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polynomial asymptotics in the paper section 4.2.1

May 6, 2015 at 7:47 PM
The paper says a worker, in order to include h(s) into the proof, has to interpolate p(x), and then divide it with t(x) to get h(x), then calculate h(s).

As a worker, I would calculate p(s), then t(s), and let h(s)=p(s)*(t(s))^(-1). Suppose this h(s) is good, this approach is much less complicated, compared to the approach described in the paper, which dictates at least O(n (log(n))^2) complexity.

I am wondering why this simpler approach to get h(s) for the worker does not work. Any help is appreciated.
May 8, 2015 at 2:24 AM
I'm not sure I understand your proposal. The worker doesn't know s, so he can only calculate p(x), h(x), etc. Your multiplication by t(x)^{-1} is the division mentioned in the paper (note that the inverse of t(x) can be cached). The interpolation is necessary because the polynomials that constitute p(x) are kept in point-value form; i.e., they're represented in terms of their values at the root values of t(x) chosen by the protocol. If you choose the root values cleverly, the interpolation only requires O(N log N).
May 10, 2015 at 6:25 AM
Thank you for taking time to respond to me! I had this question since I did not really understand the problem (even now, I still have not fully understood it). Now, I know the worker does NOT know s, t(s), vk(s), wk(s), yk(s). I thought the worker knows all these terms, then the worker can calculate p(s) easily, and find the inverse of t(s) (a number) easily, then the worker can get h(s) easily.

Now, I know the worker knows only the public evaluation key and public verification key. That's to say, the worker knows only g^s, g^t(s), etc. Then, I am wondering how hard it is to find the g^h(s) in the equation [e(g,g)]^(r_y*p(s))=e(g_y^t(s),g^h(s)).

Furthermore, you stir up a new question - About interpolation.

Since the workers knows t(x),vk(x),wk(x),yk(x), then the worker knows p(x). There is not need to interpolate p(x). Do you mean interpolation for vk(x), wk(x), yk(x)?

Thanks again.