安恒月赛-dasctf 部分writeup

1.crypto-not rsa

题目很简单，r是一个随机数，g=n+1，已知(pow(g,m,n*n)*pow(r,n,n*n))%(n*n)=c，即密文c，求解明文m。我们根据二项式定理易知可将pow(g,m,n*n)=pow(n+1,m,n*n)=pow(m*n+1,n*n)。由于n=p*q，所以我们知道phi(n*n)=p*(p-1)*q*(q-1)。我们对加密公式两侧同时加(p-1)*(q-1)次幂，可化简为(pow((m*n+1),(p-1)*(q-1),n*n))%(n*n)=pow(c,(p-1)*(q-1),n*n)从而将r化简掉。我们再借助二项式定理化简可得((p-1)*(q-1)*m*n+1)%(n*n)=pow(c,(p-1)*(q-1),n*n)。((p-1)*(q-1)*m*n)%(n*n)=pow(c,(p-1)*(q-1),n*n)-1，((p-1)*(q-1)*m*n)-(pow(c,(p-1)*(q-1),n*n)-1)=k*n*n。这时我们发现(pow(c,(p-1)*(q-1),n*n)-1)%n=0。于是我们做进一步化简可得((p-1)*(q-1)*m)%n=(pow(c,(p-1)*(q-1),n*n)-1)/n，m=((pow(c,(p-1)*(q-1),n*n)-1)/n)乘以(p-1)*(q-1)模n的逆，可将m求解，具体代码如下：

import gmpy2
c=29088911054711509252215615231015162998042579425917914434962376243477176757448053722602422672251758332052330100944900171067962180230120924963561223495629695702541446456981441239486190458125750543542379899722558637306740763104274377031599875275807723323394379557227060332005571272240560453811389162371812183549
n=6401013954612445818165507289870580041358569258817613282142852881965884799988941535910939664068503367303343695466899335792545332690862283029809823423608093
p= 80006336965345725157774618059504992841841040207998249416678435780577798937819
q = 80006336965345725157774618059504992841841040207998249416678435780577798937447
assert p*q==n
phi=(p-1)*(q-1)
c1=pow(c,phi,n*n)-1
c2=c1/n
m=(c2*gmpy2.invert(phi,n))%n
print hex(m)[2:].decode("hex")

2.crypto-ComplexEncode

首先贴上题目

from Crypto.Util.number import *
from flag import FLAG,false_flag
import gmpy2
import random
import hashlib
import base64

def rsaEncode(msg):
    f=open("out","a")
    while True:
        ran=random.randint(3, 12)
        if isPrime(ran):
            break
    e=ran*getPrime(30)
    p = getPrime(1024)
    q = getPrime(1024)
    while (not gmpy2.gcd((p-1)*(q-1),e)==ran or p*q<pow(msg,ran)):
        p = getPrime(1024)
        q = getPrime(1024)
    n=p*q
    assert(pow(msg,ran)<n)
    print("rsaEncode_init_finish!")
    dsaEncode(p)
    f.write("rsaEncode n:"+hex(n)+"\n")
    f.write("rsaEncode e:"+hex(e)+"\n")
    c=pow(msg,e,n)
    f.write("rsaEncode flag is:"+hex(c)+"\n")
    f.close()

def rsaEncode2(m):
    m=int.from_bytes(m.encode(),'big')
    f=open("out","w")
    while True:
        ran=random.randint(20, 50)
        if isPrime(ran):
            break
    e=ran*getPrime(30)
    p = getPrime(1024)
    q = gmpy2.next_prime(p)
    while (not gmpy2.gcd((p-1)*(q-1),e)==ran or p*q>pow(m,ran)):
        p = getPrime(1024)
        q = gmpy2.next_prime(p)
    n=p*q
    assert(pow(m,ran)>n)
    c=pow(m,e,n)
    f.write("n2:"+hex(n)+"\n")
    f.write("e2:"+hex(e)+"\n")
    f.write("rsaEncode2 re is:"+hex(c)+"\n")
    f.close()
    print("rsaEncode2_finish!")
    return pow(m,ran,n)

def dsaEncode(p):
    key=genkey(p)
    (r,s,k,q)=sign(rsaEncode2(false_flag),key)
    sig= r.to_bytes(205, 'big') + s.to_bytes(205, 'big') + k.to_bytes(205, 'big')+ q.to_bytes(205, 'big')
    f=open("out","a")
    f.write("dsaEncode :"+base64.b64encode(sig).decode()+"\n")
    f.close()

def genkey(x):
    # DSA
    N=1024
    L=2048-N-1
    while True:
        q = getPrime(N)
        if q-1>x:
            break
    assert(q-1>x)
    while True:
        t = random.getrandbits(L)
        p = (t * 2*q + 1)
        if isPrime(p):
            break
    e = (p-1) // q
    g = pow(2, e, p)
    y = pow(g, x, p)
    print("genkey_finish!")
    return {'y':y, 'g':g, 'p':p, 'q':q, 'x':x}

def sign(m, key):
    g, p, q, x = key['g'], key['p'], key['q'], key['x']
    k = random.randint(1, q-1)
    Hm = int.from_bytes(hashlib.md5(str(m).encode('utf-8')).hexdigest().encode(), 'big')
    r = pow(g, k, p) % q
    s = (inverse(k, q) * (Hm + x*r)) % q
    return (r, s, k, q)

if __name__ == "__main__":
    msg=int.from_bytes(FLAG.encode(),'big')
    rsaEncode(msg)

阅读题目我们发现RSA1中的p就是dsa的密钥x，而在dsa中我们易知 $s\equiv (H(m)+xr)k^{-1}modq$ ,而我们从base64的密文已知(r,s,k,q)。而我们在RSA2可以容易的解出m。于是我们倒着去做，十分容易解出原文。这里注意在RSA解密中e和phi不互质，所以我们要注意先求解 $m^{gcd(e,phi)}$ ，再解m。在dsa中直接是对RSA2中的 $m^{gcd(e,phi)}$ 做hash，所以不需要求解，而在gcd(e,phi)较小，我们可以爆破求解m。同时在RSA2中n可以使用yafu轻松分解。代码如下

import gmpy2
import base64
import hashlib
n2=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
e2=0x57e6b0c63
rsaEncode2=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
dsaEncode="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"
n=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
e=0x12a316381
flag=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
p2 = 92038254802992589334836021211021722416892343074988502820738425960842862985591620926198422759102071572852989309010179392091711882907907766877651639325642946829388178711105989373057972159393372729050507806434617955128196692311614810456775707826817847326317108693718175021812398842350298774399748620926186724103
q2 = 92038254802992589334836021211021722416892343074988502820738425960842862985591620926198422759102071572852989309010179392091711882907907766877651639325642946829388178711105989373057972159393372729050507806434617955128196692311614810456775707826817847326317108693718175021812398842350298774399748620926186723619
assert p2*q2==n2
phi=(p2-1)*(q2-1)
g2=gmpy2.gcd(e2,phi)
d2=gmpy2.invert(e2/g2,phi)
m2=pow(rsaEncode2,d2,n2)
Hm=int(hashlib.md5(str(m2).encode()).hexdigest().encode("hex"),16)
key=base64.b64decode(dsaEncode)
r=int(key[:205].encode("hex"),16)
s=int(key[205:410].encode("hex"),16)
k=int(key[410:615].encode("hex"),16)
q=int(key[615:820].encode("hex"),16)
x=(gmpy2.invert(r,q)*(k*s-Hm))%q
p=n/x
phi=(p-1)*(x-1)
g=gmpy2.gcd(e,phi)
d=gmpy2.invert(e/g,phi)
m=pow(flag,d,n)
for i in xrange(1000000):
	ss=gmpy2.iroot(i*n+m,g)
	if ss[1]:
		print ss[0]
		print hex(ss[0])[2:].decode("hex")
		break

reverse-入门逆向

我们发现加密运算c[i]=m[i]^6+1和密文"akhb~chdaZrdaZudqduvdZvvv|"，求解即可，代码如下

c="akhb~chdaZrdaZudqduvdZvvv|"
m=""
for i in c:
	m=m+chr((ord(i)-1)^6)
print m

reverse-encrypt3

我们会发现在这里是将我们的输入按字符做异或获得最后结果v10，然后将v10按字符和一串密文异或可能获得明文，这里我们尝试将0-128和密文做异或，找到解，代码如下：

c="$.#%9! qwps#&wtu!z$pws$ spwp&rqt{r!'{?"
for j in xrange(128):
	m=""
	for i in c:
		m=m+chr((ord(i)^j))
	print m

逃课的小学生

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