|WikiProject Mathematics||(Rated Stub-class, Low-priority)|
There is a slight problem here with one-way functions.
Indeed, Papadimitriou does show that one-way functions exists if and only if . However, he uses a non-standard definition of one-way function. This is explained in detail in his book. I quote from page 285: A definition of one-way functions that is closer to what we need in cryptography would replace requirement (iii) -- that inverting is worst-case difficult -- by a stronger requirement [...]
This is a big difference, and it is (as far as I know) unknown whether one-way functions are implied by .
- I've removed it. The definition isn't really "nonstandard" per-se, it is just the worst-case definition of one-wayness, used only in structural complexity instead of cryptographic complexity. I also removed a similar reference to worst-case one-way functions from the one-way function article. Blokhead 18:58, 23 October 2005 (UTC)
|y| = O(|x|c)
The article currently says in the definition "if x isn't in L, there is no certificate y with |y| = O(|x|c) such that A(x,y) = 1". What does this mean? One might try to interpret this sentence as "for all functions f such that f(x) = O(|x|c), and for any x not in L, there is no certificate y with |y| = f(x) such that A(x,y) = 1" or "there exists a function f such that f(x) = O(|x|c) and for any x not in L, there is no certificate y with |y| ≤ f(x) such that A(x,y) = 1" etc. but these don't make sense. Or is "|y| = O(|x|c)" here just a mistake and it should simply be "if x isn't in L, there is no certificate y such that A(x,y) = 1"? 184.108.40.206 (talk) 21:21, 10 April 2013 (UTC)