On the security of pseudorandomized information-theoretically secure schemes
ICITS'09 Proceedings of the 4th international conference on Information theoretic security
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The zero testing and sign determination problems of real algebraic numbers of high extension degree are important in computational complexity and numerical analysis. In this paper we concentrate on sparse cyclotomic integers. Given an integer n and a sparse polynomial f(x) = ckx^{ek}+ ck - 1x^{ek - 1}+\cdot\cdot\cdot+ c1x^{e1} over {\rm Z}, we present a deterministic polynomial time algorithm to decide whether f(\omega n) is zero or not, where \omega n denotes the n-th primitive root of unity e^{2\pi \sqrt { - 1/n} }. All previously known algorithms are either randomized, or do not run in polynomial time. As a side result, we prove that if n is free of prime factors less than k + 1, there exist k field automorphisms \sigma _{1,} \sigma _{2,}\cdot\cdot\cdot _, \sigma _k in the Galois group Gal {\text{(Q(}}\omega {\text{n)/Q)}} such that for any nonzero integers c_{1,} c_{2,}\cdot\cdot\cdot _, c_k and for any integers 0 \leqslant e1 \le e2 \le\cdots\le ek \le n, there exists i so that \left| {\sigma i} \right.(ck\omega _n^{ek}+ ck - 1\omega _n^{ek - 1}+\cdots+ c_1^{} \omega _n^{e1} \left. ) \right| \geqslant 1/2^{(k^2 \log n + k\log k)} .