On Gauss sums over dedekind domains

It is a difficult question to generalize Gauss sums to a ring of algebraic integers of an arbitrary algebraic number field. In this paper, we define and discuss Gauss sums over a Dedekind domain of finite norm. In particular, we give a Davenport–Hasse type formula for some special Gauss sums. As an application, we give some more precise formulas for Gauss sums over the algebraic integer ring of an algebraic number field (see Theorems 4.1 and 4.2).

A new twofold Cornacchia-type algorithm and its applications

<p style='text-indent:20px;'>We focus on exploring more potential of Longa and Sica's algorithm (ASIACRYPT 2012), which is an elaborate iterated Cornacchia algorithm that can compute short bases for 4-GLV decompositions. The algorithm consists of two sub-algorithms, the first one in the ring of integers <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula> and the second one in the Gaussian integer ring <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{Z}[i] $\end{document}</tex-math></inline-formula>. We observe that <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{Z}[i] $\end{document}</tex-math></inline-formula> in the second sub-algorithm can be replaced by another Euclidean domain <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z}[\omega] $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M5">\begin{document}$ (\omega = \frac{-1+\sqrt{-3}}{2}) $\end{document}</tex-math></inline-formula>. As a consequence, we design a new twofold Cornacchia-type algorithm with a theoretic upper bound of output <inline-formula><tex-math id="M6">\begin{document}$ C\cdot n^{1/4} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M7">\begin{document}$ C = \frac{3+\sqrt{3}}{2}\sqrt{1+|r|+|s|} $\end{document}</tex-math></inline-formula> with small values <inline-formula><tex-math id="M8">\begin{document}$ r, s $\end{document}</tex-math></inline-formula> given by the curves.</p><p style='text-indent:20px;'>The new twofold algorithm can be used to compute <inline-formula><tex-math id="M9">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>-GLV decompositions on two classes of curves. First it gives a new and unified method to compute all <inline-formula><tex-math id="M10">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>-GLV decompositions on <inline-formula><tex-math id="M11">\begin{document}$ j $\end{document}</tex-math></inline-formula>-invariant <inline-formula><tex-math id="M12">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> elliptic curves over <inline-formula><tex-math id="M13">\begin{document}$ \mathbb{F}_{p^2} $\end{document}</tex-math></inline-formula>. Second it can be used to compute the <inline-formula><tex-math id="M14">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>-GLV decomposition on the Jacobian of the hyperelliptic curve defined as <inline-formula><tex-math id="M15">\begin{document}$ \mathcal{C}/\mathbb{F}_{p}:y^{2} = x^{6}+ax^{3}+b $\end{document}</tex-math></inline-formula>, which has an endomorphism <inline-formula><tex-math id="M16">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> with the characteristic equation <inline-formula><tex-math id="M17">\begin{document}$ \phi^2+\phi+1 = 0 $\end{document}</tex-math></inline-formula> (hence <inline-formula><tex-math id="M18">\begin{document}$ \mathbb{Z}[\phi] = \mathbb{Z}[\omega] $\end{document}</tex-math></inline-formula>). As far as we know, none of the previous algorithms can be used to compute the <inline-formula><tex-math id="M19">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>-GLV decomposition on the latter class of curves.</p>

Quantum computing thanks to Bianchi groups

It has been shown that the concept of a magic state (in universal quantum computing: uqc) and that of a minimal informationally complete positive operator valued measure: MIC-POVMs (in quantum measurements) are in good agreement when such a magic state is selected in the set of non-stabilizer eigenstates of permutation gates with the Pauli group acting on it [1]. Further work observed that most found low-dimensional MICs may be built from subgroups of the modular group PS L(2, Z) [2] and that this can be understood from the picture of the trefoil knot and related 3-manifolds [3]. Here one concentrates on Bianchi groups PS L(2, O10) (with O10 the integer ring over the imaginary quadratic field) whose torsion-free subgroups define the appropriate knots and links leading to MICs and the related uqc. One finds a chain of Bianchi congruence n-cusped links playing a significant role [4].

The linear transformation that relates the canonical and coefficient embeddings of ideals in cyclotomic integer rings

The geometric embedding of an ideal in the algebraic integer ring of some number field is called an ideal lattice. Ideal lattices and the shortest vector problem (SVP) are at the core of many recent developments in lattice-based cryptography. We utilize the matrix of the linear transformation that relates two commonly used geometric embeddings to provide novel results concerning the equivalence of the SVP in these ideal lattices arising from rings of cyclotomic integers.