LINE CTF 2022 | Baby crypto revisited

#LINECTF2022

#uninterested_challenge_list

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LINE CTF 2021 | babycrypto 4のやりなおし。one of the most kasu challenge I've ever seen

from hashlib import sha1

# https://neuromancer.sk/std/secg/secp160r1
p = 0xffffffffffffffffffffffffffffffff7fffffff
K = GF(p)
a = K(0xffffffffffffffffffffffffffffffff7ffffffc)
b = K(0x1c97befc54bd7a8b65acf89f81d4d4adc565fa45)
E = EllipticCurve(K, (a, b))
G = E(0x4a96b5688ef573284664698968c38bb913cbfc82, 0x23a628553168947d59dcc912042351377ac5fb32)
E.set_order(0x0100000000000000000001f4c8f927aed3ca752257 * 0x01)

n = int(E.order())


r = []
s = []
k = []
h = []
with open("Babycrypto_revisited_b1f108dea290b83253b80443260b12c3cadc0ed7.txt") as f:
    for line in f:
        a, b, c, d = [int(x, 16) for x in line.strip().split(" ")]
        r.append(a)
        s.append(b)
        k.append(c)
        h.append(d)

R = E.lift_x(Integer(r[0]))
P = inverse_mod(r[0], n) * (s[0]*R - h[0]*G)

nonce_max = 2**64

def sign(msg):
    h = int(sha1(msg).hexdigest(), 16)
    while True:
        k = randint(2, nonce_max-1)
        if gcd(k, n) == 1:
            break

    r = int((h * G).xy()[0])
    s = (h + r*d) * inverse_mod(k, n) % n
    return (r, s)

N = len(r)
t = [r[i] * inverse_mod(s[i], n) % n for i in range(N)]
u = [h[i] * inverse_mod(-s[i], n) % n for i in range(N)]

B = matrix(QQ, N + 2, N + 2)
B.set_block(0, 0, matrix.identity(N) * n)
B.set_block(N, 0, matrix(QQ, 1, N, t))
B.set_block(N+1, 0, matrix(QQ, 1, N, u))
B[N,N] = nonce_max / n
B[N+1,N+1] = nonce_max

L = B.LLL()
for row in list(L):
    k1 = abs(row[0])
    x = (k1*s[0] - h[0]) * inverse_mod(r[0], n) % n
    if x*G == P:
        # print(x, k1) # find private key
        print(hex(x))