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In general it's hard to calculate the roots of a given polynomial P(x). But the other way back calculating the coefficients is much easier (Vieta's formulas).

If this is a one-way function, is it possible to create a public-key cryptosystem based on these facts? Or does it already exist?

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5  
Actually, it is easy to compute the roots of a polynomial $P(x)$ defined over a finite field. – poncho Mar 19 '14 at 16:58
1  
Ok, but what about non-finite fields? – RomeoAndJuliet Mar 19 '14 at 17:07
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With an infinite field, $P(x)$ may not be expressible in a fixed number of bits (or, depending on the field, even a bounded number of bits). That puts rather a crimp in the cryptographical applications. – poncho Mar 19 '14 at 17:44

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