> For the complete documentation index, see [llms.txt](https://docs.cooku222.kr/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.cooku222.kr/security/crypto/cryptohack/cryptohack-elliptic-curves-point-addition.md).
# \[CryptoHack] Elliptic Curves, Point Addition
```
from sympy import mod_inverse
# Define parameters
p = 9739
a = 497
b = 1768
# Elliptic curve: y^2 = x^3 + a*x + b mod p
def point_add(P, Q):
if P == Q:
return point_double(P)
if P is None:
return Q
if Q is None:
return P
x1, y1 = P
x2, y2 = Q
if x1 == x2 and (y1 + y2) % p == 0:
return None
m = ((y2 - y1) * mod_inverse(x2 - x1, p)) % p
x3 = (m**2 - x1 - x2) % p
y3 = (m * (x1 - x3) - y1) % p
return (x3, y3)
def point_double(P):
if P is None:
return None
x, y = P
m = ((3 * x**2 + a) * mod_inverse(2 * y, p)) % p
x3 = (m**2 - 2 * x) % p
y3 = (m * (x - x3) - y) % p
return (x3, y3)
# Given points
P = (493, 5564)
Q = (1539, 4742)
R = (4403, 5202)
# Compute S = P + P + Q + R
P2 = point_double(P)
PQ = point_add(P2, Q)
S = point_add(PQ, R)
```
이전문제까지는 익스 안 짜도 됐는데.. 여기서부턴 계산 복잡해져서 익스를 무조건 짜야한다
```
crypto{4215,2162}
```
---
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