sCTF 2016 Q1: Ed25519 Writeup

ed25519.py

import hashlib

b = 256
q = 2**255 - 19
l = 2**252 + 27742317777372353535851937790883648493

def H(m):
  return hashlib.sha512(m).digest()

def expmod(b,e,m):
  return pow(b, e, m)

def inv(x):
  return expmod(x,q-2,q)

d = -121665 * inv(121666)
I = expmod(2,(q-1)/4,q)

def xrecover(y):
  xx = (y*y-1) * inv(d*y*y+1)
  x = expmod(xx,(q+3)/8,q)
  if (x*x - xx) % q != 0: x = (x*I) % q
  if x % 2 != 0: x = q-x
  return x

By = 4 * inv(5)
Bx = xrecover(By)
B = [Bx % q,By % q]

def edwards(P,Q):
  x1 = P[0]
  y1 = P[1]
  x2 = Q[0]
  y2 = Q[1]
  x3 = (x1*y2+x2*y1) * inv(1+d*x1*x2*y1*y2)
  y3 = (y1*y2+x1*x2) * inv(1-d*x1*x2*y1*y2)
  return [x3 % q,y3 % q]

def scalarmult(P,e):
  if e == 0:
    return [0, 1]
  bits = map(int, bin(e)[2:])[::-1]
  x = P
  if bits[0]:
    res = x
  else:
    res = [0, 1]
  for cur in bits[1:]:
    x = edwards(x, x)
    if cur:
      res = edwards(res, x)
  return res

def encodeint(y):
  bits = [(y >> i) & 1 for i in range(b)]
  return ''.join([chr(sum([bits[i * 8 + j] << j for j in range(8)])) for i in range(b/8)])

def encodepoint(P):
  x = P[0]
  y = P[1]
  bits = [(y >> i) & 1 for i in range(b - 1)] + [x & 1]
  return ''.join([chr(sum([bits[i * 8 + j] << j for j in range(8)])) for i in range(b/8)])

def bit(h,i):
  return (ord(h[i/8]) >> (i%8)) & 1

def publickey(sk):
  h = H(sk)
  a = 2**(b-2) + sum(2**i * bit(h,i) for i in range(3,b-2))
  A = scalarmult(B,a)
  return encodepoint(A)

def Hint(m):
  h = H(m)
  return sum(2**i * bit(h,i) for i in range(2*b))

def signature(m,sk,pk):
  h = H(sk)
  a = 2**(b-2) + sum(2**i * bit(h,i) for i in range(3,b-2))
  r = Hint(''.join([h[i] for i in range(b/8,b/4)]) + m)
  R = scalarmult(B,r)
  S = (r + Hint(encodepoint(R) + pk + m) * a) % l
  return encodepoint(R) + encodeint(S)

def isoncurve(P):
  x = P[0]
  y = P[1]
  return (-x*x + y*y - 1 - d*x*x*y*y) % q == 0

def decodeint(s):
  return sum(2**i * bit(s,i) for i in range(0,b))

def decodepoint(s):
  y = sum(2**i * bit(s,i) for i in range(0,b-1))
  x = xrecover(y)
  if x & 1 != bit(s,b-1): x = q-x
  P = [x,y]
  if not isoncurve(P): raise Exception("decoding point that is not on curve")
  return P

def checkvalid(s,m,pk):
  if len(s) != b/4: raise Exception("signature length is wrong")
  if len(pk) != b/8: raise Exception("public-key length is wrong")
  R = decodepoint(s[0:b/8])
  A = decodepoint(pk)
  S = decodeint(s[b/8:b/4])
  h = Hint(encodepoint(R) + pk + m)
  if scalarmult(B,S) != edwards(R,scalarmult(A,h)):
    raise Exception("signature does not pass verification")

solve.py

import ed25519
import gmpy

def modinv_p(a, p):
  assert gmpy.is_prime(p)
  return pow(a, p-2, p)

pk = "5bfcb1cd3938f3f6f3092da5f7d7a1bdb1d694a725d0585a99208787554e110d".decode("hex")
s1 = "68299a51b6b592e2db83c26ca3594bdd81bdbb9f11c597a1deb823da7c8b9de8e2224855125b1acbeab1468bf4860c1eeb05b6d2375e2214c55bdfe808a6c106".decode("hex")
m1 = "sctf.io"
s2 = "68299a51b6b592e2db83c26ca3594bdd81bdbb9f11c597a1deb823da7c8b9de825ad01a05a0cce69258d41d42ed046956e7d4586eb21ff031bf8ac03243d5e04".decode("hex")
m2 = "2016 Q1"

# check
ed25519.checkvalid(s1, m1, pk)
ed25519.checkvalid(s2, m2, pk)

P = ed25519.decodepoint(pk)

a = 0
R1 = ed25519.decodepoint(s1[:ed25519.b/8])
S1 = ed25519.decodeint(s1[ed25519.b/8:ed25519.b/4])
h1 = ed25519.Hint(ed25519.encodepoint(R1) + pk + m1)
R2 = ed25519.decodepoint(s2[:ed25519.b/8])
S2 = ed25519.decodeint(s2[ed25519.b/8:ed25519.b/4])
h2 = ed25519.Hint(ed25519.encodepoint(R2) + pk + m2)

print R1, S1, h1
print R2, S2, h2

a = ((S1 - S2) * modinv_p(h1 - h2, ed25519.l)) % ed25519.l

r1 = (S1 - h1 * a) % ed25519.l
r2 = (S2 - h2 * a) % ed25519.l

print r1, r2

f = "the flag"
r = r1
R = ed25519.scalarmult(ed25519.B, r)
S = (r + ed25519.Hint(ed25519.encodepoint(R) + pk + f) * a) % ed25519.l
print "Flag is sctf{%s}" % ((ed25519.encodepoint(R) + ed25519.encodeint(S)).encode("base64").replace("\n", ""))

Writeup: Breaking Ed25519 with same random key.

Trivial algebraic practice.

Mon Apr 18 09:11:09 JST 2016 ~/ctf/sctf-2016-q1/crypto40/ed25519 Battery 0: Full, 100%
> time python solve.py 
[28104680254895858602950095340273610962152604410683289209133586156927824858805L, 47318893950578276168317079672385112889584626339324959120861701577279184447848L] 3056024505099303091264038272914434090600937271399321803322554667848809587426 13397442584040240938586667532063254484324168699882605334802284616584255557111053955417396865311479649066713941214800499812450892230427570948701676049481822
[28104680254895858602950095340273610962152604410683289209133586156927824858805L, 47318893950578276168317079672385112889584626339324959120861701577279184447848L] 1975756995894560183473395863515241874506266339720132698812345799672898628901 7960866852339352189900001165174010281146836611057896297646402284319532015956538563906002724988794309980855750528647709587915541142579171768379987340263132
117 117
Flag is sctf{aCmaUba1kuLbg8Jso1lL3YG9u58RxZeh3rgj2nyLneh12Mf7NAvaREdBikQrkpWSa3UT15wUmunsrceSa3CUCQ==}

real    0m0.910s
user    0m0.901s
sys     0m0.008s