Example for encryption purposes.

Take a message. Convert characters to ASCII. Concatenate the 3-digit codes
of all the characters. Break the digit string into sets of 100 digits
to get our "plaintext". We will scramble the plaintext to get a
"ciphertext", another collection of 100-digit numbers. The person on
the other end will have to unscramble each 100-digit ciphertext chunk,
then reverse the initial process by stringing these together into a long
stream of digits, taking them in threes, viewing those as ASCII codes, and
translating into characters.

So the mathematics is the pair of processes which transform 100-digit
numbers into others. In each case it's done by replacing  x  by  x^r mod N.
We can use an  N  which we make publicly known (if it's hard to factor...)
and even make public the exponent  r  used for ENcryption. The strength of
the method relies on the fact that, from this information alone, it's
hard to determine the exponent  r  used for  DEcryption. 

We choose to work with  N:=10^100+1,  although any 101-digit number will do.

Its prime factors:
[
73,
137,
401,
1201,
1601,
1676321,
5964848081,
129694419029057750551385771184564274499075700947656757821537291527196801
];


The prime decomposition of each  p-1:
[
2^3*3^2,
2^3*17,
2^4*5^2,
2^4*3*5^2,
2^6*5^2,
2^5*5*10477,
2^4*5*59*163*7753,
2^7*3*5^2*336877507*204709068163*18515344367953624441*10580572446323227392868955843


So the total set of prime divisors of  phi(N) is:
2^36
3^4
5^10
17
59
163
7753
10477
336877507
204709068163
18515344367953624441
10580572446323227392868955843

In particular, the smallest number prime to phi(N) is 7; we use this as the
encrytion exponent  r.

The decryption exponent is the inverse of  7  mod  phi(N). This inverse is:

6965979479733780200695733412491223155589206500761725425765946414664585354290201612951786057142857143