This effect can also be achieved with a

code wheel. Some prefer the wheel because it is less

eye bending. On the other hand, it requires

manual labor to operate.

Either way, they turn the alphabet into a cyclic group and add the key to the clear text message to achieve encryption.

If one is not using a one-time-key, then it may be useful to use a different group than the one shown above. One can reorder the letters, or put them into two cycles (changing the group from C26 to the Cartesian product of C2 and C13). This will make the number of possible keys effectively larger by a factor of 26!/6 as far as your adversary is concerned, and so make cracking the code more difficult. The problem with relying on this is that your adversary may be able to accumulate data over time as you use different keys, and determine what group you are using. Once that has been done, there is no further advantage to using a different square over using the original. However, this can throw off a cracker for some time.

Note that the improvement factor is less than 2*26! because for any cyclic group you choose, it is isomorphic to the same group with a different generator. For example, instead of (0,1,2,3,4...23,24,25,0) one could count (0,3,6,... 24,1,4,7,... 25,2,5,8,... 20,23,0) which would produce the same cryptographic effect but have a different ordering. C26 has 12 generators (all the odd numbers except 13). Therefore, we lose a factor of 12. Then we regain a factor of 2 because there is the same number of groups of the form C2xC13.