We investigate the structure of a function relevant to cryptography, given by f : x -> xx mod p, for p a prime. We call f the self-power map. Given x, it is easy to calculate f(x) = xx (mod p). However, it is thought to be difficult to quickly calculate f−1(xx). That is, given xx =c (mod p), for a fixed c, it is difficult to quickly solve for x. We call the problem of finding the inverse of the self-power map the Self-Power Problem. As a variation of the Discrete Logarithm Problem, the Self-Power Problem is thought to be difficult to solve and therefore considered safe for use in some versions of the ElGamal Digital Signature Algorithm. Nonetheless, utilizing functional graphs to represent the map has revealed non-random structural properties, which we describe primarily through number theory and statistics.
Friedrichsen, Matthew; Larson, Brian; and McDowell, Emily, "Structure and Statistics of the Self-Power Map" (2010). Mathematical Sciences Technical Reports (MSTR). 26.