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Concept
One-way function
Summary
In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the of a random input. Here, "easy" and "hard" are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. Not being one-to-one is not considered sufficient for a function to be called one-way (see Theoretical definition, below).
The existence of such one-way functions is still an open conjecture. Their existence would prove that the complexity classes P and NP are not equal, thus resolving the foremost unsolved question of theoretical computer science. The converse is not known to be true, i.e. the existence of a proof that P≠NP would not directly imply the existence of one-way functions.
In applied contexts, the terms "easy" and "hard" are usually interpreted relative to some specific computing entity; typically "cheap enough for the legitimate users" and "prohibitively expensive for any malicious agents". One-way functions, in this sense, are fundamental tools for cryptography, personal identification, authentication, and other data security applications. While the existence of one-way functions in this sense is also an open conjecture, there are several candidates that have withstood decades of intense scrutiny. Some of them are essential ingredients of most telecommunications, e-commerce, and e-banking systems around the world.
A function f : {0,1}* → {0,1}* is one-way if f can be computed by a polynomial time algorithm, but any polynomial time randomized algorithm that attempts to compute a pseudo-inverse for f succeeds with negligible probability. (The * superscript means any number of repetitions, see Kleene star.) That is, for all randomized algorithms , all positive integers c and all sufficiently large n = length(x) ,
where the probability is over the choice of x from the discrete uniform distribution on {0,1}n, and the randomness of .
Note that, by this definition, the function must be "hard to invert" in the average-case, rather than worst-case sense.
Official source
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