Surprisingly, only a handful of fitting mathematical frameworks have been discovered. The first one is based on the intractability of factoring large numbers, and on the intractability of modular logarithms  giving rise to RSA, and DiffieHellman, and ElGamal. Next to it stands the elliptic curves framework, based on the intractability of reversing "point addition" in the ECC way. And the third is latticebased intractability hinged on the difficulty to find a good basis to express a multidimensional point, when the number of dimensions is high enough.
For the first framework the basic idea is to use substitution tables over a large alphabet. This is featured through modular arithmetics where all integers are mapped into a final set, large as it may be. By raising a number in the set to the power of selected number (let's call it the encryption key), one generates a substituted letter for it (from the same large alphabet), but the task of finding a power value to raise by the new letter and find the former, is highly intractable for our adversary. Namely having the power value to substitute a letter for another, does not lead to the power to reverse that scheme.

