In Fuzzy vault schemes continuity is suggested in a set theoretic sense. However, they are vulnerable to collusion attacks, brute force attacks etc.,
How to make one-way functions continuous if we think about continuity in a mathematical sense. Can you give some suggestions with examples ?
In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the image 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 of a function for it to be called one-way (see Theoretical Definition, below).
The existence of such one-way functions is still an open conjecture. In fact, their existence would prove that the complexity classes P and NP are not equal, thus resolving the foremost unsolved question of theoretical computer science.[1]:ex. 2.2, page 70 The converse is not known to be true, i.e. the existence of a proof that P and NP are not equal would not directly imply the existence of one-way functions.[2]
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.
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