2022 | 第十五届全国大学生信息安全竞赛 | 华北赛区 | Crypto

Diophantine

#z3

解题思路¶

在 PoW 之后有两个判断，枚举一下就可以知道答案

Hello Everyone, let's start to learn Equations!
Now I have an elliptic curve equation: (x+1)^2 == 2(y+1)^2 - 3^2
First, tell me if there is a solution to this equation?[Y/N] Y
Great!
Then, tell me whether this equation has an infinite number of solutions?[Y/N] Y
Very well!

接下来才是重点，需要给出 \(202\) 组解 ΣΣΣ(Φ ωΦ||¡)

Now, please find me the solutions to 202 point{(x, y) where x > 0, y > 0}

一开始以为是根据 elliptic curve equation: (x+1)^2 == 2(y+1)^2 - 3^2 找对应椭圆曲线及其上的点，但是椭圆曲线一般是 \(y^2=x^3+ax+b\) 的形式，\((x+1)^2 == 2(y+1)^2 - 3^2\) 怎么变形都不像

后来意识到椭圆曲线方程可能指点的坐标为整数，直接用 z3 解方程应该行

from z3 import *
x, y = Ints('x y')
s = Solver()
s.add((x + 1) ** 2 == 2 * (y + 1) ** 2 - 9)
s.add(x > 0, y > 0)
for _ in range(202):
    s.check()
    m = s.model()
    print(f'{m[x]}, {m[y]}')
    s.add(x > m[x], y > m[y])

但是获得了几组解之后，速度就慢下来了 :( 想着画一下曲线图（其实没必要 > <），总之就是两条增长速度不一样的曲线
```
g1 = plot((x + 1) ** 2, 0, 100)
g2 = plot(2 * (x + 1) ** 2 - 9, 0, 100, linestyle="--")
(g1 + g2).show()
```
再回过头看看已经获得的解，很有规律的感觉(ŏωŏ)
```
(2, 2)
(20, 14)
(122, 86)
(716, 506)
(4178, 2954)
(24356, 17222)
```
当 \(i\ge 2\) 时，满足 \(x_i=6\times x_{i-1}-x_{i-2}+4\), \(y_i=6\times y_{i-1}-y_{i-2} + 4\)

from hashlib import md5
import itertools
import pwn

table = 'abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789'

conn = pwn.remote('39.104.61.18', 12113)
target = conn.recvline_contains('md5').decode().strip()
base = target[target.find('+') + 2:target.find(')')]
target = target[-5:]

for ch in itertools.permutations(table, 4):
    m = ''.join(ch) + base
    h = md5(m.encode()).hexdigest()
    if (h[:5] == target):
        conn.sendafter('>', f'{m[:4]}\n')
        break

conn.sendafter('equation?[Y/N]', 'Y\n')
conn.sendafter('solutions?[Y/N]', 'Y\n')

ans = [(2, 2), (20, 14)]
while len(ans) < 202:
    x, y = (ans[-1][0] * 6 - ans[-2][0] + 4), (ans[-1][1] * 6 - ans[-2][1] + 4)
    ans.append((x, y))

for r in ans:
    conn.sendafter('x=', f'{r[0]}\n')
    conn.sendafter('y=', f'{r[1]}\n')
    print(conn.recvline())
conn.interactive()

最后更新: 2022年7月23日 01:05:32

Contributors: YanhuiJessica
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Diophantine

解题思路¶

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