The following was going through my brain for the past week. If my thoughts are generally known to the cryptographic community would someone provide a link or two. Even if I’m making a classic newbie mistake.

Cryptographic internal functions operating like x=f(x) can be table lookup s boxes with the inverse s box as a separate table. A 16-bit sbox and its inverse can be implemented with a little math and no tables. Use the prime m = (2^16)+1. Then pick constants p and q such that the recursion

Seed = Seed * p % m;

touches all values, 1..2^16. Then select q such that p * q % m = 1.

p = 49374; // example values
q = 32065;
y = ((x+1) * p % m)-1; // forward
x = ((y+1) * q % m)-1; // reverse
// x+1 puts the 16-bit x into the range (1..2^16) where the recursion works.
// the -1 at the end makes the result fit in 16 bits.

There are over 32000 values of p for which the recursion touches all 2^16 values.

“Mod 0x10001” can be implemented with little more than a subtract. No actual division is required. Is this a reasonable part of a good encryption algorithm? Perhaps to replace 2 8-bit s box tables?

Getting carried away - - 32-bit (non invertible) sbox

Because m=2^32+1 is not a prime there is no full (2^32 term) sequence recursion of the form

Seed = Seed * const % m

This % m operation can also be formed with only a subtract if we don’t care to be super accurate. Or a test and subtract 1 it we want to be accurate. Thus encryption code performing

z ^= ((x+1) * const % m)-1

can be performed in decryption to recover the prior value of z.

Is this “32-bit sbox” a useful concept?