# Math name for asymetric-pair relationship?

1. Oct 19, 2011

### Mr Peanut

Math name for "asymetric-pair" relationship?

Say I have a function (F) that takes an input (P) and returns an output (E). Suppose F isn't invertible so, knowing F and E, one could not reconstruct P.

Suppose also that there is another function (R) that can take E as input and returns P... without any prior knowledge about P other than it was generated with F. (R could also be non-invertible but need not necessarily be.)

I know the computer people refer to an imperfect, practical application of this idea as public key encryption or asymmetric encryption. It assumes that prime number factors for large numbers cannot be determined analytically in practical time.

But the concept has a perfect, general, underlying mathematical interpretation.

1) Does mathematics have a name for such a pair of functions?

2) Could they, in fact, exist?

3) If they do exist; are there any examples?

2. Oct 19, 2011

### spamiam

Re: Math name for "asymetric-pair" relationship?

If I'm not mistaken, R is a left inverse of F since, as you've defined it,
$$R(F(P)) = R(E) = P$$
for any P in the domain of F. Therefore $R \circ F = \text{id}$.

For an easy example, consider the functions $f(x) = \tan(x)$ and $g(x) = \arctan{x}$ defined on the reals. Then $f(g(x)) = x$ but $g(f(x))= x+2 \pi n$.