In lectures we have just defined integrals, and said that if we take a derivative of some set of functions, it is much harder to go back to the original set of functions, if we only know the set of derivatives. However, I recently started reading about one way functions(Wikipedia, nothing serious for now) and I wonder, if thus the derivative is a one way function ?
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According to wikipedia, the existence of such one-way functions is still an open conjecture. – Dietrich Burde Apr 07 '21 at 15:34
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@DietrichBurde So that means, that we can only see from computation time if it is "hard" to invert, right ? – VLC Apr 07 '21 at 15:36
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1They are more like functions that have multiple valued inverses than like one-way functions – Moko19 Apr 07 '21 at 15:39
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1See also https://math.stackexchange.com/questions/20578/why-is-integration-so-much-harder-than-differentiation – lhf Apr 07 '21 at 16:01
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I'd say not really, because of the Risch algorithm. It can, at least in principle, compute the anti-derivative of any elementary function if that function has an elementary anti-derivative. The complete description of the algorithm takes more than 100 pages though, so it is definitely much harder than systematically computing the derivative. Maybe Wikipedia is a good starting point if you want to learn more on the subject: https://en.wikipedia.org/wiki/Risch_algorithm
Maximilian Janisch
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So basically we can say that the derivative is a "trap door" one way function and the private key is the Risch algorithm ? – VLC Apr 07 '21 at 15:41
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1@Bill I suppose so: But it is not a perfect analogy since there is no "hidden information." In principle, anybody could implement the Risch algorithm, while with RSA the private key can be truly hidden. Also, in my answer I am sweeping the following question under the rug: How fast is the Risch algorithm? Maybe there are functions which are easy to differentiate, but for which Risch takes a long time. I don't know much about this, however . – Maximilian Janisch Apr 07 '21 at 15:43