Informally, a function 
is a trapdoor one-way
function if
1. It is a one-way function, and
2. For fixed public key 
, 
is viewed as a function 
of 
that maps 
bits to 
bits. Then there is an efficient algorithm that, on input
produces 
such that 
,
for some trapdoor key 
.
is a trapdoor one-way hash function
if 
is also a one-way hash function, i.e., if
additionally
3. Given 
and 
,
it is hard to find a message 
such that 
.
It is not known if a trapdoor one-way function can be constructed from any one-way function.
An example of a trapdoor one-way function is factorization of a product of two large primes. While selecting and verifying two large primes and multiplying them together is easy, factoring the resulting product is (as far as is known) very difficult. This is the basis for RSA encryption, which is conjectured to be trapdoor one-way.
