RSA原理

RSA原理

RSA流程

• 随机生成素数 p , q

• 计算 n = p * q \(\phi(n)\) = (p-1)*(q-1)

• 找个e , 满足 1<e<\(\phi(n)\) 且 e 与 \(\phi(n)\)互素

• e * d \(\equiv\) 1 mod \(\phi(n)\) 得到 d

• 公钥 (e,n) 私钥 (d,n)

• 加密 \(c \ = \ m^e \ mod \ n\ \ \ \ \ \ 1<m<n\)

• 解密 \(m \ = \ c^d \ mod \ n\)

RSA证明

\[m^{e*d}\ mod \ n \\m^{k*\phi(n)+1}\ mod\ n\\m*m^{k*\phi(n)}\ mod \ n\\ \]

• m,n 互素

  根据欧拉定理可得 \(m*m^{k*\phi(n)}\ mod \ n\ =\ m\)

• m,n 不互素

  m = tp 或 m = tq

  不妨令 m = tp

  gcd(m,q) = 1

  由欧拉定理可得

  \[(tp)^{q-1}\ \equiv\ 1\ mod \ q \]

  变换可得

  \[(tp)^{k*(q-1)*(p-1)}\ * tp\ \equiv\ tp\ mod\ q \]

  即

  \[(tp)^{e*d}\ \equiv\ tp\ mod\ q \]

  ∴ \((tp)^{e*d}=tp+hq\)

  ∵ \(tp^{e*d} \equiv\ 0\ mod\ p\) \(gcd(p,q)=1\)

  ∴ \(h=h^{'}*p\)

  ∴ \(m^{e*d}=tp + h^{'}*p*q\)

  ∴ \(m^{e*d}\ \ \equiv\ \ tp\ \ mod\ \ p*q\)

  ∴ \(m^{e*d}\ \equiv\ m\ \ mod\ \ n\)

  ∴ \(c^d\ \equiv\ m\ \ mod\ \ n\)

代码

加密

from Crypto.Util.number import *
from gmpy2 import *

cipher = b'flag{xxxxx}'

m = bytes_to_long(cipher)
p = getPrime(1024)
q = getPrime(1024)
n = p*q
phi = (p-1)*(q-1)
e = 0x10001
d = invert(e,phi)

c = powmod(m,e,n)

解密

m = powmod(c,d,n)
cipher = long_to_bytes(m)