Give me two primes:  101, 113, now 101*113= 11413.

phi(11413) = 100*112 = 11200.

Give me a number relatively prime to 11200.

We will go with e=39, relatively prime to phi(n).

So our public key is (n,e) = (11413,39).

So a message M (0 < M < n=11413) is to be encoded.

M=808.

To send this, we need to compute M^e  % n.

808^39   %  11413

At all steps, if e is odd, I multiply ans by M (and %n).

M       e          ans
808     39         808
2323    19         5252
9393	9          5050
5959    4          5050
3838    2          5050
7474    1          909

39 = 32 + 4 + 2 + 1

808^32  * 808 ^4  * 808^2 * 808^1

Let's say the real answer is 909.

When I receive the message 909, I need to be able to decrypt it back to the original message.  So, I need the d value.  I know that e is 39.

The d-value should be such that (de)%phi(n) = 1.

I know that phi(n) is 11200, but I'm the only one who knows this because I'm the only one who knows the factors of n.  Since e is 39, what is d?

11200        0
   39   287  1
    7   5    -287
    4   1    1436
    3   1    -1723
    1        3159 = d

d = 3159

909   3159    909   % by 11413
4545  1579  11312
10908  789   5353
3939   394   5353
5454   197    808
3838    98    808
7474    49   1515
5454    24   1515
3838    12   1515
7474     6   1515
5454     3  11211
3838     1    808