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Primality testing with Gaussian periods. (English) Zbl 1429.11221

The paper provides a deterministic algorithm which decides the primality of a natural number \(n\) with complexity \((\log n)^6(2+\log\log n)^c\) bits, \(c\) a real number effectively computable. This improves a previous result due to M. Agrawal et al. [Ann. Math. (2) 160, No. 2, 781–793 (2004; Zbl 1071.11070)] with complexity that look similar, but with exponent \(21/2\) instead of \(6\). Both papers follow similar ideas using a mixture of algebraic and analytic number theory tools.
The proof needs a result from additive number theory due to D. Bleichenbacher [The continuous postage problem. Unpublished manuscript (2003)], (see Theorems 3 and 4 of the present paper) and it reasons in some ring extensions of \(\mathbb{Z}/n\mathbb{Z}\) (pseudofields, finite fields if \(n\) is a prime), extensions constructed following and improving the construction of finite fields of L. M. Adleman and H. W. Lenstra jun. [“Finding irreducible polynomials over finite fields”, in: Proceedings of the eighteenth annual ACM symposium on theory of computing, STOC 1986. New York, NY: Association for Computing Machinery (ACM), 350–355 (1986; doi:10.1145/12130.12166)], which adds to \(\mathbb{Z}/p\mathbb{Z}\) a set of Gaussian periods parametrized by a period system.
Section 1 enunciates the main result (Theorem 1) and the auxiliary results necessaries to prove it (Theorems 2 to 5) and Section 2 gives the definition of pseudofield and period system and states some properties of them whose proof is delayed to later. Then Section 3 gives the proof of Theorem 1 (see Algorithm 3.3) assuming the validity of the auxiliary results.
The rest of the paper deals with the auxiliary results. Section 4 provides the proof of Theorem 2 (which gives a method to construct finite fields) and Sections 5 to 8 the proof of the properties of pseudofields states in Section 2.
The following Sections are more analytic. Section 9 gives the proof of Theorem 3 (inspired by the result of Bleichenbacher) and Section 10 a stronger version of Theorem 4 (a number-theoretic application of Theorem 3). Sections 11 and 12 are devoted to the proof of Theorem 5 and finally Section 13 proves the existence of periodic systems.

MSC:

11Y11 Primality
11N13 Primes in congruence classes
11P70 Inverse problems of additive number theory, including sumsets

Citations:

Zbl 1071.11070
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References:

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