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I've read a paper named "How to Securely Collaborate on Data: Decentralized Threshold HE and Secure Key Update" recently and some preliminary in section II that I totally can not understand. Here is the question:

They said: For a power-of-two N, we use $\oldstyle{R}=\mathbb{Z}[ {\it X} ]/({\it X}^{\it N}+1)$ to denote the ring of integers of a number field $\mathbb{Q}[ {\it X} ]/({\it X}^{\it N}+1)$. Given a modulus $q$, $\oldstyle{R}_{\it q}=\oldstyle{R}/{\it q}\oldstyle{R}$ is the residue ring of $\oldstyle{R}$ modulo $q$. An element $a \in \mathbb{R}[{\it X}]/({\it X}^{\it N}+1)$ represented by $a(\it X)=\sum\limits_{j = 0}^{N-1}{a_j\it X_j}$ of degree $< \it N$ will be identified with its coefficient vector $(a_0,...,a_{N-1}) \in \mathbb{R}^N$. We use the notation $\begin{Vmatrix} a \end{Vmatrix}_\infty$ to denote the usual $l_\infty$-norm of $a$.

Can anyone explain each statement or expression step by step what that mean?

I really appreciate that.

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    $\begingroup$ Do you know quotient rings are, like $\mathbf Z[x]/(x^N+1)$? If not, then read about them in an abstract algebra book. It is expecting far too much to have someone explain each statement here step by step. That $R$ is the ring of integers of $\mathbf Q[x]/(x^N+1)$ is a result from algebraic number theory (a ring of integers in a cyclotomic field generated by a root of unity of $2$-power order). At a minimum you should indicate in your question what you already know about abstract algebra or algebraic number theory. $\endgroup$
    – KCd
    May 26 at 2:39
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    $\begingroup$ You're asking for quite a bit there, especially as we don't know where we have to begin. Are you familiar with the use of the symbol $\bf Z$ to stand for the integers? with ${\bf Z}[X]$ to stand for the ring of polynomials in an indeterminate $X$ with integer coefficients? with the idea of a "ring" in the first place? with the idea of a "quotient ring"? If these are not familiar to you, you are asking us to give you a semester course in abstract algebra. Better to get yourself a good textbook, if that's the case, as you have a lot to pick up. $\endgroup$
    – Gerry Myerson
    May 26 at 2:40

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