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in the original paper of Ring-LWE (https://eprint.iacr.org/2012/230.pdf), page 18, definition 3.1, they take a cyclotomic ring of integers $R$, and a prime number $q$, and then multiply an element $a$ from $R_q=R/qR$ with another element $s$ in $R_q^{\vee}=R^{\vee}/qR^{\vee}$.

My question is how they can define such multiplication?

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  • $\begingroup$ What is $R^{\vee}$? $\endgroup$
    – user23365
    Commented Dec 23, 2018 at 11:14
  • $\begingroup$ @franzlemmermeyer You can think of $R^{\vee}$ as a fractional ideal in the number field, and it contains $R$. $\endgroup$
    – C.S.
    Commented Dec 23, 2018 at 13:17
  • $\begingroup$ If you think of $R^{\vee}/qR^{\vee}$ as the ring of residue classes modulo $q$ represented by elements in some fractional ideal, then you can certainly multiply such residue classes by an arbitrary residue class $a$ modulo $q$. $\endgroup$
    – user23365
    Commented Dec 24, 2018 at 16:38

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