, ( ) :
, , . 5.2. 180. ' , . ' . , . 5.3. (. 5.4 5.5).
5.2 5.3
5.4 5.5
, , , .
, , "", T = 4 mk. n = 4 mk. ʳ (4 mk)!, . 8´8 .
5.2
, (, ) , , . . , .
|
|
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5.2.1
. . , ≠ , - . .
5.1 . , , .
5.1- - j
, , - . , - . (. 5.2) .
, 5.2, , , .
5.2 - j
, , , , , , , , . .
5.2.2
|
|
|
, . 5´5, (. 5.6).
| λ | ε | υ | ω | γ |
| ρ | ζ | δ | σ | ο |
| μ | η | β | ξ | τ |
| ψ | π | θ | α | ς |
| χ | ν | - | φ | ί |
5.6 ,
24-
, . , . ,
τ α υ ρ ο σ χ
ς φ δ μ τ ξ λ.
.
5.2.3
30- :
ŪȲ
, , .
- . .
. 5.3.
5.3 - ³
, . , 1 13 22 1 3 11 20 : ².
5.2.4
n n , . 5.3. , 30, .
: , , , , .
, , , :
5.4 -
5.4 , , .
. . ʳ 315, 14348907, 14 !
|
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5.2.5
= { 1, 2, , n }, n , , , , . 5.5, , .
5.5 -
5.5, . , .
, , . , , . , , . , , , , , , .
, 5, , , , , 13, 7, 2 3 , ' ,
(2,3,5,7,13) = 2 3 5 7 13 = 2730,
.
5.2.6
. m. m , , .
:
,b: → ,
,b: t → ,b(t),
,b(t) = at + b (mod m),
a, b , 0 ≤ a, b < m, ( ) (a, m) = 1.
, t, , at + b m.
, ,b(t) 
, a m , a m . , m = 26, a = 3, b = 5. (3,26) = 1 :
5.6 - ³
| t | |||||||||||||
| 3 t + 5 |
| t | |||||||||||||
| 3 t + 5 |
|
|
|
, :
5.7 - ³
| B | C | D | E | F | G | H | I | J | K | L | M | N | O | |
| F | I | L | O | R | U | X | A | D | G | J | M | P | S | V |
| P | Q | R | S | T | U | V | W | X | Y | Z |
| Y | B | E | H | K | N | Q | T | W | Z | C |
HOPE AVYR.
: (a, b). .
5.2.7
. .
k, 0 
k < 25, . , . DPLOMAT k = 5.
, , k:
0 1 2 3 4 5 10 15 20 25
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
D I P L O M A T
:
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
V W X Y Z D I P L O M A T B C E F G H J K N Q R S U
.
SEND MORE MONEY
HZBY TCGZ TCBZS
, '. ( ) . ,
k = 3 :
5.8 -
, . ׳ .
5.2.8
. , . , .
4´8. . , . 5.9
5.9 -
, , . , .
,
IJ
ӲƲ
, .
5.2.9
.
. 5.9.
:
³ (). , . , .
|
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:
(, , . 5.9), , . ( . .) ;
, , (, ). , (, );
, , (, ). , (, ).
:
(. 5.9) :
.
5.3
5.3.1 ³
Dz. ˲ .
. , ³ ( ).
5.3.2
, , . . , . , , ( ), , . , , ³ ( ), ³ ( )
, e ( ), 2718, :
| - | |||||||||||||||||||
| - |
, , , , , . , , , .
5.3.3
1854 볺 ӿ , . 㳺 . ӿ . , " " (.5.7), , , . ֳ ׳ . " " ͳ .
. . . . , . , . .
5.7 " "
, . ˳ 1 2 . ˳ 5 4 . , 2 4, 1 5 . , , 5 2 , , 1 4 , .
, . , . , . . :
. " " , .
³ . , 1926 ó , AT&T , ( m = 2).
{, , , D,..., Z}, (, . .), (b 0, b 1, , b 4) . k 0, k 1, k 2,... . , , 2 k.
y = x 
k.
2 - k:
x= y k = x k k.
, , ( 2) .
, , .
, , . , ' , , . ֳ .
ϳ . , .
6.1 1
. :
;
;
6.1.1 6.1.3
. , 5 , , 5 , 17 . 12.
a ≡ b (mod n)
: a (, , ) b n, k,
a ≡ b + k n.
a ≡ b (mod n), b a n..
17mod12 = 5 17≡ 5 (mod12) 5 17 12.
. - n, , - , n.
a ≡ b (mod n) c ≡ d (mod n) :
(a c) ≡ (b d) mod n, ac ≡ bd mod n
(a c) mod n ≡ (a mod n c mod n) mod n,
(ac) mod n ≡ [(a mod n)(c mod n)] mod n.
(a (b + c) mod n ≡{[a b(mod n)] +[a c (mod n)]} mod n.
. (15 + 20) mod 4 ≡ 15 mod 4 +20 mod 4 ≡(3+0) mod 4≡3 mod 4.
a ≡ b (mod n), at ≡ bt mod n, t . ij , 
mod n, 
.
. 18 ≡ 4 mod 14 t =2, 36 ≡ 8 mod 14, t = 2, , 
= 0,5 - .
:
a ≡ b (mod n),
a m ≡ b m mod n ≡(b mod n) m mod n.
. 530 mod 3 ≡ (52 mod 3)15 mod 3 ≡ (1 mod 3)15 mod 3≡ 1 mod 3.
, . , , .
n k - , 2k . .
ϳ n
mod n
, .
, 8 mod n, :
((2 mod n) 2 mod n) 2 mod n..
, 2:
= 25 (10)→ 1 1 0 0 1 (2), 25 = 24 + 23 + 20.
25 mod n = ( · 24) mod n = ( · 8· 16) mod n == [ ·(( 2) 2)2 ·((( 2)2)2)2] mod n...
1,5 k , k .
, .
() b, (, b) (, b) , b .
, (18,9) = 9; (7,3) = 1. (, b) = 1, b .
, >0, b>0, >b.
≡ mod b, >0;
b ≡ d mod , d>0;
c ≡ e mod d, e>0;
...........................
m ≡ 0 mod n, n>0.
(, b) = n..
. (21, 15).
21 ≡ 6 mod 15, 21 = 1·15 + 6;
15 ≡ 3 mod 6, 15 = 2·6 + 3;
6 ≡ 0 mod 3 6 =2·3 +0.
, (21, 15) = 3.
n>1
, . , , p q :
n = p· q
n,
(mod n) ≡ 1 mod n.
- 1≡ mod n,
n - .
