3. 仿射密码

• $K=\{(k_0,k_1)|k_0,k_1\in Z_{26},gcd(k_1,26)=1\}$$K=\{(k_0,k_1)|k_0,k_1\in Z_{26},gcd(k_1,26)=1\}$

• $E_k(m)=k_{1}m+k_0(mod26)$$E_k(m)=k_1m+k_0(mod26)$

• $D_k(c)=k_1^{-1}(c-k_0)(mod26)$$D_k(c)=k_1^{-1}(c-k_0)(mod26)$

注：$k_1{k}_1^{-1}=1mod26$$k_1k_1^{-1}=1 mod 26$

• 例：

  设M=china ,$K=(k_1,k_0)=(9,2)$$K=(k_1,k_0)=(9,2)$

  $E_k(m)=k_{1}m+k_0(mod26)=9m+2(mod26)$$E_k(m)=k_1m+k_0(mod26)=9m+2(mod26)$

  $D_k(c)=k_1^{-1}(c-k_0)(mod26)=3(c-2)(mod26)$$D_k(c)=k_1^{-1}(c-k_0)(mod26)=3(c-2)(mod26)$

  $\begin{array}{c}{E}_k(m)=9\times \left(\begin{array}{c}2\\ 7\\ 8\\ 13\\ 0\end{array}\right)+\left(\begin{array}{c}2\\ 2\\ 2\\ 2\\ 2\end{array}\right)=\left(\begin{array}{c}20\\ 13\\ 22\\ 15\\ 2\end{array}\right)mod26\Rightarrow \left(\begin{array}{c}u\\ n\\ w\\ p\\ c\end{array}\right)=C\end{array}$$E_k(m)=9\times\begin{pmatrix}2\\7\\8\\13\\0\end{pmatrix}+\begin{pmatrix}2\\2\\2\\2\\2\end{pmatrix}=\begin{pmatrix}20\\13\\22\\15\\2\end{pmatrix} mod 26 \Rightarrow \begin{pmatrix}u\\n\\w\\p\\c\end{pmatrix}=C$

  即C=umwpc

   

  $\begin{array}{c}{D}_k(c)=3\times \left(\begin{array}{c}20\\ 13\\ 22\\ 15\\ 2\end{array}\right)-\left(\begin{array}{c}6\\ 6\\ 6\\ 6\\ 6\end{array}\right)=\left(\begin{array}{c}2\\ 7\\ 8\\ 13\\ 0\end{array}\right)mod26\Rightarrow \left(\begin{array}{c}c\\ h\\ i\\ n\\ a\end{array}\right)=M\end{array}$$D_k(c)=3\times\begin{pmatrix}20\\13\\22\\15\\2\end{pmatrix}-\begin{pmatrix}6\\6\\6\\6\\6\end{pmatrix}=\begin{pmatrix}2\\7\\8\\13\\0\end{pmatrix} mod 26 \Rightarrow \begin{pmatrix}c\\h\\i\\n\\a\end{pmatrix}=M$

  恢复明文为M=china