Cryptology - Unofficial St. Mary's College of California Web Site

3.1. THE REMAINDER OPERATOR 33
Example: Encipher Stop Turn Back with a multiplication of 3.
We start as usual, by listing the letters of the message as well as their
numerical values. But rather than adding three, we multiply these values by
three.
plaintext s t o p t u r n b a c k
plainnumbers 19 20 15 16 20 21 18 14 2 1 3 11
ciphernumbers 57 60 45 48 60 63 54 42 6 3 9 33
ciphertext E H S V H K B P F C I G
Turning this ciphernumbers back into letters takes a bit of work. Replacing 6
by F and 3 by C was easy. Replacing 42 and 45 by G and J demands some
counting, and figuring out what to do with 54, 57, 60 and 63 means we have
to count through the alphabet two-and-a-half times. It’s a good thing we only
multiplied by 3 and not 13!
The answer is EHSV HKBP FCIG.
⋄
Was all the counting worth it Remember that a Caesar cipher is a weak
cipher because of a lack of mixing – letters that are adjacent in the alphabet
move to letters that are also adjacent. In the two Stop Turn Back examples
the letters rstu from the message became UVWX and YZAB, respectively, and
abc similarly became DEF and HIJ. (In fact, I picked these words because of
the many consecutive letters.) A Caesar cipher succeeds only in shifting the
aei, no, and rst-uvwxyz patterns, but does not destroy them. A better cipher
should destroy these patterns and move the letters away from their neighbors.
And our multiplication example did exactly this: the letters rstu became BEHK,
and abc became CFI. Based on this one example, it appears that multiplication
will provide more security than addition. 1
So multiplication can produce a cipher system that is better than the Caesar
ciphers – if we can find a quick way of turning large numbers into the equivalent
letters of the alphabet. This must be our next goal.
3.1 The Remainder Operator
Which ciphernumbers will be converted to the cipherletter A Of course 1. Also
27, since 27 is one more than once through the alphabet. Similarly 53 which is
one more than twice through the alphabet. And 79, 105, . . . In fact, we could
1 You may have wondered why Cryptology belongs to the field of Mathematics, rather than
that of English. After all, a cipher is supposed the hide the meaning of the message, and
surely English is about detecting the meanings of letters and words. Well, we’ve just seen the
answer. The translation a=1, b=2, c=3, ... z=26, turns the subject of Cryptology into an area
of mathematics. Similarly, once we’ve tried “add three” and “multiply by three”, almost any
mathematical device that turns one number into another may be tried out as a enciphering
method.