Code and ciphers: Julius Caesar, the Enigma and the internet

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chapter 13
Finally then
(214) 17 �(214)(611)�130 754�41�3127�2547
�2547 (mod 3127).
The encipherment of 0214 is therefore 2547.
The other blocks of the message text are enciphered in the same way,
that is, by raising each four-digit number to the power 17 (mod 3127).
The resultant cipher text is
2547 3064 2831 0063 2027 1928.
This cannot be converted back into letters since some, in fact most, digitpairs
are greater than 25.
To decrypt this cipher message the recipient (‘owner’) would raise each
four-digit number to the power of the decipherment key, 2129, (mod
3127). Since
2129�2048�64�16�1�2 11 �2 6 �2 4 �1
the computation would involve raising each four-digit number to the
powers 2 4 , 2 6 and 2 11 by repeatedly squaring and then forming appropriate
products. Using this ‘repeated squaring’ method raising the cipher
blocks to the power 2129 involves ‘only’ 14 multiplications or divisions
instead of over 4000 which a direct, ‘brute force’, calculation would
require. To show how this would be done and to confirm that 2129 is the
decipherment key in this case, here is the calculation for the decipherment
of the fourth four-digit block in the cipher message above (i.e.
0063).
We compute (63) 2129 and find the remainder when we divide by 3127.
From above,
(63) 2129 �(63) 2048 �(63) 64 �(63) 16 �(63) 1
and we proceed to make a table of 2 n th powers of 63 up to n�11 by
repeated squaring. Thus (63) 2 �3969�3127�842 so we put 842 in the
table opposite n�2. We continue in this way and the full table is shown in
Table 13.1. To get (63) 2129 (mod 3127) we now multiply together the four
right-hand numbers opposite n�0, 4, 6 and 11:
Now
(63) 2129 �(63)(523)(1500)(2822)
63�523�32 949�10�3127�1679�1679 (mod 3127),