Question

Determine whether there is a key for which the enciphering function for the shift cipher is the same as the deciphering function.

Short answer

$\mathbf{k}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{and}\mathbf{}\mathbf{k}\mathbf{=}\mathbf{13}$

Step-by-step solution

Step 1

Enciphering function of the shift cipher:

$\mathbf{f}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{p}\mathbf{+}\mathbf{k}\mathbf{)}\mathbf{mod}\mathbf{26}$

Corresponding deciphering function:

$\mathbf{f}{}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{p}\mathbf{‐}\mathbf{k}\mathbf{)}\mathbf{mod}\mathbf{26}$

Note the trivial case k = 0 also makes the enciphering function for the shift cipher is the same as the deciphering function.

Let us assume that k is not zero. We want the two function to be the same.

$\left(p+k\right)\mathbf{}\mathbf{mod}\mathbf{}\mathbf{26}\mathbf{=}\left(p‐k\right)\mathbf{}\mathbf{mod}\mathbf{}\mathbf{26}$

Subtractp from each side of the previous equation

$\mathbf{k}\mathbf{}\mathbf{mod}\mathbf{}\mathbf{26}\mathbf{=}\mathbf{‐}\mathbf{k}\mathbf{}\mathbf{mod}\mathbf{}\mathbf{26}$

Adding 26 does not have an effect on the result due to the modulo 26

$\mathbf{k}\mathbf{}\mathbf{mod}\mathbf{}\mathbf{26}\mathbf{=}\mathbf{26}\mathbf{‐}\mathbf{k}\mathbf{}\mathbf{mod}\mathbf{}\mathbf{26}$

Step 2

Since k is a number between 0 and 25 (including), 26-k is a number between1 and26 . assuming thatk is not zero, we then obtain

$\mathbf{k}\mathbf{=}\mathbf{26}\mathbf{‐}\mathbf{k}$

Add k to each side of the equation

$\mathbf{2}\mathbf{k}\mathbf{=}\mathbf{26}$

Divide each side of the equation by

$\mathbf{k}\mathbf{=}\mathbf{13}$

For the case $\mathbf{k}\mathbf{=}\mathbf{13}$ , the enciphering function for the shift cipher is then the same as the deciphering function.