
ij] – ɷɬɨɤɨɥɢɱɟɫɬɜɨɧɚɬɭɪɚɥɶɧɵɯɱɢɫɟɥɦɟɧɶɲɢɯɱɟɦz ɢɜɡɚɢɦɧɨɩɪɨɫɬɵɯɫ
ɧɢɦ
ɉɪɢɦɟɪ 29 Ɍɨɥɶɤɨ ɱɬɨ ɛɵɥɨ ɜɵɱɢɫɥɟɧɨ ɡɧɚɱɟɧɢɟ ߮
(12)
=4. ɉɨɤɚɠɟɦ
ɷɬɨɢɫɯɨɞɹɢɡɫɦɵɫɥɚɮɭɧɤɰɢɢ
ȼɵɩɢɲɟɦ ɧɚɬɭɪɚɥɶɧɵɟɱɢɫɥɚ ɦɟɧɶɲɢɟ ɞɜɟɧɚɞɰɚɬɢɢ ɜɵɱɟɪɤɧɟɦ ɢɡɧɢɯ
ɬɟɤɨɬɨɪɵɟɢɦɟɸɬɫɢɧɵɟɨɛɳɢɟɞɟɥɢɬɟɥɢɤɪɨɦɟɟɞɢɧɢɰɵɬ ɟɧɟɜɡɚɢɦɧɨ
ɩɪɨɫɬɵɟɫ
,2,3,4,,6,ૠ,8,9,10,.
ɂɬɚɤɜɡɚɢɦɧɨɩɪɨɫɬɵɟɫɞɜɟɧɚɞɰɚɬɶɸ– ɱɟɬɵɪɟɱɢɫɥɚ
, ,ૠ, .
ɆɚɥɚɹɬɟɨɪɟɦɚɎɟɪɦɚ
Ⱦɥɹɥɸɛɨɝɨ ɩɪɨɫɬɨɝɨɪ ɢɥɸɛɨɝɨɰɟɥɨɝɨܽ1ɧɟɞɟɥɹɳɟɝɨɫɹɧɚɪɜɵ
ɩɨɥɧɹɟɬɫɹ
ܽ
ିଵ
ؠ 1(mod
)
.
ɗɬɨ ɭɫɥɨɜɢɟ ɩɪɨɫɬɨɬɵɱɢɫɥɚ ɪ ɹɜɥɹɟɬɫɹɧɟɨɛɯɨɞɢɦɵɦ ɧɨ ɤ ɫɨɠɚɥɟɧɢɸ
ɧɟ ɹɜɥɹɟɬɫɹ ɞɨɫɬɚɬɨɱɧɵɦɟɫɥɢ ɫɪɚɜɧɟɧɢɟɜɟɪɧɨ ɨɬɫɸɞɚɟɳɺ ɧɟɫɥɟɞɭɟɬ ɱɬɨ
ɪ – ɩɪɨɫɬɨɟɱɢɫɥɨ
Ɂɚɦɟɬɢɦ ɱɬɨ ɛɟɪɹ ܽ < ɦɵ ɚɜɬɨɦɚɬɢɱɟɫɤɢɭɞɨɜɥɟɬɜɨɪɹɟɦ ɬɪɟɛɨɜɚɧɢɸ
ɱɬɨɛɵɚ ɧɟɞɟɥɢɥɨɫɶɧɚɪ
ɉɪɢɦɟɪ 30 ɉɪɨɢɥɥɸɫɬɪɢɪɭɟɦɦɚɥɭɸɬɟɨɪɟɦɭɎɟɪɦɚɧɚɩɪɢɦɟɪɟɩɪɨɫɬɨɝɨ
=13 ɢɧɟɞɟɥɹɳɟɝɨɫɹɧɚɧɟɝɨɰɟɥɨɝɨܽ=27
ɜɟɪɧɨɥɢɫɪɚɜɧɟɧɢɟ
27
ଵଷିଵ
ؠ 1
(
mod 13) ?
ɂɧɵɦɢɫɥɨɜɚɦɢɛɭɞɟɬɥɢɪɚɡɧɨɫɬɶ
27
ଵଶ
െ1=
ଷ)ଵଶ
(
3
െ1=3ଷെ1
ɞɟɥɢɬɶɫɹɛɟɡɨɫɬɚɬɤɚɧɚ"
ଷ
3
െ1
13
ଵ଼
(
3
=
(
=
ଵ଼
+1)(3
13
ଵ଼
3
+1)(3ଽ+1)(3
െ1
ଵ଼
(
)
3
=
+1)(3ଽ+1)(3
ଷ
െ1
13
)(
3+3ଷ+1
13
ଽ
െ1
)
.
)
=
ɉɨɫɤɨɥɶɤɭ
ଷ
3
13
െ1
27 െ1
=
13
=2,
ɬɨɢɩɟɪɜɨɧɚɱɚɥɶɧɚɹɞɪɨɛɶɢɦɟɟɬɰɟɥɨɟɡɧɚɱɟɧɢɟɱɬɨɢɬɪɟɛɨɜɚɥɨɫɶɩɨɤɚɡɚɬɶ
Ɍɟɨɪɟɦɚɗɣɥɟɪɚ
Ⱦɥɹɥɸɛɨɝɨ ɧɚɬɭɪɚɥɶɧɨɝɨm ɢɥɸɛɨɝɨɰɟɥɨɝɨܽ1ɜɡɚɢɦɧɨɩɪɨɫɬɨɝɨɫ
mɜɵɩɨɥɧɹɟɬɫɹ
)
ܽ
ఝ(
ؠ 1(mod ݉
)
.
Ɍɟɨɪɟɦɚ ɗɣɥɟɪɚɹɜɥɹɟɬɫɹ ɨɛɨɛɳɟɧɢɟɦ ɦɚɥɨɣ ɬɟɨɪɟɦɵɎɟɪɦɚ ɩɨɫɤɨɥɶɤɭ
ɭɫɥɨɜɢɟɩɪɨɫɬɨɬɵɪ ɜɩɨɫɥɟɞɧɟɣɡɚɦɟɧɹɟɬɫɹɛɨɥɟɟɦɹɝɤɢɦɬɪɟɛɨɜɚɧɢɟɦɰɟɥɨɫɬɢ
mɇɨɟɫɥɢ ݉ = ɬ ɟɹɜɥɹɟɬɫɹ ɩɪɨɫɬɵɦɬɨ ɩɨ ɫɜɨɣɫɬɜɭɮɭɧɤɰɢɢ ɗɣɥɟɪɚ
߮(݉)= ߮()= െ1
ɉɪɢɦɟɪ ɉɪɨɢɥɥɸɫɬɪɢɪɭɟɦɬɟɨɪɟɦɭɗɣɥɟɪɚ ɧɚɩɪɢɦɟɪɟɧɚɬɭɪɚɥɶɧɨɝɨ
݉ =11 ɢɜɡɚɢɦɧɨɩɪɨɫɬɨɝɨɫɧɢɦܽ =7
ɜɟɪɧɨɥɢɫɪɚɜɧɟɧɢɟ
)
7
ఝ(ଵଵ
ؠ 1
(
mod 11) ?
ɂɧɵɦɢɫɥɨɜɚɦɢɛɭɞɟɬɥɢɪɚɡɧɨɫɬɶ
)
ఝ(ଵଵ
7
െ1=7ଵെ1
ɞɟɥɢɬɶɫɹɛɟɡɨɫɬɚɬɤɚɧɚ"
ଵ
7
െ1
11
ହ
(
=
)(
7
െ1
7ହ+1
11
)
=
(
16807 െ1
)(
16807 + 1
11
)
16806 ή16808
=
11
= 16806 ή 1528 .
ɉɨɥɭɱɢɥɨɫɶɞɟɥɟɧɢɟɧɚɰɟɥɨɱɬɨɢɬɪɟɛɨɜɚɥɨɫɶɩɨɤɚɡɚɬɶ
ɉɪɟɞɫɬɚɜɥɟɧɧɵɯ ɫɜɟɞɟɧɢɣ ɢɡ ɬɟɨɪɢɢ ɱɢɫɟɥ ɞɨɫɬɚɬɨɱɧɨ ɱɬɨɛɵ ɩɨɞɪɨɛɧɨ
ɪɚɫɫɦɨɬɪɟɬɶɚɥɝɨɪɢɬɦ56$ ɚɫɢɦɦɟɬɪɢɱɧɨɝɨɲɢɮɪɨɜɚɧɢɹ
ȺɅȽɈɊɂɌɆRSA
ȻɨɛɠɟɥɚɟɬɩɨɥɭɱɚɬɶɫɟɤɪɟɬɧɵɟɫɨɨɛɳɟɧɢɹɨɬȺɥɢɫɵɫɢɫɩɨɥɶɡɨɜɚɧɢɟɦɚɥ
ɝɨɪɢɬɦɚ56$ ɚɫɢɦɦɟɬɪɢɱɧɨɝɨɲɢɮɪɨɜɚɧɢɹɂɯɞɟɣɫɬɜɢɹ
ɒɚɝȻɨɛɚɈɧɛɟɪɺɬɞɜɚɛɨɥɶɲɢɯɩɪɨɫɬɵɯɱɢɫɥɚɪ ɢq ɢɜɵɱɢɫɥɹɟɬɢɯ
ɩɪɨɢɡɜɟɞɟɧɢɟ
݊ = ήݍ .
ʛːˑˉˋ˕ˈˎˋ ɪ ɢqɞɨɥɠɧɵɯɪɚɧɢɬɶɫɹȻɨɛɨɦ ɜɫɬɪɨɝɨɣɬɚɣɧɟɚɢɯɩɪɨ
ɢɡɜɟɞɟɧɢɟn ɬɚɣɧɨɣ ɧɟɹɜɥɹɟɬɫɹ
ɉɪɢɫɨɜɪɟɦɟɧɧɨɦɛɵɫɬɪɨɞɟɣɫɬɜɢɢɗȼɆɞɥɹɭɫɬɨɣɱɢɜɨɫɬɢɲɢɮɪɚɤɜɡɥɨɦɭ
ɬɪɟɛɭɟɬɫɹɱɬɨɛɵɪ ɢq ɛɵɥɢɱɢɫɥɚɦɢɩɨɪɹɞɤɚɧɟɦɟɧɟɟ10
ଵ
ɜɞɟɫɹɬɢɱɧɨɣɫɢ
ɫɬɟɦɟɫɱɢɫɥɟɧɢɹɢɥɢɡɚɧɢɦɚɥɢɧɟ ɦɟɧɟɟɪɚɡɪɹɞɨɜɩɪɢɡɚɩɢɫɢɜ ɞɜɨɢɱɧɨɣ
ɫɢɫɬɟɦɟɬ ɟɪ ɢq – ɩɨɪɹɞɤɚ ɛɢɬɢɥɢɄɛɢɬ
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