Public key: $(e,n) = (17, 1 066 601)$
Private key: $d = 939 293$
Message: 452423293428
Table:
Digital signature:
I want to decrypt and encrypt the message by:
$P = C^d (mod 1 066 601)$
$C = P^e (mod 1 066 601)$
My book says, I have to group the resulting numbers into blocks of 2m decimal digits, where 2m is the largest positive even integer such that all blocks of number equivalents corresponding to m letters (viewed as a single integer with 2m decimal digits) are less than n, e.g. if $6565<n<656565$ then $m = 2$
So:
$C \equiv 452423^{939293} \equiv 593991 (mod 1 066 601)$
$C \equiv 293428^{939293} \equiv 85260 (mod 1 066 601)$
I also tried blocks of 5 and 7 and 5 works!
Why do you have to group the resulting number into blocks of 2m decimal digits, where 2m is the largest positive even integer?
You do not have to do so, but it is a simple method that works.
You can partition your message as you like, the only requirements are that the largest partition is less that $n$ and, of course, you have an algorithm/protocol which can construct and reconstruct the partitions.
In your example you could encrypt/decrypt each character separately, but this would be inefficient and less secure.
Edit: Your partitions have to be less than the modulus $n$, because if you decrypt an encrypted message with $m^d \pmod n$ then the result is less that $n$. And you loose security with small partitions because you might take the $e^{th}$ root for smaller $e,$ e.g. if $e=3$ and if you encrypt 'x' you have $33^3 = 35937$ and you can decrypt without knowing $d$, simply by computing $35937^{1/3}= 33.$