Chapter Questions
In the encryption method using linear shift sequences (15.2), prove that the following system of equations$$\left[\begin{array}{llll}v_k & v_{k+1} & \cdots & v_{k+m-1} \\v_{k+1} & v_{k+2} & \cdots & v_{k+m} \\\cdots \cdots \cdots & \cdots \cdots & \cdots & \cdots \\v_{k+m+1} & v_{k+m+2} & \cdots & v_{k+2 m-2}\end{array}\right]\left[\begin{array}{c}h_0 \\h_1 \\\vdots \\h_m\end{array}\right]=\left[\begin{array}{c}v_{k_m} \\v_{k+m+1} \\\vdots \\v_{k+2 m-1}\end{array}\right]$$holds and makes it possible to find $h(x)$ whenever we know $v_k \ldots v_{k+2 m-1}$.
Prove that the prime-number method in 15.4 has the property that anyone who determines the secret number $\varphi(n)$ knows the factorization of $n$ : since $p q=n$ is known, and $p+q=n-\varphi(n)+1$, it is easy to find $p$ and $q$.