Caesar Cipher

Caesar Cipher: ROT

MOD Arithmetic can be used determine a secret key

Our definition of congruent integers can be very helpful if we are to determine whether a fancy encryption like ROT1253 - which uses the encoding function f : P  P+1253 - will produce the same encryption as ROT0 (which we don’t want to use) or ROT1 or …. ROT25 ? Here is how, plain and simple: 1253=48*26+5 where the remainder of 5 tells us that we have to turn our disk 5 positions to the right (“ROT5”-encryption) to obtain our cipher letters. We ignored the redundant 48 full turns. I.e. when encoding the plain letter Y, the shift by 5 to the right yields the cipher letter d.

CONCLUSION:
In order to determine which encryption “ROT N” yields we simply have to divide N by the modulus 26. The obtained remainder R – which must be a number between 0 and 25 – tells us the number of shifts to encrypt since N is congruent to R mod 26:  N = R mod 26.

Awesome! We have discovered a criterion that helps us to determine immediately how many position shifts we have to do given any ROT number as a secret key. For example, ROT5, ROT31, ROT57, ROT 161 and ROT1253 encryptions eventually end up 5 positions to the right. ROT3, ROT29, ROT55, ROT81, ROT107 encryptions eventually end up 3 positions to the right and thus shift the same plain letters to the same cipher letters. Furthermore, a shift of 5 letters to the right is identical to a 21-position-shift to the left (we denote it as ROT(-21)) and also to a 47-position-shift to the left (ROT(-47)) on the Caesar Disk.

On Caesar Disks:

                   …=ROT(-47)= ROT(-21)==ROT5=ROT31=ROT57=ROT 83=…
  ...=ROT(-49)= ROT(-23)==ROT3=ROT29=ROT55=ROT 81=….