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[原创]关于题目more-calc的解法和拓展
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发表于: 2020-12-20 16:19
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看雪似乎不支持数学公式的渲染(还是我没找到?)
数学公式我都用图片替代了,好难受.
由于p高达2048bit,所以极有可能m < p
当m < p 时,下式成立
于是,我们相当与把q给消去了
构造出了新的rsa加密且满足:
然后一个简单的解密即可
且满足∀ m < pi ,(1 <= i <= k) ,则
等效于
其中 i 满足1 <= i <= n
即只需要知道n中的一个素数即可解密
题中首先生成6个素数,并且n由其中三个素数相乘得到
这道题,我们只需要想办法恢复素数集就可以解出了
我们知道每一组的密文和素数在素数集的位置
为了方便描述,我们令p中的素数分别为:p0,p1,p2,p3,p4,p5
于是,我们可以得到如下20个等式:(其实'+'和'-'是一样的,为了好看,统一写作'+')
以第一,第二组方程为例:
可以改写为:
作差可得:
对上式取最大公因式可得p0 * p1
(当然也有可能出现k * pi * pj的情况,不过k通常很小)
组合多组数据再次求公约数就可以解出原本的素数了
实际上,用密文的第一组数据的话,可以只求出前三个素数
由于每一个素数都是1024bit,所以有极大概率m < pi ,(0<= i <= 5)
显然有以下两个式子:
c1−c0=p0∗p1(k1∗p3−k0∗p2)
c7−c0=p0∗(k0∗p1∗p2−k1∗p3∗p4)
两式有公约数p0,可以直接解出p0来作为n
import gmpy2
from Crypto.Util.number import *
flag = b"flag{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}"
p = getStrongPrime(2048)
for i in range(1, (p+1)//2):
s += pow(i, p-2, p)
s = s % p
q = gmpy2.next_prime(s)
n = p*q
e = 0x10001
c = pow(bytes_to_long(flag), e, n)
print(p)
print(c)
import gmpy2
from Crypto.Util.number import *
flag = b"flag{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}"
p = getStrongPrime(2048)
for i in range(1, (p+1)//2):
s += pow(i, p-2, p)
s = s % p
q = gmpy2.next_prime(s)
n = p*q
e = 0x10001
c = pow(bytes_to_long(flag), e, n)
print(p)
print(c)
c = m ^ e mod n
c' = c mod p = m ^ e mod p
c = m ^ e mod n
c' = c mod p = m ^ e mod p
#!/usr/bin/pythonimport gmpy2
from Crypto.Util.number import long_to_bytes
p = 27405107041753266489145388621858169511872996622765267064868542117269875531364939896671662734188734825462948115530667205007939029215517180761866791579330410449202307248373229224662232822180397215721163369151115019770596528704719472424551024516928606584975793350814943997731939996459959720826025110179216477709373849945411483731524831284895024319654509286305913312306154387754998813276562173335189450448233216133842189148761197948559529960144453513191372254902031168755165124218783504740834442379363311489108732216051566953498279198537794620521800773917228002402970358087033504897205021881295154046656335865303621793069
c = 350559186837488832821747843236518135605207376031858002274245004287622649330215113818719954185397072838014144973032329600905419861908678328971318153205085007743269253957395282420325663132161022100365481003745940818974280988045034204540385744572806102552420428326265541925346702843693366991753468220300070888651732502520797002707248604275755144713421649971492440442052470723153111156457558558362147002004646136522011344261017461901953583462467622428810167107079281190209731251995976003352201766861887320739990258601550606005388872967825179626176714503475557883810543445555390014562686801894528311600623156984829864743222963877167099892926717479789226681810584894066635076755996423203380493776130488170859798745677727810528672150350333480506424506676127108526488370011099147698875070043925524217837379654168009179798131378352623177947753192948012574831777413729910050668759007704596447625484384743880766558428224371417726480372362810572395522725083798926133468409600491925317437998458582723897120786458219630275616949619564099733542766297770682044561605344090394777570973725211713076201846942438883897078408067779325471589907041186423781580046903588316958615443196819133852367565049467076710376395085898875495653237178198379421129086523
e = 0x10001
d = gmpy2.invert(e,p-1)
m = gmpy2.powmod(c,d,p)
print(long_to_bytes(m))
#!/usr/bin/pythonimport gmpy2
from Crypto.Util.number import long_to_bytes
p = 27405107041753266489145388621858169511872996622765267064868542117269875531364939896671662734188734825462948115530667205007939029215517180761866791579330410449202307248373229224662232822180397215721163369151115019770596528704719472424551024516928606584975793350814943997731939996459959720826025110179216477709373849945411483731524831284895024319654509286305913312306154387754998813276562173335189450448233216133842189148761197948559529960144453513191372254902031168755165124218783504740834442379363311489108732216051566953498279198537794620521800773917228002402970358087033504897205021881295154046656335865303621793069
c = 350559186837488832821747843236518135605207376031858002274245004287622649330215113818719954185397072838014144973032329600905419861908678328971318153205085007743269253957395282420325663132161022100365481003745940818974280988045034204540385744572806102552420428326265541925346702843693366991753468220300070888651732502520797002707248604275755144713421649971492440442052470723153111156457558558362147002004646136522011344261017461901953583462467622428810167107079281190209731251995976003352201766861887320739990258601550606005388872967825179626176714503475557883810543445555390014562686801894528311600623156984829864743222963877167099892926717479789226681810584894066635076755996423203380493776130488170859798745677727810528672150350333480506424506676127108526488370011099147698875070043925524217837379654168009179798131378352623177947753192948012574831777413729910050668759007704596447625484384743880766558428224371417726480372362810572395522725083798926133468409600491925317437998458582723897120786458219630275616949619564099733542766297770682044561605344090394777570973725211713076201846942438883897078408067779325471589907041186423781580046903588316958615443196819133852367565049467076710376395085898875495653237178198379421129086523
e = 0x10001
d = gmpy2.invert(e,p-1)
m = gmpy2.powmod(c,d,p)
print(long_to_bytes(m))
c = m ^ e mod n
m = c ^ d mod n
c = m ^ e mod n
m = c ^ d mod n
c' = m ^ e mod pi
m = c ^ d mod pi
c' = m ^ e mod pi
m = c ^ d mod pi
from Crypto.Util.number import getPrime
from FLAG import flag
flag = int(flag.hex(), 16)
num = 6
e = 0x10001
p = [getPrime(1024) for _ in range(num)]
f = open('cipherlist.txt', 'w')
for i in range(num):
for j in range(i + 1, num):
for k in range(j + 1, num):
n = p[i] * p[j] * p[k]
f.write(hex(pow(flag, e, n)) + '\n')
from Crypto.Util.number import getPrime
from FLAG import flag
flag = int(flag.hex(), 16)
num = 6
e = 0x10001
p = [getPrime(1024) for _ in range(num)]
f = open('cipherlist.txt', 'w')
for i in range(num):
for j in range(i + 1, num):
for k in range(j + 1, num):
n = p[i] * p[j] * p[k]
f.write(hex(pow(flag, e, n)) + '\n')
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