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sCTF 2016 Q1: Ed25519 Writeup
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")
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", ""))
@elliptic-shiho

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elliptic-shiho commented Apr 18, 2016

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
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