非反馈 feedback shift register¶
Introduction¶
In order to make the sequence of key stream output as complex as possible, a nonlinear feedback shift register is used. There are three common types.
- Nonlinear combination generator that uses a nonlinear combination function for the output of multiple LFSRs
- Nonlinear filter generator that uses a nonlinear combination function for the contents of an LFSR
- Clock generator that uses the output of one (or more) LFSRs to control the clock of another (or multiple) LFSRs
Nonlinear Combination Generator¶
Introduction¶
The combination generator is generally shown below.
Geffe¶
Here we take Geffe as an example. Geffe contains 3 linear feedback shift registers, and the nonlinear combination function is
F(x_1,x_2,x_3)=(x_1 \and x_2) \oplus (\urcorner x_1 \and x_3)=(x_1 \and x_2) \oplus ( x_1 \and x_3)\oplus x_3
2018 Strong Net Cup streamgame3¶
Simply look at the topic
from flag import flag
assert flag.startswith("flag{")
assert flag.endswith("}")
assert len(flag)==24
def lfsr(R,mask):
output = (R << 1) & 0xffffff
i=(R&mask)&0xffffff
lastbit=0
while i!=0:
lastbit^=(i&1)
i=i>>1
output^=lastbit
return (output,lastbit)
def single_round(R1,R1_mask,R2,R2_mask,R3,R3_mask):
(R1_NEW,x1)=lfsr(R1,R1_mask)
(R2_NEW,x2)=lfsr(R2,R2_mask)
(R3_NEW,x3)=lfsr(R3,R3_mask)
return (R1_NEW,R2_NEW,R3_NEW,(x1*x2)^((x2^1)*x3))
R1=int(flag[5:11],16)
R2=int(flag[11:17],16)
R3=int(flag[17:23],16)
assert len (bin (R1) [2:]) == 17
assert len (bin (R2) [2:]) == 19
assert len(bin(R3)[2:])==21
R1_mask=0x10020
R2_mask=0x4100c
R3_mask=0x100002
for fi in range(1024):
print fi
tmp1mb = ""
for i in range(1024):
tmp1kb = ""
for j in range(1024):
tmp=0
for k in range(8):
(R1,R2,R3,out)=single_round(R1,R1_mask,R2,R2_mask,R3,R3_mask)
tmp = (tmp << 1) ^ out
tmp1kb+=chr(tmp)
tmp1mb + = tmp1kb
f = open("./output/" + str(fi), "ab")
f.write(tmp1mb)
f.close()
It can be seen that the program is very similar to the Geffe generator. Here we use the related attack method to attack. We can count the output of the last class Geffe generator when the three LFSR outputs are different, as follows.
| x_1 | x_2 | x_3 | F(x_1,x_2,x_3) |
| ----- | ----- | ----- | ---------------- |
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |
it can be discovered
- Geffe's output has the same probability as x_1 0.75
- Geffe's output has the same probability of x_2 as 0.5
- Geffe's output has the same probability of x_3 as 0.75
This shows that the output is very relevant to the first and third. Therefore, we can violently enumerate the output of the first and third LFSRs to determine the number equal to the output of Geffe-like, and if it is about 75%, it can be considered correct. The second is a direct violent enumeration.
The script is as follows
#for x1 in range(2):
# for x2 in range(2):
# for x3 in range(2):
# print x1,x2,x3,(x1*x2)^((x2^1)*x3)
#n = [17,19,21]
#cycle = 1
#for i in n:
# cycle = cycle*(pow(2,i)-1)
#print cycle
def lfsr(R, mask):
output = (R << 1) & 0xffffff
i = (R & mask) & 0xffffff
lastbit = 0
while i != 0:
lastbit ^= (i & 1)
i = i >> 1
output ^= lastbit
return (output, lastbit)
def single_round(R1, R1_mask, R2, R2_mask, R3, R3_mask):
(R1_NEW, x1) = lfsr(R1, R1_mask)
(R2_NEW, x2) = lfsr(R2, R2_mask)
(R3_NEW, x3) = lfsr(R3, R3_mask)
return (R1_NEW, R2_NEW, R3_NEW, (x1 * x2) ^ ((x2 ^ 1) * x3))
R1_mask = 0x10020
R2_mask = 0x4100c
R3_mask = 0x100002
n3 = 21
n2 = 19
n1 = 17
def guess(beg, end, num, mask):
ansn = range(beg, end)
data = open('./output/0').read(num)
data = ''.join(bin(256 + ord(c))[3:] for c in data)
now = 0
true = 0
for i in ansn:
r = i
cnt = 0
for j in range(num * 8):
r, lastbit = lfsr(r, mask)
lastbit = str(lastbit)
cnt += (lastbit == data[j])
if cnt > now:
now = cnt
res = i
print now, res
return res
def bruteforce2(x, z):
data = open('./output/0').read(50)
data = ''.join(bin(256 + ord(c))[3:] for c in data)
for y in range(pow(2, n2 - 1), pow(2, n2)):
R1, R2, R3 = x, y, z
flag = True
for i in range(len(data)):
(R1, R2, R3,
out) = single_round(R1, R1_mask, R2, R2_mask, R3, R3_mask)
if str(out) != data[i]:
flag = False
break
if y % 10000 == 0:
print 'now: ', x, y, z
if flag:
print 'ans: ', hex(x)[2:], hex(y)[2:], hex(z)[2:]
break
R1 = guess(pow(2, n1 - 1), pow(2, n1), 40, R1_mask)
print R1
R3 = guess(pow(2, n3 - 1), pow(2, n3), 40, R3_mask)
print R3
R1 = 113099
R3 = 1487603
bruteforce2(R1, R3)
The results are as follows
➜ 2018-CISCN-start-streamgame3 git:(master) ✗ python exp.py
161 65536
172 65538
189 65545
203 65661
210 109191
242 113099
113099
157 1048576
165 1048578
183 1048580
184 1049136
186 1049436
187 1049964
189 1050869
190 1051389
192 1051836
194 1053573
195 1055799
203 1060961
205 1195773
212 1226461
213 1317459
219 1481465
239 1487603
1487603
now: 113099 270000 1487603
now: 113099 280000 1487603
now: 113099 290000 1487603
now: 113099 300000 1487603
now: 113099 310000 1487603
now: 113099 320000 1487603
now: 113099 330000 1487603
now: 113099 340000 1487603
now: 113099 350000 1487603
now: 113099 360000 1487603
years old: 1b9cb 5979c 16b2f3
Thus flag is flag{01b9cb05979c16b2f3}.
topic¶
-
2017 WHCTF Bornpig
-
2018 Google CTF 2018 Betterzip
Reference¶
-
https://www.rocq.inria.fr/secret/Anne.Canteaut/MPRI/chapter3.pdf
-
http://data.at.preempted.net/INDEX/articles/Correlation_Attacks_Geffe.pdf
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