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

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A ciphertext message produced by the knapsack cryptosystem employing the super increasing sequence 1,3,5,11,351,3,5,11,35, modulus m=73m=73, and multiplier a=5a=5, is

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

The knapsack sequence was arrived at by multiplying with 55 the original plaintext code. To reverse the process, we multiply with xx such that

5x1(mod73)\begin{align*} 5x\equiv 1 \pmod{73} \end{align*}

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

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

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

11=01+03+05+111+0353=01+13+05+011+03554=01+13+15+111+13551=01+03+15+111+1355=01+03+15+011+03536=11+03+05+011+135\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:

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

Converting this to letters we arrive at the plaintext message

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

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