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A CM construction for curves of genus 2 with \(p\)-rank 1. (English) Zbl 1218.11060
Let \(p\) be a prime number and \(k=\mathbb{F}_{p^2}\) the finite field of \(p^2\) elements. The paper contains algorithms to construct genus two curves over \(k\) whose Jacobian \(J\) has \(p\)-rank one, the group \(J(k)\) has a subgroup of prescribed prime order \(r\), and the embedding degree of \(J\) with respect to \(r\) is also prescribed. Similar constructions were known for the ordinary and supersingular Jacobians, but for the \(p\)-rank one case no such construction existed yet.
The crucial ingredient of the algorithms is an accurate analysis of the arithmetic properties of Weil numbers of \(p\)-rank one simple abelian surfaces over \(k\), and the non-Galois CM quartic fields generated by them. Once the right Weil numbers are constructed, the curves are obtained by standard variants of the CM method. The \(p\)-rank one Weil numbers lead to specific problems concerning class polynomials that are dealt with in the paper as well. The algorithms allow the construction of curves with parameters in the range of cryptographic applications in a few seconds on a standard PC.

MSC:
11G10 Abelian varieties of dimension \(> 1\)
11G20 Curves over finite and local fields
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94A60 Cryptography
Software:
Echidna; ECPP
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