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Elliptic periods and primality proving. (English) Zbl 1310.11118
From the text: We construct extension rings with fast arithmetic using isogenies between elliptic curves. As an application, we give an elliptic version of the AKS primality criterion.
The main restriction of classical Kummer theory is that not every ring \(R\) has a primitive \(d\)-th root of unity. One may look for an auxiliary extension \(R'\supset R\) that contains such a primitive root, but this may result in many complications and a great loss of efficiency. Another approach, already experimented in the context of normal bases for finite fields extensions, consists in replacing the multiplicative group \(\mathbb G_m\) by some well chosen elliptic curve \(E\) over \(R\). We then look for a section \(T\in E(R)\) of exact order \(d\). Because elliptic curves are many, we increase our chances to find such a section. We call the resulting algebra \(S\) a ring of elliptic periods because of the strong analogy with classical Gauss periods.
The first half of the present work is devoted to the explicit study of Kummer theory of elliptic curves and, more specifically, to the algebraic and algorithmic description of the residue algebras constructed. The resulting elliptic functions and equations are not quite as simple as binomials. Still they can be described very explicitly and quickly, e.g. in quasi-linear time in the degree \(d\). The geometric situation is summarized by Theorem 1 and the \(R\)-algebra \(S\) of elliptic periods is described by Theorem 2.
The second half of the paper proposes an elliptic version of the AKS primality criterion. A general, context free, primality criterion in the style of Berrizbeitia is first given in Theorem 3. This criterion involves an \(R\)-algebra \(S\) where \(R = \mathbb Z/n\mathbb Z\) and \(n\) is the integer to be tested for primality. If we take \(S\) to be \(R[x]=(x^d-\alpha)\), we recover results by Berrizbeitia and his followers. If we take \(S\) to be a ring of elliptic periods, we obtain the elliptic primality criterion of Corollary 2.
11Y11 Primality
11Y16 Number-theoretic algorithms; complexity
14H52 Elliptic curves
Full Text: DOI arXiv
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