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| from sage.all import * from Crypto.Util.number import * import random
Griffin = flagct_val = 9412139614776358237302032187121621394441766988270
def recover_p_a_b(points_matrix): flat_points = [] for row in points_matrix: for pt in row: if pt is not None and pt != 0: flat_points.append(pt)
candidates = [] for _ in range(20): sample = random.sample(flat_points, 4) x, y = [], [] for pt in sample: x.append(Integer(pt[0])) y.append(Integer(pt[1]))
lhs = [y[i]**2 - x[i]**3 for i in range(4)]
# 利用 (LHS0 - LHS1)(x2 - x3) - (LHS2 - LHS3)(x0 - x1) 消除 a, b val = (lhs[0] - lhs[1]) * (x[2] - x[3]) - (lhs[2] - lhs[3]) * (x[0] - x[1]) if val != 0: candidates.append(abs(val))
p_recovered = candidates[0] for val in candidates[1:]: p_recovered = gcd(p_recovered, val)
if not is_prime(p_recovered): p_recovered = p_recovered.factor()[-1][0]
print(f"[+] Recovered p: {p_recovered}")
# 恢复 a, b pt0, pt1 = flat_points[0], flat_points[1] x0, y0 = Integer(pt0[0]), Integer(pt0[1]) x1, y1 = Integer(pt1[0]), Integer(pt1[1])
Fp = GF(p_recovered) # a = ( (y0^2 - x0^3) - (y1^2 - x1^3) ) / (x0 - x1) try: a_rec = (Fp(y0**2 - x0**3) - Fp(y1**2 - x1**3)) / (Fp(x0) - Fp(x1)) b_rec = Fp(y0**2 - x0**3) - a_rec * Fp(x0) except ZeroDivisionError: print("[-] Points collision, retrying a/b recovery...") return recover_p_a_b(points_matrix)
return p_recovered, Integer(a_rec), Integer(b_rec)
p, a, b = recover_p_a_b(Griffin) print(f"[+] Curve parameters: p = {p}, a = {a}, b = {b}")
E = EllipticCurve(GF(p), [a, b]) G = E.lift_x(Integer(3137)) n = Integer(G.order()) print(f"[+] ord(G)=n={n}") print(f"[+] factor(n)={factor(n)}")
fac = list(factor(n)) prime_factors = [int(pp) for (pp, _) in fac] q = max(prime_factors) print(f"[+] choose q={q}")
co = int(n) // q Gq = co * G if int(Gq.order()) != q: raise ValueError("subgroup order mismatch for q")
tbl = {} P = E(0) for k in range(q): if P.is_zero(): tbl[None] = k else: tbl[(int(P[0]), int(P[1]))] = k P += Gq
def dlog_mod_q_point(pt): Q = co * pt if Q.is_zero(): return 0 return tbl[(int(Q[0]), int(Q[1]))]
rows = len(Griffin) cols = len(Griffin[0]) A = Matrix(GF(q), rows, cols)
print("[*] computing all logs mod q for Griffin (290x80) ...") for i in range(rows): for j in range(cols): x, y = Griffin[i][j] A[i, j] = dlog_mod_q_point(E(Integer(x), Integer(y))) if (i+1) % 25 == 0: print(f" done {i+1}/{rows}")
def find_circuit_indices(A, max_trials=80000, support_cap=24, seed=42): rng = random.Random(int(seed)) all_idx = list(range(A.nrows())) for t in range(1, max_trials+1): basis_idx = rng.sample(all_idx, 80) B = A.matrix_from_rows(basis_idx) Bt = B.transpose() if Bt.rank() != 80: continue invBt = Bt.inverse()
outside = [i for i in all_idx if i not in set(basis_idx)]
for j in outside: v = A.row(j) c = (invBt * v.column()).column(0) supp = [k for k in range(80) if c[k] != 0] if len(supp) <= support_cap: return sorted(set([basis_idx[k] for k in supp] + [j]))
return None
circuit = find_circuit_indices(A)
RS = A.matrix_from_rows(circuit).row_space() hawk = [i for i in range(rows) if A.row(i) in RS]
hawk = list(hawk)
H = A.matrix_from_rows(hawk) # 40x80 over GF(q) col_basis = H.column_space().basis()
B = matrix.zero(40) for i in range(40): B[i, i] = q B = list(B) for item in col_basis: B.append([int(item[i]) for i in range(40)]) Bint = Matrix(ZZ, B) L = Bint.LLL() M = L.BKZ(block_size=32) print("BKZ_done") cand = []
for r in M.rows(): v = [int(x) % q for x in r] # centered norm filter vc = [(x - q if x > q//2 else x) for x in v] if any(t != 0 for t in vc) and max(abs(int(t)) for t in vc) <= 6000: cand.append(v)
def try_affine_to_1_256(v_modq, q): F = GF(q) v = [F(int(x)) for x in v_modq] v0, v1 = v[0], v[1] for X0 in range(1, 257): for X1 in range(1, 257): if X0 == X1: continue denom = F(X0 - X1) if denom == 0: continue a_ = (v0 - v1) / denom if a_ == 0: continue b_ = v0 - a_ * F(X0) inva = a_**(-1) xs = [int((vk - b_) * inva) for vk in v] if all(1 <= x <= 256 for x in xs) and len(set(xs)) == 40: return xs return None
def validate_xs(xs_list, H, q, checks=20): # for several columns, interpolate deg<20 from first 20 points then verify last 20 F = GF(q) R = PolynomialRing(F, 'X') X = R.gen() xsF = [F(int(x)) for x in xs_list] for j in range(min(checks, H.ncols())): ys = [H[i, j] for i in range(40)] pts = list(zip(xsF[:20], ys[:20])) f = R(0) for i, (xi, yi) in enumerate(pts): num = R(1); den = F(1) for k, (xk, _yk) in enumerate(pts): if i == k: continue num *= (X - xk) den *= (xi - xk) f += yi * num * (den**(-1)) for xk, yk in zip(xsF[20:], ys[20:]): if f(xk) != yk: return False return True
xs_base = None for v in cand: xs_try = try_affine_to_1_256(v, q) if xs_try and validate_xs(xs_try, H, q, checks=20): xs_base = xs_try break vneg = [(-Integer(x)) % q for x in v] xs_try = try_affine_to_1_256(vneg, q) if xs_try and validate_xs(xs_try, H, q, checks=20): xs_base = xs_try break
if xs_base is None: raise ValueError("failed to recover a valid xs_base")
print("[+] xs_base recovered.")
print("[*] Step5: full DLP mod n for 40 points (col0) ...")
T = [] for pr in prime_factors: cof = int(n) // pr Gp = cof * G if int(Gp.order()) != pr: raise ValueError("prime subgroup order mismatch") t = {} Q = E(0) for k in range(pr): if Q.is_zero(): t[None] = k else: t[(int(Q[0]), int(Q[1]))] = k Q += Gp T.append((pr, cof, t))
def dlog_full_mod_n(P): rs = [] ms = [] for pr, cof, t in T: Q = cof * P if Q.is_zero(): r = 0 else: r = t[(int(Q[0]), int(Q[1]))] rs.append(Integer(r)) ms.append(Integer(pr)) return Integer(crt(rs, ms)) % n
Y = [] for i in range(len(hawk)): ridx = hawk[i] x, y = Griffin[ridx][0] P = E(Integer(x), Integer(y)) Y.append(dlog_full_mod_n(P)) if (i+1) % 10 == 0: print(f" done {i+1}/40")
print("[+] y-values ready.")
print("[*] Step6: enumerate xs variants and collect candidates ...")
V = [] x0 = list(map(int, xs_base)) lo = 1 - min(x0) hi = 256 - max(x0) for d in range(lo, hi + 1): V.append((0, d, [x + d for x in x0]))
xf = [257 - x for x in x0] lo = 1 - min(xf) hi = 256 - max(xf) for d in range(lo, hi + 1): V.append((1, d, [x + d for x in xf]))
def roots_one_prime(xs_use, pr): F = GF(pr) R = PolynomialRing(F, 'X') X = R.gen() pts = list(zip(xs_use[:20], Y[:20])) f = R(0) for i in range(20): xi = F(int(pts[i][0])) yi = F(int(pts[i][1]) % pr) num = R(1) den = F(1) for j in range(20): if i == j: continue xj = F(int(pts[j][0])) num *= (X - xj) den *= (xi - xj) f += yi * num * (den**(-1)) for i in range(40): if f(F(int(xs_use[i]))) != F(int(Y[i]) % pr): return None eq = f - F(int(flagct_val) % pr) rts = eq.roots() return [int(rt) for rt, _m in rts]
S = set() O = []
for t in range(len(V)): mode, d, xs_use = V[t] R0 = [] ok = True for pr in prime_factors: rts = roots_one_prime(xs_use, pr) if not rts: ok = False break R0.append((pr, rts)) if not ok: continue C = [0] mod = 1 for pr, rts in R0: N = [] for a0 in C: for b0 in rts: N.append(int(crt(Integer(a0), Integer(b0), Integer(mod), Integer(pr))) % (mod * pr)) mod *= pr C = sorted(set(N)) for x in C: x = int(x) if x not in S: S.add(x) O.append((x, mode, d)) if (t+1) % 20 == 0: print(f" processed {t+1}/{len(V)} variants, unique_cands={len(S)}")
cand_list = sorted(S) print(f"[+] FINAL unique candidates = {len(cand_list)}") print("[*] done.")
from tqdm import * import time for i in trange(len(cand_list)): n = Integer(29808324682087298967547021317914008861362873223757) r = Integer(cand_list[i])
# ---------------------------- # 2) FLAG template bytes # ---------------------------- # FLAG = b"alictf{" + uuid4_ascii + b"}" # uuid4 pattern: 8-4-4-4-12 (36 chars) # version: xxxxxxxx-xxxx-4xxx-yxxx-xxxxxxxxxxxx # y in {8,9,a,b}
prefix = b"alictf{" suffix = b"}"
L = len(prefix) + 36 + len(suffix) # 44 bytes assert len(prefix) == 7 and len(suffix) == 1
uuid_start = len(prefix) # 7 uuid_end = uuid_start + 36 # 43 (exclusive)
# positions of '-' inside UUID (0-based within UUID string) dash_pos = [8, 13, 18, 23]
# UUID indices layout: # 0..7 hex, 8 '-', 9..12 hex, 13 '-', 14..17 hex, 18 '-', 19..22 hex, 23 '-', 24..35 hex version_pos = 14 # must be '4' variant_pos = 19 # in {8,9,a,b}
# known bytes array (None = unknown) known = [None] * L
# fill prefix/suffix for i, b in enumerate(prefix): known[i] = b known[L - 1] = suffix[0]
# fill dashes for dp in dash_pos: known[uuid_start + dp] = ord('-')
# fill version '4' known[uuid_start + version_pos] = ord('4')
# variant will be constrained later variant_position_global = uuid_start + variant_pos
# ---------------------------- # 3) Collect unknown positions # ---------------------------- unknown_hex_positions = [] for i in range(L): if known[i] is None: if i == variant_position_global: continue unknown_hex_positions.append(i)
# ---------------------------- # 4) Build modular equation target # ---------------------------- offset_val = Integer(0) for i in range(L): if known[i] is not None: offset_val += Integer(known[i]) * (Integer(256) ** (L - 1 - i))
# We want: sum(unknown_bytes * 256^k) == (r - offset_val) mod n target_eq = (r - offset_val) % n
# ---------------------------- # 5) Lattice modeling # ---------------------------- # For each unknown hex byte: # x = 48 + 49*b + rr # b in {0,1}, rr in [0..9] # This covers '0'..'9' (b=0) and 'a'..'j' (b=1); later we filter to 'a'..'f'. # # Variant byte is one of { '8','9','a','b' }. # Model variant as: 56 + t + 41*u, where u,t in {0,1} # u=0 => 56+t -> '8'/'9' # u=1 => 97+t -> 'a'/'b' (since 56+41=97)
m_hex = len(unknown_hex_positions)
# variable ordering: # 0..m_hex-1 : 2*b_i # m_hex..2*m_hex-1 : 2*rr_i # 2*m_hex : 2*u # 2*m_hex+1 : 2*t num_vars = 2 * m_hex + 2 dim = num_vars + 2 # + modulus row + embedding row
M = Matrix(ZZ, dim, dim)
# weights W_b = 9 W_r = 1 W_eq = 2**300
# Fill columns for each hex unknown for j, pos in enumerate(unknown_hex_positions): power = Integer(256) ** (L - 1 - pos)
# 2*b_j M[j, j] = 2 * W_b M[j, dim - 1] = 49 * power * W_eq
# 2*rr_j idx_r = m_hex + j M[idx_r, idx_r] = 2 * W_r M[idx_r, dim - 1] = 1 * power * W_eq
# Variant u,t pos_v = variant_position_global power_v = Integer(256) ** (L - 1 - pos_v)
idx_u = 2 * m_hex idx_t = 2 * m_hex + 1
M[idx_u, idx_u] = 2 * W_b M[idx_u, dim - 1] = 41 * power_v * W_eq
M[idx_t, idx_t] = 2 * W_b M[idx_t, dim - 1] = 1 * power_v * W_eq
# Modulus row M[dim - 2, dim - 1] = n * W_eq
# ---------------------------- # 6) CVP embedding row (targets) # ---------------------------- # marker col: dim-2 M[dim - 1, dim - 2] = 1
# Targets for b and rr for j in range(m_hex): M[dim - 1, j] = -1 * W_b # want 2b close to 1 for j in range(m_hex): M[dim - 1, m_hex + j] = -9 * W_r # want 2rr close to 9
# Targets for u,t M[dim - 1, idx_u] = -1 * W_b M[dim - 1, idx_t] = -1 * W_b
# Equation target: # target_eq currently corresponds to sum(all unknown bytes * 256^k) # Subtract bases: # - For each hex unknown: base 48 # - For variant: base 56 hex_base_sum = sum([48 * (Integer(256) ** (L - 1 - pos)) for pos in unknown_hex_positions]) variant_base = 56
target_eq2 = (target_eq - hex_base_sum - variant_base * power_v) % n
M[dim - 1, dim - 1] = -target_eq2 * W_eq
# ---------------------------- # 7) Run BKZ # ---------------------------- print("[*] bytes length =", L) print("[*] unknown hex count =", m_hex) print("[*] Running BKZ-40 ...") t0 = time.time() Lred = M.BKZ(block_size=26) print("[*] BKZ done in %.2fs" % (time.time() - t0))
# ---------------------------- # 8) Extract solution # ---------------------------- def is_hex_char(x: int) -> bool: return (48 <= x <= 57) or (97 <= x <= 102)
found = None
for row in Lred: marker = row[dim - 2] if abs(marker) != 1: continue
sign = 1 if marker == 1 else -1
out = known[:] ok = True
# hex unknowns for j, pos in enumerate(unknown_hex_positions): raw_b = (row[j] // W_b) * sign val_2b = raw_b + 1 if val_2b & 1: ok = False break b_j = val_2b // 2
raw_r = (row[m_hex + j] // W_r) * sign val_2r = raw_r + 9 if val_2r & 1: ok = False break rr_j = val_2r // 2
byte = 48 + 49 * b_j + rr_j if not is_hex_char(int(byte)): ok = False break out[pos] = int(byte)
if not ok: continue
# variant u,t raw_u = (row[idx_u] // W_b) * sign v2u = raw_u + 1 if v2u & 1: continue u = v2u // 2
raw_t = (row[idx_t] // W_b) * sign v2t = raw_t + 1 if v2t & 1: continue t = v2t // 2
variant_byte = 56 + t + 41 * u if int(variant_byte) not in [ord('8'), ord('9'), ord('a'), ord('b')]: continue out[variant_position_global] = int(variant_byte)
if any(x is None for x in out): continue
# verify modulo m_candidate = Integer(0) for x in out: m_candidate = m_candidate * 256 + x
if m_candidate % n == r: found = bytes(out) break
if found: print("[+] FLAG =", found.decode()) break else: print("[-] Not found. Try: increase BKZ block_size (e.g. 50/60) or tune W_eq/W_b.")
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