, RSA, , -, , .
:
y2 + a × x × y + b × y = x 3 + c × x 2 + d × x + e.
, : y 2 + y = x 3 x 2. . 4.1 . , . (0, 0), (1, 1), (1, 0) D (0, 1).
. 4.1. . |
:
O, , ;
, ;
, O.
.
1. O : + O = .
2. , P 1 = (x, y) P 2 = (x, y) ( O), 1 + 2 = O 1 = 2.
3. P Q (. . 4.2) , . P Q, , S. P + Q + S = , , P + Q = S, P + Q = T.
4. - P Q, S = P S = Q .
5. Q, Q S . Q + Q =
2 × Q = S.
. 4.2. . |
, . k k .
, . , :
.
Ep (a, b). b
.
. , 0 £ £ , . , , . , Ep (a, b) . , y, ( , y = 0). (x, y) Ep (a, b).
|
|
Ep (a, b) :
1. + O = .
2. = (x, y), + (x, y) = O. (x, y) . , (x, y) Ep (a, b).
3. = (x 1, y 1) Q = (x 2, y 2), x 1 ¹ x 2, P + Q = (x 3, y 3) :
,
,
.
4. = (x 1, y 1) Q = (x 2, y 2), x 1 = x 2, y 1 = y 2 ¹ 0, P + Q = (x 3, y 3) :
,
,
.
λ , P = (x 1, y 1) Q = (x 2, y 2). P = Q , λ.
/ . , , Pm (x, y). Ep (a, b) G .
n, Q Q = n × G.
: k Cm, :
.
Cm :
.
Pm k ´ Q, k, , Q , k ´ Q. k, k ´ G. k ´ G , , . , k, , .
k, G k ´ G. , : P Q Ep (a, b), k < p , P = k ´ Q.