通过公钥解密密文思路（256bits RSA）

256bit RSA公钥安全系数极低，只需要几分钟即可破解密文，本文综合其他文章记录了一次解密256bits RSA加密的密文的过程，仅作为备忘。

1.分解公钥，分解出n与e:

　　1.1使用openssl（红色标记是e与n）

 qi@zhuandshao:~/download/iscc-ctf/RSA$ openssl rsa -pubin -text -modulus -in public.pem

 Public-Key: ( bit)

 Modulus:

 :a4:::de:fd::8b:::b4:e2:eb:1e:c9:

 bf::a6:1c:d9:c3:b5:a0:a7::::1e:eb:2f:

 b8::a7

13 Exponent: 65537 (0x10001)　　　　　　　　　　　　　　　 #e
14
15 Modulus=A41006DEFD378B7395B4E2EB1EC9BF56A61CD9C3B5A0A73528521EEB2FB817A7 #n

 writing RSA key

 -----BEGIN PUBLIC KEY-----

 MDwwDQYJKoZIhvcNAQEBBQADKwAwKAIhAKQQBt79N4tzlbTi6x7Jv1amHNnDtaCn

 NShSHusvuBenAgMBAAE=

 -----END PUBLIC KEY-----

 qi@zhuandshao:~/download/iscc-ctf/RSA$

1.2使用脚本

 from Crypto.PublicKey import RSA

 pub = RSA.importKey(open('xxx\public.pem').read())

 n = long(pub.n)

 e = long(pub.e)

 print n

 print e

2.使用msieve来对n来分解因式p、q:(红色标记部分)

 qi@zhuandshao:~/download/iscc-ctf/RSA$ msieve 0XA41006DEFD378B7395B4E2EB1EC9BF56A61CD9C3B5A0A73528521EEB2FB817A7 -v

 Msieve v. 1.54 (SVN )

 Wed May  :: 

 random seeds:  1225946d

 factoring  ( digits)

 no P-/P+/ECM available, skipping

 commencing quadratic sieve (-digit input)

 using multiplier of 

 using generic 32kb sieve core

 sieve interval:  blocks of size 

 processing polynomials in batches of 

 using a sieve bound of  ( primes)

 using large prime bound of  ( bits)

 using trial factoring cutoff of  bits

 polynomial 'A' values have  factors

 restarting with  full and  partial relations

  relations ( full +  combined from  partial), need 

 sieving complete, commencing postprocessing

 begin with  relations

 reduce to  relations in  passes

 attempting to read  relations

 recovered  relations

 recovered  polynomials

 attempting to build  cycles

 found  cycles in  passes

 distribution of cycle lengths:

 length  : 

 length  : 

 largest cycle:  relations

 matrix is  x  (5.3 MB) with weight  (29.92/col)

 sparse part has weight  (29.92/col)

 filtering completed in  passes

 matrix is  x  (4.0 MB) with weight  (33.55/col)

 sparse part has weight  (33.55/col)

 saving the first  matrix rows for later

 matrix includes  packed rows

 matrix is  x  (2.6 MB) with weight  (24.46/col)

 sparse part has weight  (17.67/col)

 commencing Lanczos iteration

 memory use: 2.7 MB

 lanczos halted after  iterations (dim = )

 recovered  nontrivial dependencies

87 p39 factor: 258631601377848992211685134376492365269------------------->p
88
89 p39 factor: 286924040788547268861394901519826758027------------------->q

 elapsed time ::

 qi@zhuandshao:~/download/iscc-ctf/RSA$

3.使用脚本来生成私钥文件(修改红色部分)

 import math

 import sys

 from Crypto.PublicKey import RSA

 keypair = RSA.generate(1024)

11 keypair.p = 258631601377848992211685134376492365269           #msieve求解的ｐ
12
13 keypair.q = 286924040788547268861394901519826758027　　　      #msieve求解的q　　　　　
14
15 keypair.e = 65537                                             #分解出的e

 keypair.n = keypair.p * keypair.q

 Qn = long((keypair.p-1) * (keypair.q-1))

 i = 1

 while (True):

 x = (Qn * i ) + 1

 if (x % keypair.e == 0):

 keypair.d = x / keypair.e

 break

 i += 1

 private = open('private.pem','w')

 private.write(keypair.exportKey())

 private.close()

4.使用生成的privete.pem私钥文件对密文解密

　

 　openssl rsautl -decrypt -in flag.enc -inkey private.pem -out flag

附录：

1.linux下安装msieve

sourceforgot上下载软件源代码包：

https://sourceforge.net/projects/msieve/

解压后

 $ cd msieve-code/

 $make

 to build:

 make all

 add 'WIN=1 if building on windows

 add 'WIN64=1 if building on 64-bit windows

 add 'ECM=1' if GMP-ECM is available (enables ECM)

 add 'CUDA=1' for Nvidia graphics card support

 add 'MPI=1' for parallel processing using MPI

 add 'BOINC=1' to add BOINC wrapper

 add 'NO_ZLIB=1' if you don't have zlib

 $ make all ECM= #根据自己的配置进行选择

应该会报错ｇｍｐ.h不存在，安装高精度数学库就可以啦。

2.linux安装gmp(高精度数学库)　

环境：ubuntu
17.04

源代码：https://gmplib.org/

下载gmp-5.0.1的源代码，解压至gmp-5.0.1目录。

#lzip -d gmp-6.1..tar.lz
#tar -xvf gmp-6.1..tar

su切换至超级用户权限。
./configure
--prefix=/usr  --enable-cxx

提示：
checking
for suitable m4… configure: error:
 No
usable m4 in $PATH or /usr/5bin (see config.log for
reasons).
根据提示查看config.log日志文件，发现文件太大，何处找原因呢？
没有办法，直接google搜索上面的英文提示。
居然马上就找到了资料解决这个问题，原来是缺少m4软件包。
查了一下m4是一个通用的宏处理器，由Brian
Kernighan 和Dennis
Ritchie设计。
apt-get
install build-essential
m4
安装完毕，其中的build-essential是ubuntu下用来解决安装g++/gcc编译环境依赖关系的软件包。

开始编译，安装gmp数学库。

 ./configure --prefix=/usr  --enable-cxx
 make
 make check
 make install 

参考资料：

　　1.256-bitRSA破解-实验吧

　　2.[翻译]初学者向导―GGNFS和MSIEVE分解因数-『外文翻译』-看雪安全论坛：http://bbs.pediy.com/thread-156206.htm

　　3.ubuntu10.4下安装和使用GMP高精度数学库：http://blog.csdn.net/bingqingsuimeng/article/details/12748341