How to shift each letter of the string by a given number of letters?

If you want to rotate also the capitals you could use something like this:

cat data.txt | tr '[a-z]' '[n-za-m]' | tr '[A-Z]' '[N-ZA-M]'

where data.txt has whatever you want to rotate.

Do you mean something like ROT13:

pax$ echo 'hello there' | tr '[a-z]' '[n-za-m]'
uryyb gurer

pax$ echo 'hello there' | tr '[a-z]' '[n-za-m]' | tr '[a-z]' '[n-za-m]'
hello there

For a more general solution where you want to provide an arbitrary rotation (0 through 26), you can use:

#!/usr/bin/bash

dual=abcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyz
phrase='hello there'
rotat=13
newphrase=$(echo $phrase | tr "${dual:0:26}" "${dual:${rotat}:26}")
echo ${newphrase}

$ alpha=abcdefghijklmnopqrstuvwxyz
$ rot=3
$ sed "y/${alpha}/${alpha:$rot}${alpha::$rot}/" <<< 'foobar'
irredu