#### Question

The ciphertext message produced by the knapsack cryptosystem employing the superincreasing sequence 1, 3, 5, 11, 35, modulus m = 73, and multiplier a = 5 is 55, 15, 124, 109, 25, 34. Obtain the plaintext message.

#### Solution

Verified#### Step 1

1 of 2A ciphertext message produced by the knapsack cryptosystem employing the super increasing sequence $1,3,5,11,35$, modulus $m=73$, and multiplier $a=5$, is

$\begin{align*} 55,15,124,109,25,34 \end{align*}$

The knapsack sequence was arrived at by multiplying with $5$ the original plaintext code. To reverse the process, we multiply with $x$ such that

$\begin{align*} 5x\equiv 1 \pmod{73} \end{align*}$

Such an $x$ is given by $x=44$. We multiply each number by $x$ modulo $m$ to get:

$\begin{align*} 11,3,54,51,5,36 \end{align*}$

Now, express all these numbers by the superincreasing sequence above:

$\begin{align*} 11= 0\cdot 1+ 0\cdot 3+0\cdot 5+1\cdot 11+0\cdot 35\\ 3= 0\cdot 1+ 1\cdot 3+0\cdot 5+0\cdot 11+0\cdot 35\\ 54= 0\cdot 1+ 1\cdot 3+1\cdot 5+1\cdot 11+1\cdot 35\\ 51= 0\cdot 1+ 0\cdot 3+1\cdot 5+1\cdot 11+1\cdot 35\\ 5= 0\cdot 1+ 0\cdot 3+1\cdot 5+0\cdot 11+0\cdot 35\\ 36= 1\cdot 1+ 0\cdot 3+0\cdot 5+0\cdot 11+1\cdot 35 \end{align*}$

This gives the binary sequence:

$\begin{align*} 00010\ 01000\ 01111\ 00111\ 00100\ 10001 \end{align*}$

Converting this to letters we arrive at the plaintext message

$\begin{align*} \text{CIPHER} \end{align*}$