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[原创]Crpyto-RSA stereotyped message atack-(UCTF2021-a bit weird)
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发表于: 2021-3-19 13:45 10512
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前段时间做UCTF2021的Crypto题时,遇到的一道RSA stereotyped message攻击的题,主要涉及的知识有RSA 、Coppersmith、与 and
和或 or
的理解、移位操作。由于公式原因,若感兴趣可参考本人博客:https://fishni.github.io/2021/03/17/Crypto-03-RSA%20Stereotyped%20message%20attack-(UCTF2021-a%20bit%20weird)/
要传递的message格式:
hello ,my name is xxx
在使用RSA对这个message加密,我们知道密文ciphertext和公钥(N,e=3),如何求原始的message
我们用b'\x00'
替换消息中的x
,这样就有了$(m+x)^e$ mod n=c
问题变为如何找到$x$
Coppersmith解决了这个问题,更多可参考:Comppersmith
I found this weird RSA looking thing somewhere. Can you break it for me? I managed to find x for you, but I don't know how to solve it without d...
依据msg.txt,知道
我们知道m&x,x
但并恢复不了m
,看来我们必须解密c才能得到$m | x$
由此,可得知以下关于m|x
故有以下已知信息:
这可以归为RSA中stereotyped messages类型攻击,我们能使用Coppersmith恢复a
使用场景:知道消息的部分关键位数,使用该方法可以找到消息的其他位
RSA的一个模型:$m^e=c (mod N)$,这个模型可以解决$c=(m+x)^e$,你知道明文的部分m,但是不知道x
如果寻找的是消息的$N^{1/e}$,并且是个较小的根,应该可以较快找到
多项式为$f(x)=(m+x)^e-c$ 想发现模N的一个根。
代码实现:
您可以使用这些值,直到它找到根。默认值应该是一个很好的开始。如果你想调整:
获取x的低440位的sage脚本,可在线运行sage脚本https://sagecell.sagemath.org/
sage脚本运行结果:
https://github.com/cscosu/ctf-writeups/tree/master/2021/utctf/A_Bit_Weird
https://github.com/mimoo/RSA-and-LLL-attacks/blob/master/coppersmith.sage
#main.py
from
Crypto.Util
import
number
from
secret
import
flag
import
os
length
=
2048
p, q
=
number.getPrime(length
/
/
2
), number.getPrime(length
/
/
2
)
N
=
p
*
q
e
=
3
m
=
number.bytes_to_long(flag)
x
=
number.bytes_to_long(os.urandom(length
/
/
8
))
c
=
pow
(m|x, e, N)
print
(
'N ='
, N);
print
(
'e ='
, e);
print
(
'c ='
, c);
print
(
'm&x ='
, m&x);
#main.py
from
Crypto.Util
import
number
from
secret
import
flag
import
os
length
=
2048
p, q
=
number.getPrime(length
/
/
2
), number.getPrime(length
/
/
2
)
N
=
p
*
q
e
=
3
m
=
number.bytes_to_long(flag)
x
=
number.bytes_to_long(os.urandom(length
/
/
8
))
c
=
pow
(m|x, e, N)
print
(
'N ='
, N);
print
(
'e ='
, e);
print
(
'c ='
, c);
print
(
'm&x ='
, m&x);
N
=
13876129555781460073002089038351520612247655754841714940325194761154811715694900213267064079029042442997358889794972854389557630367771777876508793474170741947269348292776484727853353467216624504502363412563718921205109890927597601496686803975210884730367005708579251258930365320553408690272909557812147058458101934416094961654819292033675534518433169541534918719715858981571188058387655828559632455020249603990658414972550914448303438265789951615868454921813881331283621117678174520240951067354671343645161030847894042795249824975975123293970250188757622530156083354425897120362794296499989540418235408089516991225649
e
=
3
c
=
6581985633799906892057438125576915919729685289065773835188688336898671475090397283236146369846971577536055404744552000913009436345090659234890289251210725630126240983696894267667325908895755610921151796076651419491871249815427670907081328324660532079703528042745484899868019846050803531065674821086527587813490634542863407667629281865859168224431930971680966013847327545587494254199639534463557869211251870726331441006052480498353072578366929904335644501242811360758566122007864009155945266316460389696089058959764212987491632905588143831831973272715981653196928234595155023233235134284082645872266135170511490429493
# m&x
m_x
=
947571396785487533546146461810836349016633316292485079213681708490477178328756478620234135446017364353903883460574081324427546739724
x
=
15581107453382746363421172426030468550126181195076252322042322859748260918197659408344673747013982937921433767135271108413165955808652424700637809308565928462367274272294975755415573706749109706624868830430686443947948537923430882747239965780990192617072654390726447304728671150888061906213977961981340995242772304458476566590730032592047868074968609272272687908019911741096824092090512588043445300077973100189180460193467125092550001098696240395535375456357081981657552860000358049631730893603020057137233513015505547751597823505590900290756694837641762534009330797696018713622218806608741753325137365900124739257740
N
=
13876129555781460073002089038351520612247655754841714940325194761154811715694900213267064079029042442997358889794972854389557630367771777876508793474170741947269348292776484727853353467216624504502363412563718921205109890927597601496686803975210884730367005708579251258930365320553408690272909557812147058458101934416094961654819292033675534518433169541534918719715858981571188058387655828559632455020249603990658414972550914448303438265789951615868454921813881331283621117678174520240951067354671343645161030847894042795249824975975123293970250188757622530156083354425897120362794296499989540418235408089516991225649
e
=
3
c
=
6581985633799906892057438125576915919729685289065773835188688336898671475090397283236146369846971577536055404744552000913009436345090659234890289251210725630126240983696894267667325908895755610921151796076651419491871249815427670907081328324660532079703528042745484899868019846050803531065674821086527587813490634542863407667629281865859168224431930971680966013847327545587494254199639534463557869211251870726331441006052480498353072578366929904335644501242811360758566122007864009155945266316460389696089058959764212987491632905588143831831973272715981653196928234595155023233235134284082645872266135170511490429493
# m&x
m_x
=
947571396785487533546146461810836349016633316292485079213681708490477178328756478620234135446017364353903883460574081324427546739724
x
=
15581107453382746363421172426030468550126181195076252322042322859748260918197659408344673747013982937921433767135271108413165955808652424700637809308565928462367274272294975755415573706749109706624868830430686443947948537923430882747239965780990192617072654390726447304728671150888061906213977961981340995242772304458476566590730032592047868074968609272272687908019911741096824092090512588043445300077973100189180460193467125092550001098696240395535375456357081981657552860000358049631730893603020057137233513015505547751597823505590900290756694837641762534009330797696018713622218806608741753325137365900124739257740
#简单统计一下,m&x和x的位数
len
(
bin
(m_x)[
2
:]),
len
(
bin
(x)[
2
:])
#简单统计一下,m&x和x的位数
len
(
bin
(m_x)[
2
:]),
len
(
bin
(x)[
2
:])
# x比m多的位数
2048
-
440
# x比m多的位数
2048
-
440
dd
=
f.degree()
beta
=
1
epsilon
=
beta
/
7
mm
=
ceil(beta
*
*
2
/
(dd
*
epsilon))
tt
=
floor(dd
*
mm
*
((
1
/
beta)
-
1
))
XX
=
ceil(N
*
*
((beta
*
*
2
/
dd)
-
epsilon))
roots
=
coppersmith_howgrave_univariate(f, N, beta, mm, tt, XX)
dd
=
f.degree()
beta
=
1
epsilon
=
beta
/
7
mm
=
ceil(beta
*
*
2
/
(dd
*
epsilon))
tt
=
floor(dd
*
mm
*
((
1
/
beta)
-
1
))
XX
=
ceil(N
*
*
((beta
*
*
2
/
dd)
-
epsilon))
roots
=
coppersmith_howgrave_univariate(f, N, beta, mm, tt, XX)
# Coppersmith implementation
import
time
debug
=
True
# display matrix picture with 0 and X
def
matrix_overview(BB, bound):
for
ii
in
range
(BB.dimensions()[
0
]):
a
=
(
'%02d '
%
ii)
for
jj
in
range
(BB.dimensions()[
1
]):
a
+
=
'0'
if
BB[ii,jj]
=
=
0
else
'X'
a
+
=
' '
if
BB[ii, ii] >
=
bound:
a
+
=
'~'
print
(a)
def
coppersmith_howgrave_univariate(pol, modulus, beta, mm, tt, XX):
"""
Coppersmith revisited by Howgrave-Graham
finds a solution if:
* b|modulus, b >= modulus^beta , 0 < beta <= 1
* |x| < XX
"""
#
# init
#
dd
=
pol.degree()
nn
=
dd
*
mm
+
tt
#
# checks
#
if
not
0
< beta <
=
1
:
raise
ValueError(
"beta should belongs in (0, 1]"
)
if
not
pol.is_monic():
raise
ArithmeticError(
"Polynomial must be monic."
)
#
# calculate bounds and display them
#
"""
* we want to find g(x) such that ||g(xX)|| <= b^m / sqrt(n)
* we know LLL will give us a short vector v such that:
||v|| <= 2^((n - 1)/4) * det(L)^(1/n)
* we will use that vector as a coefficient vector for our g(x)
* so we want to satisfy:
2^((n - 1)/4) * det(L)^(1/n) < N^(beta*m) / sqrt(n)
so we can obtain ||v|| < N^(beta*m) / sqrt(n) <= b^m / sqrt(n)
(it's important to use N because we might not know b)
"""
if
debug:
# t optimized?
print
(
"\n# Optimized t?\n"
)
print
(
"we want X^(n-1) < N^(beta*m) so that each vector is helpful"
)
cond1
=
RR(XX^(nn
-
1
))
print
(
"* X^(n-1) = "
, cond1)
cond2
=
pow
(modulus, beta
*
mm)
print
(
"* N^(beta*m) = "
, cond2)
print
(
"* X^(n-1) < N^(beta*m) \n-> GOOD"
if
cond1 < cond2
else
"* X^(n-1) >= N^(beta*m) \n-> NOT GOOD"
)
# bound for X
print
(
"\n# X bound respected?\n"
)
print
(
"we want X <= N^(((2*beta*m)/(n-1)) - ((delta*m*(m+1))/(n*(n-1)))) / 2 = M"
)
print
(
"* X ="
, XX)
cond2
=
RR(modulus^(((
2
*
beta
*
mm)
/
(nn
-
1
))
-
((dd
*
mm
*
(mm
+
1
))
/
(nn
*
(nn
-
1
))))
/
2
)
print
(
"* M ="
, cond2)
print
(
"* X <= M \n-> GOOD"
if
XX <
=
cond2
else
"* X > M \n-> NOT GOOD"
)
# solution possible?
print
(
"\n# Solutions possible?\n"
)
detL
=
RR(modulus^(dd
*
mm
*
(mm
+
1
)
/
2
)
*
XX^(nn
*
(nn
-
1
)
/
2
))
print
(
"we can find a solution if 2^((n - 1)/4) * det(L)^(1/n) < N^(beta*m) / sqrt(n)"
)
cond1
=
RR(
2
^((nn
-
1
)
/
4
)
*
detL^(
1
/
nn))
print
(
"* 2^((n - 1)/4) * det(L)^(1/n) = "
, cond1)
cond2
=
RR(modulus^(beta
*
mm)
/
sqrt(nn))
print
(
"* N^(beta*m) / sqrt(n) = "
, cond2)
print
(
"* 2^((n - 1)/4) * det(L)^(1/n) < N^(beta*m) / sqrt(n) \n-> SOLUTION WILL BE FOUND"
if
cond1 < cond2
else
"* 2^((n - 1)/4) * det(L)^(1/n) >= N^(beta*m) / sqroot(n) \n-> NO SOLUTIONS MIGHT BE FOUND (but we never know)"
)
# warning about X
print
(
"\n# Note that no solutions will be found _for sure_ if you don't respect:\n* |root| < X \n* b >= modulus^beta\n"
)
#
# Coppersmith revisited algo for univariate
#
# change ring of pol and x
polZ
=
pol.change_ring(ZZ)
x
=
polZ.parent().gen()
# compute polynomials
gg
=
[]
for
ii
in
range
(mm):
for
jj
in
range
(dd):
gg.append((x
*
XX)
*
*
jj
*
modulus
*
*
(mm
-
ii)
*
polZ(x
*
XX)
*
*
ii)
for
ii
in
range
(tt):
gg.append((x
*
XX)
*
*
ii
*
polZ(x
*
XX)
*
*
mm)
# construct lattice B
BB
=
Matrix(ZZ, nn)
for
ii
in
range
(nn):
for
jj
in
range
(ii
+
1
):
BB[ii, jj]
=
gg[ii][jj]
# display basis matrix
if
debug:
matrix_overview(BB, modulus^mm)
# LLL
BB
=
BB.LLL()
# transform shortest vector in polynomial
new_pol
=
0
for
ii
in
range
(nn):
new_pol
+
=
x
*
*
ii
*
BB[
0
, ii]
/
XX
*
*
ii
# factor polynomial
potential_roots
=
new_pol.roots()
print
(
"potential roots:"
, potential_roots)
# test roots
roots
=
[]
for
root
in
potential_roots:
if
root[
0
].is_integer():
result
=
polZ(ZZ(root[
0
]))
if
gcd(modulus, result) >
=
modulus^beta:
roots.append(ZZ(root[
0
]))
#
return
roots
# Public key
N
=
13876129555781460073002089038351520612247655754841714940325194761154811715694900213267064079029042442997358889794972854389557630367771777876508793474170741947269348292776484727853353467216624504502363412563718921205109890927597601496686803975210884730367005708579251258930365320553408690272909557812147058458101934416094961654819292033675534518433169541534918719715858981571188058387655828559632455020249603990658414972550914448303438265789951615868454921813881331283621117678174520240951067354671343645161030847894042795249824975975123293970250188757622530156083354425897120362794296499989540418235408089516991225649
length_N
=
2048
ZmodN
=
Zmod(N);
e
=
3
# Obscuring term (was called `x` previously)
obs
=
15581107453382746363421172426030468550126181195076252322042322859748260918197659408344673747013982937921433767135271108413165955808652424700637809308565928462367274272294975755415573706749109706624868830430686443947948537923430882747239965780990192617072654390726447304728671150888061906213977961981340995242772304458476566590730032592047868074968609272272687908019911741096824092090512588043445300077973100189180460193467125092550001098696240395535375456357081981657552860000358049631730893603020057137233513015505547751597823505590900290756694837641762534009330797696018713622218806608741753325137365900124739257740
length_obs
=
440
# Ciphertext
C
=
6581985633799906892057438125576915919729685289065773835188688336898671475090397283236146369846971577536055404744552000913009436345090659234890289251210725630126240983696894267667325908895755610921151796076651419491871249815427670907081328324660532079703528042745484899868019846050803531065674821086527587813490634542863407667629281865859168224431930971680966013847327545587494254199639534463557869211251870726331441006052480498353072578366929904335644501242811360758566122007864009155945266316460389696089058959764212987491632905588143831831973272715981653196928234595155023233235134284082645872266135170511490429493
# Obsuring term with lower 440 bits zero'd (was called 'b' previously)
obs_clean
=
(obs >> length_obs) << length_obs
# Problem to equation.
# The `x` here was called `a` previously, which we are now solving for.
P.<x>
=
PolynomialRing(ZmodN)
pol
=
(obs_clean
+
x)^e
-
C
dd
=
pol.degree()
beta
=
1
# b = N
epsilon
=
beta
/
7
# <= beta / 7
mm
=
ceil(beta
*
*
2
/
(dd
*
epsilon))
# optimized value
tt
=
floor(dd
*
mm
*
((
1
/
beta)
-
1
))
# optimized value
XX
=
ceil(N
*
*
((beta
*
*
2
/
dd)
-
epsilon))
# optimized value
# Coppersmith
roots
=
coppersmith_howgrave_univariate(pol, N, beta, mm, tt, XX)
print
()
print
(
"Solutions"
)
print
(
"We found:"
,
str
(roots))
# Coppersmith implementation
import
time
debug
=
True
# display matrix picture with 0 and X
def
matrix_overview(BB, bound):
for
ii
in
range
(BB.dimensions()[
0
]):
a
=
(
'%02d '
%
ii)
for
jj
in
range
(BB.dimensions()[
1
]):
a
+
=
'0'
if
BB[ii,jj]
=
=
0
else
'X'
a
+
=
' '
if
BB[ii, ii] >
=
bound:
a
+
=
'~'
print
(a)
def
coppersmith_howgrave_univariate(pol, modulus, beta, mm, tt, XX):
"""
Coppersmith revisited by Howgrave-Graham
finds a solution if:
* b|modulus, b >= modulus^beta , 0 < beta <= 1
* |x| < XX
"""
#
# init
#
dd
=
pol.degree()
nn
=
dd
*
mm
+
tt
#
# checks
#
if
not
0
< beta <
=
1
:
raise
ValueError(
"beta should belongs in (0, 1]"
)
if
not
pol.is_monic():
raise
ArithmeticError(
"Polynomial must be monic."
)
#
# calculate bounds and display them
#
"""
* we want to find g(x) such that ||g(xX)|| <= b^m / sqrt(n)
* we know LLL will give us a short vector v such that:
||v|| <= 2^((n - 1)/4) * det(L)^(1/n)
* we will use that vector as a coefficient vector for our g(x)
* so we want to satisfy:
2^((n - 1)/4) * det(L)^(1/n) < N^(beta*m) / sqrt(n)
so we can obtain ||v|| < N^(beta*m) / sqrt(n) <= b^m / sqrt(n)
(it's important to use N because we might not know b)
"""
if
debug:
# t optimized?
print
(
"\n# Optimized t?\n"
)
print
(
"we want X^(n-1) < N^(beta*m) so that each vector is helpful"
)
cond1
=
RR(XX^(nn
-
1
))
print
(
"* X^(n-1) = "
, cond1)
cond2
=
pow
(modulus, beta
*
mm)
print
(
"* N^(beta*m) = "
, cond2)
print
(
"* X^(n-1) < N^(beta*m) \n-> GOOD"
if
cond1 < cond2
else
"* X^(n-1) >= N^(beta*m) \n-> NOT GOOD"
)
# bound for X
print
(
"\n# X bound respected?\n"
)
print
(
"we want X <= N^(((2*beta*m)/(n-1)) - ((delta*m*(m+1))/(n*(n-1)))) / 2 = M"
)
print
(
"* X ="
, XX)
cond2
=
RR(modulus^(((
2
*
beta
*
mm)
/
(nn
-
1
))
-
((dd
*
mm
*
(mm
+
1
))
/
(nn
*
(nn
-
1
))))
/
2
)
print
(
"* M ="
, cond2)
print
(
"* X <= M \n-> GOOD"
if
XX <
=
cond2
else
"* X > M \n-> NOT GOOD"
)
# solution possible?
print
(
"\n# Solutions possible?\n"
)
detL
=
RR(modulus^(dd
*
mm
*
(mm
+
1
)
/
2
)
*
XX^(nn
*
(nn
-
1
)
/
2
))
print
(
"we can find a solution if 2^((n - 1)/4) * det(L)^(1/n) < N^(beta*m) / sqrt(n)"
)
cond1
=
RR(
2
^((nn
-
1
)
/
4
)
*
detL^(
1
/
nn))
print
(
"* 2^((n - 1)/4) * det(L)^(1/n) = "
, cond1)
cond2
=
RR(modulus^(beta
*
mm)
/
sqrt(nn))
print
(
"* N^(beta*m) / sqrt(n) = "
, cond2)
print
(
"* 2^((n - 1)/4) * det(L)^(1/n) < N^(beta*m) / sqrt(n) \n-> SOLUTION WILL BE FOUND"
if
cond1 < cond2
else
"* 2^((n - 1)/4) * det(L)^(1/n) >= N^(beta*m) / sqroot(n) \n-> NO SOLUTIONS MIGHT BE FOUND (but we never know)"
)
# warning about X
print
(
"\n# Note that no solutions will be found _for sure_ if you don't respect:\n* |root| < X \n* b >= modulus^beta\n"
)
#
# Coppersmith revisited algo for univariate
#
# change ring of pol and x
polZ
=
pol.change_ring(ZZ)
x
=
polZ.parent().gen()
# compute polynomials
gg
=
[]
for
ii
in
range
(mm):
for
jj
in
range
(dd):
gg.append((x
*
XX)
*
*
jj
*
modulus
*
*
(mm
-
ii)
*
polZ(x
*
XX)
*
*
ii)
for
ii
in
range
(tt):
gg.append((x
*
XX)
*
*
ii
*
polZ(x
*
XX)
*
*
mm)
# construct lattice B
BB
=
Matrix(ZZ, nn)
for
ii
in
range
(nn):
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