Hard-core predicate

In cryptography, a hard-core predicate of a one-way function f is a predicate b (i.e., a function whose output is a single bit) which is easy to compute given x but is hard to compute given f(x). In formal terms, there is no probabilistic polynomial-time algorithm that computes b(x) from f(x) with probability significantly greater than one half over random choice of x.
A hard-core function can be defined similarly.
A hard-core predicate captures "in a concentrated sense" the hardness of inverting f.
While a one-way function is hard to invert, there are no guarantees about the feasibility of computing partial information about the preimage c from the image f(x). For instance, while RSA is conjectured to be a one-way function, the Jacobi symbol of the preimage can be easily computed from that of the image.
It is clear that if a one-to-one function has a hard-core predicate, then...